Daf Ditty Pesachim 30: What’S a Ma’Shehu Between Friends? Avogadro’S Limit

Daf Ditty Pesachim 30: What’s a Ma’shehu between friends? Avogadro’s Limit

Certain issurim (forbidden things) can become batel (halachically nullified) if they were unintentionally mixed in heter (permitted things).

In order for this to happen, the heter must outnumber the issur . Depending on what the issur is, the necessary measurement may be 1/60 (taste), 1/100 (terumah) 1/200 (kilayei hakerem and orla) or just a simple rov (majority).

However, some things "never" become batel; they are assur even b'mashehu, that is, even in the smallest amount. The classic example of this is chametz (leaven) on Pesach, which is assur even in the minutest proportions.

Rava teaches something unique about chametz in the world of prohibited foods. Most other prohibited foods can become nullified when it is a small fraction (1/60th) of the food mixture, however chametz b’mashehu…diluted in the Kinneret lake makes drinking its waters problematic on Pesach.

In Israel, some Haredi rabbis have instructed adherents to drink only bottled mineral water on Passover because people have been known to throw bread into the Sea of Galilee, Israel’s main source of drinking water.

Washington Post March 30th 2012

Afloat, adrift, a flight, a wing A cock, a quail, the promise of spring And the riverbank talks of the Waters of March It's the promise of life, it's the joy in your heart

The Waters of March, Antônio Carlos Jobim

Rava said: The halakha is that with regard to the prohibition against eating a mixture of leavened bread during its time of prohibition, i.e., during Passover, regardless of whether it is mixed with its own type or with another type, it is forbidden, even if any amount becomes mixed in, in accordance with the opinion of Rav. Not during its time of prohibition, but rather after the conclusion of Passover, regardless of whether the leavened bread was mixed with its own type or with another type, it is permitted, even when it gives flavor to the mixture. This is in accordance with the opinion of Rabbi Shimon, who states that leavened bread owned by a Jew during Passover is permitted after Passover.

The Gemara asks: Did Rava actually say this, that according to Rabbi Shimon, leavened bread owned by a Jew on Passover is permitted after Passover? But didn’t Rava himself say that Rabbi

4 Shimon imposed a penalty forbidding one from deriving benefit from leavened bread owned by a Jew during Passover, since he transgressed the prohibition it shall not be seen and the prohibition it shall not be found?

The Gemara resolves this challenge: This penalty applies only to leavened bread that is in its pure unadulterated form, but with regard to a mixture, no, one does not impose a penalty, even though the leavened bread is still extant. The Gemara adds: And Rava follows his line of reasoning, stated elsewhere, that indicates that he rules in accordance with the opinion of Rabbi Shimon, as Rava said: While we were studying in Rav Naḥman’s house, on the evening when the seven days of Passover had passed, he said to us: Go and buy leavened bread from the gentiles who baked it on that day, the last day of Passover. Based on this story, it is clear that he maintained that one may eat leavened bread that was owned by a gentile during Passover.

Rav said: Earthenware pots in which leavened bread was cooked during Passover should be broken, as some small quantity of the flavor of the leavened bread was absorbed into the pot. It is therefore prohibited to cook in them again, as the forbidden flavor of this leavened bread would be transmitted to the new food. The Gemara asks: And why was Rav so stringent with regard to these pots? Let him retain the pots until after the conclusion of Passover and prepare mixtures of another type of food in them. Even Rav maintains that when a small bit of leavened bread is mixed with another type of food after Passover, the mixture is permitted. The Gemara explains that he did allow this due to a rabbinic decree that perhaps one will come to prepare a mixture of the same type in these pots, causing their contents to become prohibited. He therefore instructed that one destroy these pots in order to avoid this pitfall.

5 And Shmuel said: They need not be broken. Rather, he should retain them until after its time, i.e., the conclusion of Passover, and then he may prepare food of either the same type or another type in them.

And Shmuel follows his line of reasoning, stated elsewhere, as Shmuel said to the pot merchants, who would dramatically raise their prices after Passover: Level the prices for your pots. And if you do not bring your prices down, I will teach you that the halakha is in accordance with the opinion of Rabbi Shimon, that one is permitted to derive benefit from leavened bread after Passover. This ruling would lead people to retain their vessels and desist from purchasing new vessels after Passover, and consequently the merchants would lose business.

The Gemara asks: Let him indeed teach this ruling to them, for Shmuel holds that the halakha is in accordance with the opinion of Rabbi Shimon, as was previously mentioned. As such, why did he not publicize his opinion on the matter? The Gemara answers: It was Rav’s locale, and therefore it would not be appropriate for Shmuel to publicly present a position with which Rav did not agree. However, when he saw the merchants raising their prices in an unfair manner, he nevertheless threatened to make his opinion on the matter public.

The Gemara recounts: There was a certain oven that they smeared with grease from meat, and consequently the flavor of meat was absorbed into it, and then they used it to bake bread. Rava bar Ahilai prohibited eating this bread even with salt, and not just with milk. This status would apply forever to bread baked in this oven, even during subsequent baking, lest one eat such bread with kutaḥ, a dairy-based seasoning, which would be a violation of the prohibition against eating meat with milk.

Tosafos

'סות ה"ד רמא אבר אתכלה ץמח נמזב ו יב ן ימב נ ו יבו ן אלש ימב נ ו רוסא לכב אוהש ברכ אוהש לכב רוסא ו נ ימב אלש ן יבו ו נ ימב ן יב ו נמזב ץמח אתכלה אבר רמא ה"ד 'סות

Tosfos argues with Rashi over Miyn be'Miyno, and elaborates. He also queries the text ''be'Cholshehu ke'Rav.

קספ י"שר ןילוחב ןימ'ד ונימב והשמב ' יברכ .הדוהי בכ המ ויב י' ןלח יש

Halachah (Opinion #1): Rashi in Chulin (Daf 109. DH 've'Su') Paskens 'Miyn be'Miyno be'Mashehu' like Rebbi Yehudah.

רמאד 'פב לכ רשבה ילוח( ן ).טק ןימ' ונימב אל 'ליטב .'יבא ויבןמ )ט

Who says in Perek Kol ha'Basar (Chullin 109.) 'Miyn be'Miyno Lo Bateil'.

ףאו יבר רמאק םתה ' ןיארנ ירבד יבר 'הדוהי , ללכמ אתכלהד יכה .יהאכה למ הוי יר רד יר םהרא

And also Rebbi says there that 'Rebbi Yehudah's words appear correct'.

דועו איבה היאר אהמ גילפד י ייבא אברו קרפב ארתב ז"עד ףד( ).וס אלח' ארמחד אלחו ,'ארכישד ארימח' ארימח' ,'ארכישד אלחו ארמחד אלח' ).וס ףד( ז"עד ארתב קרפב אברו ייבא י חד יטי ו ימח אר רעשד י ' - אבר רמא והשמב ; בא י י רמא ב נ ןתו םעט - םושמ רמד רבס רתב 'אמש' ןנילזא רמו רבס רבס רמו ןנילזא 'אמש' רתב רבס רמד םושמ רתב ' אמעט ' א ז ל י נ ן .ןנילזא 'אע ת

He brings an additional proof from the Machlokes Abaye and Rava in the last Perek of Avodah- Zarah (Daf 66.) regarding 'vinegar of wine and vinegar of beer', and 'a dough of wheat and a dough of barley'' - where Rav says 'Mashhu' and Abaye, 'be'Nosein Ta'am' - because one of them goes after 'the name (what it is called)', and the other, after 'the taste'.

אמלא יב ן ייבאל יב ן אברל ןימ ימב נ ו והשמב . המ וניב י בל ב יא ןי

Conclusion: So we see, that both Abaye and Rava hold 'Miyn be'Miyno be'Mashehu'.

,דועו קספ אבר אתעמשב ימ' ן ימב נ ו והשמב ' , .ברכ , ' והשמב ו נ ימב ן ימ' אתעמשב אבר קספ ,דועו

Moreover (he explains), Rava in our Sugya holds 'Miyn be'Miyno be'Mashehu'.

ו ת"ר רמוא לכד ולא םניא יאר ,תו ןיאד הכלה אל 'רכ הדוהי אלו .ברכ אלו הדוהי 'רכ אל הכלה ןיאד ,תו יאר םניא ולא לכד רמוא ת"ר

Rabeinu Tam however, refutes all of Rashi's proofs, maintaining that the Halachah is neither like Rebbi Yehudah nor like Rav.

איההו 'פד ארתב 'סמד ז"ע ולפ אתג יבאד י ,אברו יצמ רמימל ארימחד אלחו לבטד ו י י ן ךסנ ודומד 'ר י נחו ן ל"רו םהש םהש ל"רו ן נחו י 'ר ודומד ךסנ ן י י ו לבטד אלחו ארימחד רמימל יצמ ,אברו והשמב . המ

Refutation #1: Because in the case in the last Perek of Avodah-Zarah regarding the Machlokes between Abaye and Rava, it may well be that Rebbi Yochanan and Resh Lakish concede that wine and beer of Tevel and Yayin Nesech are Asur be'Mashehu.

7 אתעמשדו יא 'רג והשמב' 'ברכ , ל"י קסופש ץמחב ,ברכ אלו ימעטמ ' - ברד ל"סד לכב ירוסיא ן הרותבש ימב' נ ו מ' רתש רסאלבלס ר והשמב ' , אברו תיל היל אלא אקוד ץמחב חספב םושמ ארמוח .ץמחד ארמוח םושמ חספב ץמחב אקוד אלא היל תיל אברו , ' והשמב

Refutation #2: And as for our Sugya, even if we have the text 'be'Mashehu like Rav', it could be that the Gemara Paskens like Rav, not for the same reason - but because whereas Rav holds 'be'Miyno be'Mashehu' by all Issurim in the Torah, Rava only says it by Chametz on Pesach, due to the Chumra of Pesach.

םגו בר יאדוהי ןואג קספ ,ןכ אקודד ץמחב קספ אבר ברכ . . . בכארקפץח קד ,כקפןא אויב ג

And also Rav Yehuda Ga'on Paskens that it is on Pesach exclusively that Rava holds like Rav ...

לאשד ו הימקמ לכ' ירוסיא ן הרותבש והשמב ברכ וא נב ןתו םעט ?י"רכ םעט ןתו נב וא ברכ והשמב הרותבש ן ירוסיא לכ' הימקמ

Because they asked him whether by all Issurim in the Torah, we say Ma’shehu like Rav or be'Nosein Ta'am like Rebbi Yochanan?

בישהו רמאדמ אבר ץמח' נמזב ו יב ן ימב נ ו יב ן אלש ימב נ ו והשמב ' , כ"א אהב נ י והנ הכלה ברכ והשמב , לבא ראשב ראשב לבא , והשמב ברכ הכלה והנ י נ אהב כ"א , ' והשמב ו נ ימב אלש ן יב ו נ ימב ן יב ןירוסיא ט"נב 'רכ ןנחוי .ל"רכו נו ר "ב יוי

To which he replied - 'Since Rava ruled 'be'Miyno be'Mashehu' specifically with regard to Chametz in its time, it is clear that his ruling is confined to Chametz on Pesach, but that regarding other Issurim, the Halachah is Rebbi Yochanan and Resh Lakish.

Summary

Rav Avrohom Adler writes:1

Chametz mixtures

Rava rules that a chametz mixture on Pesach is prohibited in any proportion, whether the mixture is of the same or different species, like Rav. If the mixture occurred after Pesach, whether into the same or different species, it is permitted, like Rabbi Shimon, who says that chametz after Pesach is only Rabbinically prohibited.

Even though Rava says that Rabbi Shimon prohibits chametz after Pesach from benefit as a fine, this only applies to the actual chametz, but not when it’s in a mixture. The Gemora says that this ruling is consistent with Rava’s statement that when he was in Rav Nachman’s house, once Pesach was over, he would tell them to go buy chametz from non-Jews, even though it was baked on Pesach, since there is only a Rabbinic prohibition on chametz that was improperly owned by a Jew, but no Torah prohibition on chametz from Pesach.

1 http://dafnotes.com/wp-content/uploads/2015/11/Pesachim_30.pdf

8 Earthenware chametz pots

Rav says that earthenware vessels of chametz must be broken on Pesach, since the chametz inside them is prohibited. The Gemora asks why one may not keep them and use them after Pesach for foods that are different than chametz species, since Rav said such a mixture is permitted after Pesach. The Gemora answers that we are concerned that he will end up using them for the same species, and we therefore prohibit its use altogether. Shmuel says that one may keep these pots and use them after Pesach for any type of food, as he says any mixture after Pesach is permitted. The Gemora says that this is consistent with Shmuel’s warning the pot merchants that if they overcharge for their pots, he will teach everyone that we rule like Rabbi Shimon, and therefore people don’t need new pots after Pesach. The Gemora asks why he didn’t teach that, as he rules that way, and the Gemora answers that he didn’t want to do so in Rav’s locale, since Rav ruled differently.

Mark Kerzner writes:2

If chametz is mixed with some permitted food, it is still prohibited to be eaten on Passover - so says Rav. Of course, if you can feel its taste, then it is the same as chametz itself, but Rav was referring even to minute amounts.

Why does the regular law of nullification of a small amount does not apply?

This is a special decree: since normally people are accustomed to eating chametz, and since eating it leads to being cut off from one’s spiritual source, then it is better to distance people from even small admixtures of chametz.

However, Shmuel does not see a reason for this additional decree, and according to him a small admixture of chametz that is not noticeable is in fact allowed on Passover. Even Shmuel does not allow nullification if the mixtures are of the same kind, for example, bread mixed with matzah. Finally, Rabbi Yochanan is even more lenient: he allows nullification of even bread with matzah.

Correspondingly, they will differ for chametz after Passover: some will consider the mixture with its own kind forever forbidden, if owned by a Jew on Passover, while others will allow mixtures after the time of the prohibition has passed.

Rava lived after them, and therefore his view is decisive: chametz mixture is forbidden on Passover and permitted afterwards

2 http://talmudilluminated.com/pesachim/pesachim30.html

9 Preparing Knives for Passover

Steinzaltz (OBM) writes:3

How do you prepare your kitchen for Pesaḥ? Do you “kasher” your silverware or do you have a separate set for the holiday? This is the conversation that is quoted by the Gemara between Ravina and Rav Ashi on this topic:

Ravina: “What do you do with knives in preparation for Pesaḥ?” Rav Ashi: “I craft new ones.” Ravina: “How about people who cannot afford to purchase new utensils? What should they do?” Rav Ashi: “I didn’t mean that I actually make new ones. What I meant was that I refurbish my knives every year. I put mud around the wooden handle so that it should not be ruined, and I put the metal part of the knife into fire. Then I remove the clay and put the handle into boiling water.”

The Gemara concludes that utensils can be made kosher for Passover by means of putting them in boiling water of a keli rishon – a pot that is directly on a flame – based on the principle ke-bolo kakh polto – something that absorbs taste will expel it when subjected to the same situation. Since kitchen utensils are often used in a boiling pot over a flame, that is the level of heat necessary to remove whatever had been absorbed.

One type of material from which hametz cannot be removed is earthenware. According to the Gemara, once an earthenware pot is used for cooking, what had been absorbed can never be fully removed.

One of the issues raised by the Gemara is whether glazed earthenware pots will have the same rule as simple ones. In this photograph of an earthenware jar that was found in Dura Europos in Babylonia, the difference between glazed and simple earthenware finishes is apparent.

From the discussion in the Gemara it is clear that glazing was done with different materials that gave different color finishes, each of which allowed for a different amount of absorption by the pot.

All of this is only if the utensil had been used for hametz in a situation where heat was applied. If, however, something was only used for cold food, then we do not assume that anything had been absorbed and it can be used on Pesaḥ.

3 https://steinsaltz.org/daf/pesahim30/

10 MIXTURES OF CHAMETZ ON PESACH

Rav Mordechai Kornfeld writes:4

Rava concludes that the Halachah follows the opinion of Rav, who rules that even a minute amount of Chametz forbids a mixture when it falls into a non-Chametz food on Pesach. A "Ma’shehu" (any amount) of Chametz is forbidden "Bein b'Mino, Bein she'Lo b'Mino," whether it became mixed with the same type of food or with a different type of food.

Why is the law with regard to a mixture of Chametz more stringent than the law with regard to all other prohibitions? In every other case of a prohibited food item, when a prohibited food becomes mixed with a similar type of food (but one that is permitted), it prohibits the mixture only when it comprises a majority (Rov) of the mixture. When a prohibited liquid becomes mixed with a permitted liquid, it prohibits the mixture only "b'Nosen Ta'am," when enough of the forbidden item is present such that its taste is noticeable in the mixture.5

RASHI (29b, DH she'Lo b'Mino) writes that Rav maintains that in the case of all prohibited foods, a mixture of "Min she'Lo b'Mino" is prohibited "b'Nosen Ta'am," and a mixture of "Min b'Mino" is prohibited "b’mashehu." The only difference between Chametz and other prohibitions is that Chametz is also prohibited "b’mashehu" when the mixture is "Min she'Lo b'Mino." In the case of other prohibited foods that fell into a mixture of "Min she'Lo b'Mino," the mixture is prohibited only "b'Nosen Ta'am."

Rashi explains that this difference is due to the severity of the prohibition of Chametz. Chametz is more severe than other prohibitions because it is punishable with Kares. Moreover, people are not accustomed to refraining from Chametz (11a), as they eat it throughout the year. Rav therefore maintains that even a mixture of "Min she'Lo b'Mino" is prohibited "b’mashehu," like a mixture of "Min b'Mino."

TOSFOS (DH Amar Rava) argues (above) with Rashi, who says that a mixture "Min b'Mino" of any other prohibition is also prohibited "b’mashehu."

Tosfos asserts that a mixture of "Min b'Mino" is not prohibited "b’mashehu," but only "b'Rov," when the prohibited item comprises a majority of the mixture. This is the opinion of the Rabanan. The reason Chametz is prohibited "b’mashehu," even in a mixture of "Min b'Mino," is because of the severity of Chametz, as Rashi explains.

Because of the severity of the prohibition, the Rabanan decreed that both "Min b'Mino" and "Min she'Lo b'Mino" are prohibited "b’mashehu."

4 https://www.dafyomi.co.il/pesachim/insites/ps-dt-030.htm 5 This amount is defined practically as 1/60th; see REMA, Yoreh De'ah 98:1

11 RAMBAM

Although chametz on Pesach is forbidden by Scriptural Law, it is not governed by these general principles, for this mixture is not forbidden forever. For after Pesach, the entire mixture will be permitted, as we explained. Therefore the slightest amount [of chametz] causes [a mixture] to become forbidden, whether [it becomes mixed] with a substance of its own type or of another type.6

RAMBAM (Hilchos Ma'achalos Asuros 16:9) above, and the RAN explain that the reason Chametz is prohibited "b’mashehu" is because Chametz is a "Davar she'Yesh Lo Matirin," it is a forbidden item which eventually will become permitted. Since one can wait until after Pesach to eat the mixture, the mixture is prohibited "b’mashehu" on Pesach.

Rava rules like Rebbi Shimon who maintains that after Pesach, Chametz that was owned by a Jew is prohibited only mid'Rabanan. The Rabanan did not prohibit Chametz after Pesach when the Chametz is mixed with another food. Therefore, since one can wait to eat the mixture until after Pesach, when the mixture will be permitted, on Pesach it is prohibited "b’mashehu."

RAN casts doubt on this reasoning. Even though a mixture of Chametz is permitted after Pesach, Chametz that is distinct and not mixed is indeed prohibited (mid'Rabanan) after Pesach. Therefore, Chametz cannot be called a "Davar she'Yesh Lo Matirin."

MORDECHAI (#553) suggests another reason for why Chametz is not considered a "Davar she'Yesh Lo Matirin." The Chametz does not become indefinitely permitted after Pesach, because it will become forbidden at a later time -- next Pesach. Therefore, it cannot be called a "Davar she'Yesh Lo Matirin."

RABEINU TAM (cited by Tosfos DH Amar Rava) and the BA'AL HA'ME'OR have a different text in the Gemara. According to their text, Rava does not say that a mixture of Chametz is "forbidden b’mashehu," but merely that it is "forbidden." Rava means to say that it is forbidden "b'Nosen Ta'am," in accordance with the opinion of Rebbi Yochanan. Accordingly, there is no difference between Chametz and all other prohibitions. This is also the opinion of the SHE'ILTOS.

6 Although even the tiniest amount of chametz causes an entire mixture to be forbidden, chametz was not mentioned by Avodah Zarah 73b together with wine poured as a libation and tevel. The reason is that the prohibition of the mixture of chametz is motivated by a different rationale (Kessef Mishneh).

HALACHAH:

SHULCHAN ARUCH (OC 447:1)

If chometz became mixed in before Pesach began, it is batel (nullified) in 60 (i.e. less than 1.67%). It does not return and become revivified on Pesach and thereby forbidden.

The Rema's gloss: our custom is like the first opinion: in all mixtures that are wet with wet.

But we forbid if the thing is dry that is mixed (or there is concern that it mixed), such as bread that fell into wine even though it was taken out from the wine, it's prohibited on Pesach because of the concern that maybe crumbs remained in the wine which give taste on Pesach.

13 Shulchan Aruch rules like Tosfos, that mixtures of all other Isurim of "Min b'Mino" are prohibited only "b'Shishim" (when the Isur is more than 1/60th of the Heter), and a mixture with Chametz (both "Min b'Mino" and "Min she'Lo b'Mino") is prohibited "b’mashehu," due to the severity of the Isur.

However, the MISHNAH BERURAH (447:2) adds that when there are pressing reasons to be lenient, one may rely on the view of the She'iltos and use a mixture with Chametz as long as it is not "Nosen Ta'am."

With regard to whether Chametz is considered a "Davar she'Yesh Lo Matirin," both opinions are cited by the REMA (YD 102:4) and the SHACH (YD 102:14).

There is a practical difference between the two reasons for why a mixture of Chametz is forbidden even "b’mashehu" -- the severity of the prohibition of Chametz, or the fact that Chametz is a "Davar she'Yesh Lo Matirin." If the reason is because of the severity of the prohibition of Chametz, then the mixture with a "Ma’shehu" is forbidden only when the punishment of Kares applies. On Erev Pesach after the sixth hour and before nightfall, Chametz is forbidden but there is no punishment of Kares. As a result, a mixture of "Min b'Mino" will not be forbidden "b’mashehu" on Erev Pesach before nightfall, according to Tosfos. However, according to the Rambam, a mixture of Chametz on Erev Pesach will be forbidden "b’mashehu," because it is a "Davar she'Yesh Lo Matirin," which will become permitted after Pesach. (MAGID MISHNEH, Hilchos Chametz u'Matzah 1:5)

The SHULCHAN ARUCH (OC 447:2) above, rules that a mixture of Chametz on Erev Pesach is permitted if the quantity of Chametz is only a "Ma’shehu."

It is evident from this ruling that the Shulchan Aruch does not agree with the Rambam, and he maintains that Chametz is not a "Davar she'Yesh Lo Matirin."

TA’AROVET CHAMETZ

Rav David Brovsky writes:7

GENERALLY, REGARDING ORDINARY PROHIBITED FOODS (MA’AKHALOT ASSUROT), WE DISTINGUISH BETWEEN DIFFERENT TYPES OF MIXTURES: YAVESH BE-YAVESH AND LACH BE-LACH.

A MIXTURE OF YAVESH BE-YAVESH REFERS TO A MIXTURE OF DRY SUBSTANCES THAT WERE NOT COOKED TOGETHER. FOR EXAMPLE, A MIXTURE OF PIECES OF KOSHER AND NON-KOSHER MEAT THAT HAVE NOT BEEN COOKED TOGETHER IS CONSIDERED A "DRY MIXTURE."

THERE IS A FURTHER DISTINCTION BETWEEN A MIXTURE OF SIMILAR PIECES AND DISSIMILAR PIECES. A MIXTURE OF SIMILAR PIECES (MIN BE-MINO) REFERS TO A MIXTURE IN WHICH THE PIECES OF KOSHER AND NON-KOSHER MEAT ARE FROM THE SAME SPECIES OF ANIMALS AND HAVE SIMILAR TASTES, SUCH AS SLABS OF KOSHER AND NON-KOSHER BEEF. A MIXTURE OF DISSIMILAR PIECES (MIN BE-SHEINO MINO) REFERS TO A MIXTURE IN WHICH THE PIECES ARE FROM DIFFERENT SPECIES OF ANIMALS AND THEREFORE HAVE DIFFERENT TASTES, SUCH AS BEEF AND PORK.

IN A MIXTURE OF YAVESH BE-YAVESH, WE ARE NOT CONCERNED WITH A PHYSICAL MELDING OF THE PIECES OR WITH THE TRANSFER OF "TASTE" FROM ONE PIECE TO ANOTHER, BUT RATHER WITH THE CHANCE THAT ONE MAY EAT A PIECE OF NON-KOSHER MEAT. IF THE MAJORITY OF THE PIECES ARE KOSHER, MAY ONE EAT A NON-IDENTIFIED PIECE FROM THE MIXTURE? WHAT IS THE HALAKHIC IDENTITY OF A PARTICULAR PIECE CHOSEN FROM THE MIXTURE? THE STAKES ARE HIGH, BEING THAT ONE MAY ACTUALLY EAT AN ENTIRE NON- KOSHER PIECE OF MEAT. BUT THERE IS ALSO A CHANCE THAT ONE MAY NOT PARTAKE OF ANY NON-KOSHER SUBSTANCE IF THE MAJORITY OF THE PIECES ARE KOSHER. THE GENERAL HALAKHIC PRINCIPLE OF STATISTICAL MAJORITY, BASED ON SHEMOT 23:2 (ACHAREI RABIM LE-HATOT), LEGISLATES THAT "CHAD BETREI BATEL," ONE NON-KOSHER PIECE CAN BE VIEWED AS "BATEL" (NULLIFIED) IN THE MAJORITY OF KOSHER PIECES, AND THE MIXTURE MAY BE CONSUMED.

ANOTHER TYPE OF MIXTURE IS KNOWN AS LACH BE-LACH, A "WET MIXTURE." IN THIS SCENARIO, THE NON-KOSHER SUBSTANCE MELDS WITH THE KOSHER SUBSTANCE AND THEY CANNOT BE SEPARATED FROM ONE ANOTHER. THIS CAN OCCUR WHEN TWO LIQUIDS, KOSHER AND NON-KOSHER,

7 https://www.etzion.org.il/en/laws-pesach-defining-chametz-2-%D6%A0mixtures

15 ARE MIXED TOGETHER. TA’AM (TASTE) CAN ALSO BE TRANSFERRED, UNDER CERTAIN CIRCUMSTANCES, IF A SOLID COMES INTO CONTACT WITH A LIQUID. FOR EXAMPLE, IF MILK FALLS ONTO A PIECE OF MEAT OR MEAT FALLS INTO A SOUP AND IS REMOVED BUT THE TASTE REMAINS, TA’AM MAY BE TRANSFERRED UNDER THE PROPER CONDITIONS. QUITE OFTEN, THE ENTIRE MIXTURE WILL ASSUME THE TASTE OF THE NON-KOSHER ENTITY. WHAT IS THE STATUS OF THE ENTIRE MIXTURE?

WE MUST ONCE AGAIN DISTINGUISH BETWEEN A MIXTURE OF SIMILAR TASTING SUBSTANCES (A PIECE OF NON-KOSHER BEEF IN A KOSHER BEEF SOUP) AND A MIXTURE OF DISSIMILAR TASTING FOODS (NON-KOSHER MEAT IN A TOMATO SOUP). ON THE ONE HAND, IN A MIN BE-SHEINO MINO (DISSIMILAR) MIXTURE, THE TASTE OF THE PRIMARY, KOSHER SUBSTANCE MAY DILUTE THE TASTE OF THE MINOR, NON-KOSHER SUBSTANCE UNTIL THE TASTE OF THE MINORITY NON-KOSHER SUBSTANCE MAY NO LONGER BE NOTICEABLE. IN THIS CASE, IT IS POSSIBLE TO ACTUALLY DETERMINE WHETHER THE TASTE OF THE PROHIBITED SUBSTANCE IS STILL NOTICEABLE. ON THE OTHER HAND, IN A MIXTURE OF SIMILAR TASTING SUBSTANCES, A MIN BE-MINO MIXTURE, THE TASTE OF THE PROHIBITED SUBSTANCE IS NOT DIFFERENT FROM THAT OF THE PROHIBITED ONE AND THEREFORE IS UNDISTINGUISHABLE.

CHAZAL TEACH (CHULLIN 97B) THAT A MIXTURE OF DISSIMILAR SUBSTANCES (MIN BE-SHEINO MINO) REQUIRES SIXTY TIMES MORE HETER (PERMITTED SUBSTANCE) THAN ISSUR (PROHIBITED SUBSTANCE) IN ORDER TO PERMIT THE MIXTURE, WHILE A MIXTURE OF SIMILAR TASTING SUBSTANCES (MIN BE-MINO) IS THE SUBJECT OF TALMUDIC DEBATE. IN SUCH A CASE, THE RABBANAN MAINTAIN THAT THE MIXTURE IS PERMITTED ONCE THERE IS A MAJORITY OF HETER. R. YEHUDA DISAGREES AND ARGUES THAT, ON THE CONTRARY, THE MIXTURE IS NOT PERMITTED EVEN IF THE SLIGHTEST AMOUNT OF ISSUR HAS BEEN INTRODUCED.

ISSUR MASHEHU- CHAMETZ ON PESACH CANNOT BE NULLIFIED:

REGARDING A CHAMETZ LACH BE-LACH MIXTURE OF DISSIMILAR SUBSTANCES, THE CHAKHAMIM AND R. ELIEZER (PESACHIM 42A) DEBATE WHETHER THERE IS A SPECIAL PROHIBITION CONCERNING A TA’AROVET CHAMETZ. HOWEVER, REGARDLESS OF THE SPECIFIC PROHIBITIONS AND PUNISHMENTS INVOLVED, THE UNIQUENESS OF CHAMETZ GENERATES AN IMPORTANT PRACTICAL DIFFERENCE BETWEEN CHAMETZ MIXTURES AND OTHER MIXTURES. WHILE CHAMETZ THAT BECOMES MIXED WITH OTHER FOODS BEFORE PESACH IS NULLIFIED BY PERMISSIBLE MATTER SIXTY TIMES GREATER IN VOLUME (BITTUL BA-SHISHIM), WHEN CHAMETZ IS MIXED WITH PERMISSIBLE MATTER DURING THE FESTIVAL, IT CAN NEVER BE NULLIFIED. EVEN THE SMALLEST AMOUNT (MASHEHU) PROHIBITS THE ENTIRE MIXTURE. AS RAV TEACHES (PESACHIM 29B):

16 RAV SAID: CHAMETZ, IN ITS TIME [DURING PESACH], WHETHER [MIXED] WITH ITS OWN KIND OR WITH A DIFFERENT KIND, IS FORBIDDEN; WHEN NOT IN ITS TIME, [IF MIXED] WITH ITS OWN KIND, IT IS FORBIDDEN; [IF WITH] A DIFFERENT KIND, IT IS PERMITTED… THIS REFERS TO A MINUTE QUANTITY [OF CHAMETZ].

THE RISHONIM QUESTION WHY THE LAWS OF CHAMETZ ARE SO MUCH STRICTER THAN THAT OF ORDINARY MIXTURES TO THE DEGREE THAT A MIXTURE OF DISSIMILAR SUBSTANCES, WHICH IS USUALLY BATEL BA-SHISHIM, IS NOT BATEL AT ALL IN THE CASE OF CHAMETZ.

RASHI (PESACHIM 29B, S.V. SHE-LO) EXPLAINS THAT THE SAGES ADDED THIS EXTRA STRINGENCY TO CHAMETZ AND NOT TO OTHER PROHIBITED SUBSTANCES SUCH AS CHEILEV (PROHIBITED FATS) AND DAM (BLOOD), WHICH ARE ALSO PUNISHABLE BY KARET, BECAUSE IN THOSE CASES “ONE IS ACCUSTOMED TO SEPARATE ONESELF FROM THEM. CHAMETZ, HOWEVER, ONE IS NOT ACCUSTOMED TO SEPARATE ONESELF FROM, AS ONE EATS IT THROUGHOUT THE YEAR.” IN HIS OPINION (FOLLOWING THE VIEW OF R. YEHUDA), A REGULAR MIXTURE OF SIMILAR SUBSTANCES (MIN BE-MINO) IS ALSO NOT SUBJECT TO BITTUL. THE CHAMETZ STRINGENCY IS THUS REALLY MANIFESTS IN A CASE OF MIN BE-SHEINO MINO. THE ROSH (AVODA ZARA 5:30) CONCURS.

RAMBAM (HILKHOT MA'AKHALOT ASSUROT 16:9) OFFERS A DIFFERENT REASON. HE WRITES THAT CHAMETZ CANNOT BE NULLIFIED BECAUSE IT IS A "DAVAR SHE-YESH LO MATIRIN," A PROHIBITED ITEM THAT WILL EVENTUALLY BECOME PERMITTED. THE TALMUD (BEITZA 3B) TEACHES THAT A NON-KOSHER ITEM THAT WILL BECOME PERMITTED AFTER TIME, SUCH AS “CHADASH” (THE NEW WHEAT THAT IS PROHIBITED UNTIL AFTER THE FIRST DAY OF PESACH), CANNOT BE NULLIFIED IN A MIXTURE. RASHI EXPLAINS THAT ONE SHOULD NOT RELY UPON BITTUL IF ONE CAN SIMPLY WAIT UNTIL THE ITEM WILL BE PERMITTED. THEREFORE, IN OUR CASE, SINCE ONE COULD SIMPLY WAIT UNTIL AFTER PESACH IS OVER AND THEN EAT THE CHAMETZ, THERE IS NO JUSTIFICATION FOR RELYING UPON BITTUL.

THE RISHONIM CHALLENGE THIS ASSERTION FOR NUMEROUS REASONS. THE MORDEKHAI (PESACHIM 553), FOR EXAMPLE, SUGGESTS THAT SINCE CHAMETZ WILL BECOME PROHIBITED AGAIN THE NEXT YEAR, IT CANNOT BE CONSIDERED TO BE A DAVAR SHE-YESH LO MATIRIN. (THE RAN [NEDARIM 52A] OFFERS AN ENTIRELY DIFFERENT UNDERSTANDING OF DAVAR SHE-YESH LO MATIRIN.)

THE ACHARONIM SUGGEST A NUMBER OF PRACTICAL DIFFERENCES BETWEEN RASHI, WHO ATTRIBUTES THIS STRINGENCY TO THE SEVERITY OF THE PROHIBITION OF CHAMETZ, AND THE RAMBAM, WHO DEFINES CHAMETZ AS A DAVAR SHE-YESH LO MATIRIN.

17 FOR EXAMPLE, CHAMETZ IS PROHIBITED MI-DEORAITA ON EREV PESACH FROM NOON UNTIL NIGHTFALL, ALTHOUGH THE PUNISHMENT OF KARET IS NOT INCURRED. THE SHULCHAN ARUKH (447:2) RULES THAT IF CHAMETZ IS MIXED INTO A MIXTURE ON EREV PESACH, EVEN AFTER THE SIXTH HOUR, THE CHAMETZ IS STILL BATEL BA-SHISHIM. THE MISHNA BERURA (15) EXPLAINS THAT SINCE THE PROHIBITION IS LESS SEVERE THAN THE PROHIBITION OF CHAMETZ DURING PESACH, IT CAN BE NULLIFIED BEFORE PESACH.

SIMILARLY, THE ARUKH HA-SHULCHAN (447:3) SUGGESTS THAT CHAMETZ NUKSHEH SHOULD BE BATEL BA-SHISHIM, AS ONE WHO EATS IT DOES NOT INCUR KARET. SINCE THESE PROHIBITED CONSUMPTIONS ARE LESS SEVERE IN NATURE, CHAZAL DID NOT APPLY THE PROHIBITION OF MIXTURES TO THEM, AS PER RASHI’S SUGGESTION. THE ARUKH HA-SHULCHAN OFFERS OTHER DIFFERENCES BETWEEN THE TWO OPINIONS AS WELL.

RABBEINU TAM (TOSAFOT 30A, S.V. AMAR RAV) AND THE SHE’ILTOT (TZAV, SHE’ILTA 3:80) DISAGREE WITH THE OPINIONS CITED ABOVE AND ASSERT THAT CHAMETZ IS ACTUALLY BATEL BA-SHISHIM EVEN DURING PESACH. TOSAFOT REPORT, HOWEVER, THAT RABBEINU TAM DID NOT ACTUALLY ACCEPT THIS OPINION IN PRACTICE. THE SHULCHAN ARUKH (447:1) RULES THAT CHAMETZ IS NOT BATEL IF IT IS MIXED WITH PERMITTED SUBSTANCES DURING PESACH.

IS IT ALSO PROHIBITED TO DERIVE BENEFIT FROM A MIXTURE THAT CONTAINS A MINUTE AMOUNT OF CHAMETZ?

THE RA’AVAD, RAMBAN, AND RASHBA RULE THAT ONE MAY DERIVE BENEFIT FROM SUCH A MIXTURE; CHAZAL ONLY PROHIBITED EATING IT. THE RIF, RAMBAM, RASHI AND TOSAFOT, HOWEVER, DISAGREE, AND FORBID DERIVING BENEFIT FROM THIS MIXTURE AS WELL. THE SHULCHAN ARUKH (447:1) RULES ACCORDINGLY.

WHICH MIXTURES ARE SUBJECT TO THIS STRINGENCY?

THE RISHONIM DISAGREE AS TO WHETHER THIS STRINGENCY ALSO APPLIES TO A MIXTURE OF DRY SUBSTANCES. FOR EXAMPLE, IF A PIECE OF CHAMETZ BECAME MIXED IN WITH PIECES OF MATZA AND CANNOT BE DISCERNED, WHAT IS THE STATUS OF THE MIXTURE? GENERALLY, A SUBSTANCE OF ISSUR IN A YAVEH BA-YAVESH MIXTURE IS BATEL BE-ROV, NULLIFIED WHEN MIXED WITH A MAJORITY OF KOSHER FOOD. THE RIF (AVODA ZARA 30A) IMPLIES THAT THE STRINGENCY REGARDING CHAMETZ APPLIES TO DRY MIXTURES AS WELL; EVEN A MASHEHU OF CHAMETZ IN A DRY MIXTURE IS NOT BATEL. THE ROSH (AVODA ZARA 5:30), HOWEVER, LIMITS ITS APPLICATION TO WET MIXTURES. THE SHULCHAN ARUKH (447:9) CITES TWO OPINIONS REGARDING THIS

18 QUESTION, AND THE MISHNA BERURA (95) CONCLUDES THAT THE CONSENSUS OF THE ACHARONIM IS TO ACCEPT THE FIRST, MORE STRINGENT OPINION.

THE BEIT YOSEF (467) CITES EARLIER AUTHORITIES WHO CAUTION AGAINST DRINKING WATER FROM WELLS DURING PESACH, ESPECIALLY FROM WELLS SHARED WITH NON-JEWS, LEST THERE BE PIECES OF CHAMETZ IN THE WELL. ONE SHOULD CAREFULLY FILTER OUT GRAINS OF WHEAT WITH A CLOTH. THE MISHNA BERURA (467:62, 67) WRITES THAT WHILE ONE MAY BE LENIENT REGARDING GRAINS OF WHEAT, IF BREAD FELL INTO A PIT OF WATER, THE ENTIRE PIT IS PROHIBITED.

IN A SIMILAR VEIN, IN RECENT YEARS, POSKIM HAVE QUESTIONED WHETHER THE WATER WHICH COMES FROM THE KINNERET SHOULD BE PROHIBITED, LEST PIECES OF BREAD CAST INTO THE KINNERET DURING PESACH PROHIBIT THE ENTIRE LAKE.

R. ASHER WEISS, IN HIS HAGGADA SHEL PESACH - MINCHAT ASHER, CITES NUMEROUS OPINIONS THAT ASSERT THAT ONE NEED NOT BE CONCERNED WITH CHAMETZ WHICH FALLS INTO A RIVER, AS OPPOSED TO A CONTAINED BODY OF WATER. AFTER DISCUSSING THE RELEVANT VIEWS OF THE ACHARONIM, HE NOTES THAT ACCORDING TO MEKOROT (ISRAEL’S NATIONAL WATER COMPANY), ALL WATER THAT IS BROUGHT FROM THE KINNERET IS FIRST EXPOSED TO A POOL OF FISH, WHICH CONSUME ORGANIC MATERIAL, THEN TO A CHEMICAL PROCESS THAT BREAKS DOWN ORGANIC MATERIAL, AND FINALLY TO CHLORINATION, WHICH KILLS BACTERIA AND PURIFIES THE WATER. HE CONCLUDES:

IN MY HUMBLE OPINION, THE REAL ANSWER TO THIS QUESTION LIES IN A POWERFUL RATIONALE, WHICH STATES THAT A RIVER CANNOT BE PROHIBITED BY A MASHEHU [OF CHAMETZ]. BECAUSE IF SO, THE ENTIRE OCEAN COULD BE PROHIBITED BY A CRUMB OF BREAD WHICH FELL IN, AND ALL OF THE WORLD’S WATER WOULD BE PROHIBITED! AND THIS IS SIMPLY UNACCEPTABLE, AND LOGIC DOES NOT TOLERATE THIS.

THE POPULAR WORK PISKEI TESHUVOT (467:14) ALSO DISCUSSES THIS ISSUE AND CITES FIVE REASONS TO BE LENIENT.

19 BEMASHEHU: THE LIMITS OF DILUTION

Assuming for the sake of argument that chametz thrown in the Kinneret on Pesach is prohibited even in the infinitesimal amount of "mashe-hu," there remains the additional issue of whether the water derived from the Kinneret is indeed an admixture ("ta'aruvot").

Let up posit, for example, the ton of chametz was deposited as a solid. Who is to say that some of this chametz dissolved and diffused into every glassful that is drawn from the sea? Similarly, assume that the chametz was poured into the Kinneret in the form of liquid (perhaps a waste of good scotch).

My recollection is that there is an issue in the gemara (in the case of tevilah, for example, where the gemara countenances the possibility that someone who immersed himself into a lake where a barrel of wine fell did not have a valid tevilah because the wine may not have diffused), and there is a machloket at least as to whether we are to assume that diffusion occurred.

Chametz on Pesach-The Kinneret (Sea of the Galilee)

20 Rav Ovadia Yosef writes:8

Question: You have explained in a previous Halacha that Chametz on Pesach can never be nullified by any ratio. If so, how is it permissible to drink water from lakes, such as the Kinneret, in which there are certainly crumbs of Chametz?

Answer: We have explained that Chametz on Pesach is not nullified even by a ratio of one to one- thousand. This means that if even a tiny crumb of Chametz falls into a large pot filled with food on Pesach, the entire pot of food becomes forbidden. This is something that we never find by any other prohibition, for when a small amount of prohibited food falls into a larger amount of permissible food, it is usually nullified by a ratio of one to sixty. For instance, if a drop of milk accidentally spills into a large pot of meat soup, the soup is permissible for consumption since there is certainly at least sixty times more than the drop of milk in the soup. Our Sages ruled stringently with regards to Chametz that it cannot be nullified at all since the Torah was also extremely stringent about the prohibition of Chametz on Pesach, for one who consumes Chametz incurs the “Karet” punishment in addition to the fact that the Torah prohibited even ownership of Chametz on Pesach.

Thus, most places have an issue for most centralized water sources contain some Chametz particles in them. Based on this, it should be forbidden to drink such water on Pesach, for this should be considered a Chametz mixture. There are indeed pious individuals who act stringently in this regard and fill up large tanks with enough water to last them the entire Pesach before Pesach begins and they do not use water from the tap on Pesach for the above reason.

Indeed, over forty years ago, people began raising awareness that there were several restaurants along the banks of the Kinneret (Sea of the Galilee) in northern Israel which would throw bread into the lake on Pesach itself. When Maran zt”l served as Chief Rabbi of Israel, he went and inspected this on his own. When he realized that the claims were actually correct, he penned a response on this topic.

This issue was first discussed approximately two-hundred years ago by Hagaon Harav Yehoshua Heschel of Tarnipol, author of the Sefer Yehoshua, and he writes that in Jewish communities along rivers, non-Jews throw bread into the river all throughout Pesach and this creates a problem of Chametz mixtures. He writes that this was the case in the city of Dubno along whose river ran a water-powered mill and the non-Jews would dump the chaff and bran into the river. Similarly, the leftover beer they produced there from barley would be spilled into the river as well. If so, the river should thereby constitute a Chametz mixture.

The great author of the Sefer Yehoshua inquired about this from Hagaon Harav Yaakov Loberbaum of Lisa, author of the Chavot Da’at, and the later ruled leniently on the matter since the water was constantly flowing at a fast pace and the Chametz did not have the opportunity be steeped in it; thus, this cannot be considered a Chametz mixture. Hagaon Harav Yehoshua Heschel himself added a different reason for leniency which was that the only reason why the Sages ruled that Chametz is not even nullified by a ratio of one to one-thousand is because they were concerned about a situation where the Chametz might actually add flavor to the mixture. However, with

8 Yabia Omer, Volume 7, Chapter 43: http://halachayomit.co.il/en/default.aspx?HalachaID=4538

21 regards to a tremendous river where it is impossible for Chametz to add any flavor to the water, they did not enact that the Chametz should not be nullified.

Nevertheless, because these rationales were novel, the author of the Sefer Yehoshua writes that he is hesitant to rely on these leniencies with regards to Chametz on Pesach. On the other hand, the author of the Yad Yehuda writes that the ruling of the Sefer Yehoshua is correct in that our Sages never enacted this decree in a situation where the Chametz would never be able to add flavor to the mixture.

Nonetheless, Maran Rabbeinu Ovadia Yosef zt”l writes that it is difficult to rely on this reason alone to rule leniently. However, he adds that there is another great reason for leniency based on what Maran Ha’Shulchan Aruch rules that a courtyard where there are birds need not be searched for Chametz, for even if a piece of Chametz was left over, the birds would certainly eat it. (Although Maran Ha’Shulchan Aruch implements this rule only with regards to doubtful Chametz, nevertheless, here there is even more room for leniency.) Based on this, in the Kinneret where it is almost certain that the fish eat whatever is thrown into it quickly, there is no reason to be concerned about any Chametz mixture.

Furthermore, there is even a disagreement among the Poskim as to the validity of the law that Chametz on Pesach is not even nullified by a ratio of one to one-thousand, for according to the She’iltot of Rav Achai Gaon and the Ba’al Ha’Ma’or, Chametz, even on Pesach, is nullified by a ratio of one to sixty like all other forbidden foods. With all of these reasons (and several others) combined, Maran zt”l rules that there is room for leniency regarding drinking water from sources contaminated by small amounts of Chametz on Pesach.

The Avogadro constant is named after the Italian scientist Amedeo Avogadro (1776–1856), who, in 1811, first proposed that the volume of a gas (at a given pressure and temperature) is proportional to the number of atoms or molecules regardless of the nature of the gas. The name Avogadro's number was coined in 1909 by the physicist Jean Perrin, who defined it as the number of molecules in exactly 32 grams of oxygen. The goal of this definition was to make the mass of a mole of a substance, in grams, be numerically equal to the mass of one molecule relative to the mass of the hydrogen atom; which, because of the law of definite proportions, was the natural unit of atomic mass, and was assumed to be 1/16 of the atomic mass of oxygen.

23 Diluting a solution to Avogadro’s limit9

How many atoms/molecules are in a mole? There are precisely 6.022 × 10^23 units (Avogadro’s number). While many are familiar with this number, the true enormity of its size is hard to fully grasp. In this article, I explore the concept of Avogadro’s limit. Through this, I convey the immense multiplicity of molecules making up the world around us.

Avogadro’s limit

Avogadro’s limit refers to the concentration below which a solution becomes so dilute it is unlikely that any of the original molecules are present. More precisely, we can define the limit as the concentration at which there is only one molecule per litre of solution. Since there are 6.022 × 10^23 atoms in a mole, this gives a limit of 1.66 × 10^-24 mol/L.

How to dilute a solution to Avogadro’s limit

The true size of Avogadro’s constant is clarified by how challenging it is to dilute a solution to Avogadro’s limit. I illustrate this by exploring here how a solution of table salt (sodium chloride) at Avogdro’s limit could be prepared, starting with a single grain of salt.

First, we must estimate the mass of a grain of salt. An average grain of regular table salt has dimensions of roughly 0.3 mm × 0.3 mm × 0.3 mm [1]. This corresponds to a volume of 2.7 × 10^-5 mL. The density of NaCl is 2.16 g/mL and so the grain contains roughly 5.83 x 10^- 5 g. The molecular weight of NaCl is 58.44 g/mol, giving 9.98 × 10^-7 moles (n). Now we calculate what volume (V) we’d have to dissolve the salt grain in to reach Avogadro’s limit concentration (C).

To put this in prospective, this is roughly the volume of the Black Sea (5.47 × 10^17 L) [2]. Imagine dissolving a single grain of salt into such a volume! At this concentration, 1 L of the resulting solution typically would contain a single molecule of NaCl. Obviously, this would not be a very practical way to prepare such a solution. (Of course, attaining water truly free of NaCl would itself be unachievable. This concept works only as a thought experiment).

Reaching Avogadro’s limit by serial dilution

A much more efficient way to achieve the required, remarkable dilution is to do a series of repeated dilutions. Though more steps are required, much less water is needed. Each dilution

9 https://www.sciencesnail.com/science/diluting-a-solution-to-avogadros-limit

24 decreases the initial concentration (Ci) to a final concentration (Cf) in proportion to the dilution factor (D).

For instance, suppose we dissolve the single grain of table salt into 1 L of water. This gives a molarity of roughly 10^-6 mol/L. We proceed to dilute 500 mL of this solution into a fresh 500 mL of water. How many times would we have to repeat this process to reach Avogadro’s limit? Each dilution achieves a 2-fold dilution factor.

We would have to repeat this serial dilution 59 times to reach Avogadro’s limit! This is a lot, though it is certainly easier than using a Black Sea’s volume of water. The serial dilution can be made more efficient by using a higher dilution factor:

Suppose we instead dilute our initial 1 L salt grain solution by adding 100 mL of it into a fresh 900 mL of water. Each dilution now achieves a 10-fold dilution factor.

We’d now only have to repeat this dilution 18 times to go below Avogadro’s limit. I hope that imagining the physical preparation of these solutions will give my readers pause at the size of Avogadro’s number.

Avogadro’s limit in homeopathy

It is interesting to consider Avogadro’s limit in the context of homeopathy. This pseudoscientific approach to alternative medicine is based on the bizarre premise that dilution enhances the potency of medicine. Usually homeopathic preparations are diluted well below Avogadro’s Limit, commonly to around 10^-60 mol/L (37 orders of magnitude below the limit!). Such a preparation would certainly contain none of the original substance. Not surprisingly, a wealth of scientific evidence suggests that homeopathic remedies’ benefits are merely due to the placebo effect [3].

The principles of homeopathy hopefully surprise and confuse you, as it goes against our most basic understanding of physics, chemistry and biology. All scientifically understood drugs exert their effects through physical or chemical interactions with the biological system. Surely a drug cannot interact with/modulate a biological system if none is administered. The most commonly proposed mechanism for homeopathy is the concept of “water memory,” that water molecules are permanently imprinted with the memory of compounds they’ve previously interacted with. Proponents even claim that this information is transmitted to new water molecules upon dilution. Of course, there are serious problems with these ideas. The structural features of water have been heavily studied and are well understood. Water molecules dynamically associate in a hydrogen-bonding network. Presumably, it is through this that the water could “remember.” Unfortunately for homeopathy though, it seems that water is rather forgetful. Experimentally, this network’s has been shown to be remarkably short-lived with all memory gone after only 50 femtoseconds [4].

The concept of water memory does not even require a scientific rebuttal to expose its absurdity, however. If water is imprinted with all chemicals it has ever meet, then just about all water on Earth would already have memory of near every conceivable medicine or toxin. Furthermore, recall the central premise that potency increases with dilution. It follows that the addition and subsequent dilution of a specific medicine would be ineffective at developing that particular activity over all others already present.

The probability of finding a molecule below Avogadro’s limit

When doing calculations of moles or concentration, we normally needn’t consider probability at all. This is broadly true under regular concentration scales as long as the solution is well- mixed. The abundance of atoms allows the law of large numbers to hold. There are so many molecules that the probabilistic aspects of where any particular one is located averages out, macroscopically the solution is perfectly homogenous. For instance, a small portion taken of a 1 mol/L solution would also be at 1 mol/L.

An interesting aspect of Avogadro’s limit is that here the law of large numbers no longer applies. Calculations of moles or concentrations near or below Avogadro’s limit must consider probability. Under these conditions, finding a molecule of interest in a given volume

26 of a solution is probabilistic. Not only this, there is even uncertainty in the concentration of a prepared solution, as long as it is made by serial dilution. I am unaware of any attempts to account for the probabilistic nature of this situation, and so I conclude this article by developing a numerical solution below.

Suppose we have a solution of a molecule of interest at concentration C with volume VT. We withdraw a sample of volume VS. What is the probability of the sample containing nS molecules?

Broadly, the probability of an event is given by the ratio of how many ways there are for the desired outcome to occur over the total number of possible outcomes. We can calculate the probability of the stated event by developing expressions for these two quantities. In doing so, it is helpful to divide the total solution into imaginary Vs-sized portions, with the molecules randomly distributed among them.

We can evaluate the total number of possible outcomes from our random sample by noticing that this is equivalent to a classic combinatorics problem: sorting x identical objects into rdistinct groups. The number of possible outcomes is given by the following combination [5]:

First, the number of objects is given by the total number of molecules (nT) in the solution:

The number of distinct groups is the number of imaginary Vs-sized portions we divided the solution into:

The total number of outcomes is given by:

Next, how many ways are there for our sample volume to specifically contain nS molecules? This is actually an equivalent combinatorics problem to the above. However, now we must begin with our selected sample volume containing precisely nS molecules. This leaves nSfewer objects to arrange and one less group to distribute them among. We now have:

Therefore, the total number of desired outcomes is given by:

Now we calculate the probability of our desired outcome by taking the ratio of the desired outcomes to all possible outcomes:

This equation robustly allows calculation of the probability of withdrawing nS molecules from a solution of known concentration. To gain further insight on Avogadro’s limit, we can use it to evaluate the probability of our sample containing at least one molecule of interest. The probability equation simplifies greatly in the case of no molecules (nS = 0):

It is now easy to calculate the probability of there being at least one molecule of interest in the sample through the complement:

To clarify the effect of concentration on the probability of a molecule of interest being present, the equation is plotted below for VT=10 L and VS=1 L.

At concentrations well above Avogadro’s limit (1.66 x 10^-24 mol/L) the probability of obtaining at least one molecule tends towards unity, but falls off quickly below the limit. The exact curve shape depends on both total and sample volumes, which together affect how much uncertainty there is in the outcome.

Is Tap Water Chametz On Pesach?

Rav Shlomo Zalman Auerbach writes:10

Chametz on Pesach is assur even a Mashehu, a tiny drop. If so how can we drink water from rivers or lakes. Surely someone poured into it something chametz.

Rav Shlomo Zalman Auerbach (Halichos Shlomo 1:4:5) said that there is no problem since the chametz cannot be detected with the senses of a person.

Rav Shlomo Zalman held (D'var Halacha "vuv") that "Mashehu" is also a shiur and does not mean any microscopic amount. The shiur of Mashehu is if a person were able to isolate the chametz in his cup of water it would be visible or somehow detectable through his human senses. If not, then it is less than the shiur of Mashehu and is mutar on Pesach . The Orchos Halacha (19) brings that while Rav Shlomo Zalman's father in law, Rav Aryeh Leib Ruchamkin was still alive,

10 http://www.revach.net/halacha/tshuvos/Rav-Shlomo-Zalman-Auerbach-Is-Tap-Water-Chametz-On-Pesach/3561

29 Rav Shlomo Zalman would only drink from the water that was drawn from the well before Pesach since before Pesach Chametz is batel.

However after his father-in-law's petira he stopped keeping this chumra.

The Importance of Small Things in Space and Time11

Small Physical Things:

Assur B'mashehu

רמא אבר אתכלה ץמח נמזב ו יב ן ימב נ ו יב ן אלש ימב נ ו רוסא והשמב . דומלת ילבב , תכסמ םיחספ ל םיחספ תכסמ , ילבב דומלת

Our Daf Rava teaches that the law is that chametz at the time of Pesach, whether it is mixed with other chametz or with different substances, even a trace of chametz prohibits the entire mixture.

Background:

Rava teaches something unique about chametz in the world of prohibited foods. Most other prohibited foods can become nullified when it is a small fraction (1/60th) of the food mixture. This means that effectively, one tiny grain of chametz can render an entire pot of stew prohibited.

Why might this be the case? What is so "dangerous" about chametz that we must obliterate every trace of it to an extent far beyond the standard with prohibited foods?

More Background: The Torah further prohibits even having possession of chametz, as well as a much harsher 'punishment' for its consumption than other prohibited foods. How might this affect the stringency? What does this teach us about chametz?

Even More Background: The prohibition of chametz is time-bound. In other words, throughout the year, chametz could be glatt-kosher, but for only the week of Pesach, its prohibition is the strongest in the entire world of prohibited foods. How do you think the time aspect influences the strength of the prohibition, if at all?

What does this teach us about how we relate to things in time?

A 'Flash' of Inspiration and Pesach

'ר קודצ ןהכה ,(ןילופ) תקדצ קידצה א תישאר תסינכ םדאה תדובעל 'ה ךירצ תויהל ןוזפחב ומכ וניצמש חספב םירצמ םירצמ חספב וניצמש ומכ ןוזפחב תויהל ךירצ 'ה תדובעל םדאה תסינכ תישאר א קידצה תקדצ ,(ןילופ) ןהכה קודצ 'ר היהש לכאנ זפחב ו ן אלו חספ .תורוד נפמ י הלחתההש קתנל צע ומ לכמ תואת םלוע הזה אוהש רשוקמ םהב ךירצ רומשל רומשל ךירצ םהב רשוקמ אוהש הזה םלוע תואת לכמ ומ

11 http://nesiya.org/wp-content/uploads/2016/02/Pesach_2015-1.pdf

30 עגרה ררועתמש וב וצר ן 'ה ופחלו ז לע ותוא עגר רהמל תאצל םהמ ילוא י .לכו רחאו ךכ בוש ךלי יתמב נ תו טאלו ידכ ן ד או ונימ ל בשך חו .כ לאםמתצ הלער תאל זוחו ' ןור ברותשער : חספ ד ו ר ו ת ס

A person's entry into the service of God must be with haste, as we find that the Paschal offering brought in Egypt was eaten in haste, which was not the case with the Paschal offering brought in later generations. Because when a person begins to sever himself from all the physical desires of this world to which he is attached, he must guard the moment in which the will of God stirs up within him, and make haste in that moment to leave them, perhaps he will succeed. Afterwards, he can once again proceed with moderation and slowness as is the law regarding the Paschal offering brought in later generations.

Rav Tzadok HaKohen Tzidkat ha-Tzadik 1

What is Rav Tzadok teaching here? He is referencing the difference in the Torah of how the Israelites participated in the first Pesach, as they left Egypt, in contradistinction to the subsequent celebrations of Pesach in the desert.

The primary difference that Rav Tzadok is highlighting is that when leaving Egypt, the Israelites performed the mitzvah with haste and alacrity, and that component was not involved in the later years. Rav Tzadok is teaching that this is connected to the moment that one has a flash of inspiration. What do you think? How do you respond to flashes of inspiration? Of creativity? How do you capture them and make them meaningful? Can we live lives of constant inspiration?

We want to suggest that all three of these aspects are deeply connected. The incredible stringency of the prohibition of chametz may be connected to the unique nature of the time-bound prohibition of something that is familiar and permitted at all other times.

Since we are accustomed to eating chametz throughout the year, it requires extra caution and awareness during this one week. The things that I am most likely to take for granted since they are present in my daily life throughout the year, are the things that I may forget to pay attention to. The prohibition of chametz even at the level of one tiny grain is calling to us to pay attention to the details in our daily lives.

To pay attention to those things that are precious, yet we may take for granted. Each item, each moment of our daily routines could become great moments of potential, when I focus my awareness on them.12 This brings us to Rav Tzadok's teaching with which he opens his entire treatise about spiritual growth. He is speaking to the very human experience of a tiny flash of inspiration and how to capture it.

That flash is a precious gift which may not come again, and we should not delay when it comes. If we spoke above about the routine and familiar and the potential that exists there, Rav Tzadok is

12 This is not to suggest that chametz itself is so precious to us, rather it is training us to have consciousness in our daily routines of how unique and precious each moment could be.

31 drawing our awareness to that which is outside of our familiar routine. "Seize the day!" says Rav Tzadok and start building—don't let this moment pass you by!

Our lives are full of many precious things, people and moments that we may take for granted. There are those that are familiar to us as we meet them regularly. There are also the ideas and people that we only encounter in rare flashes of opportunity.

Both of these categories offer incredible opportunity for growth and building inclusive and diverse communities and lives. The focus on the details of Pesach challenges us to re-evaluate how we think about each small detail and opportunity.

Pesach teaches us to pay close attention to the small things. Through the prohibition of chametz, I am sensitized to the small things in my life that are precious, and through the suggestion of seizing and capturing the small flashes of inspiration, I am able to build a life of meaning and growth when a moment of opportunity arrives, inviting me to take a step (or a leap!) in my growth and development.

The Mashehu that makes all the difference!

The Meor Einayim from Chernobyl in Parshas Tzav regarding the fact that chametz is assur even b'mashehu (and isn't batel even 1 in a thousand):13

13 http://dixieyid.blogspot.com/2007/04/only-difference-between-chametz-and.html

The letters in the words "chametz" and "matza" are almost exactly the same. Two out of the three letters in each word (in Hebrew) are the same: mem and tzadi. The only difference is that Chametz has a Ches. And Matza has a "hay."

And the only difference between those two letter is just a little line connecting the vertical line to the horizonal line on the top left side of the letter. That little line, that Mashehu of a line is so small that you would think it wouldn't matter. But that Mashehu makes all the difference between the word being "chametz," or "matza." That is why chametz is assur even b'mashehu.

The yetzer hara doesn't usually try to get us to do aveiros that are open and obvious to us as aveiros. First, it tries to blur the distinction between them by making the aveira seem to us to be a mitzva. That is how it tricks us. And often the distinctions can be very slight (outwardly) (only a Mashehu) but that Mashehu makes the difference between aveira and mitzva, between chametz and matza.

May be be zoche to be makpid on that Mashehu seemingly small difference between the yetzer tov and the yetzer hara, and may we be zoche to be meva'air and mevatel all of our inner chametz!

Thinking of the waters of the Kinneret and the ratio of chametz be’Mashehu….

An analysis of Waters of March14

Waters of March (Portuguese: Águas de Março) is a song composed by Brazilian composer Antônio Carlos Jobim.

Jobim reflects on life through this unconventional style, comparing it to nature and the passage of seasons.

https://www.youtube.com/watch?v=6MNknFy2gdQ

14 https://www.sciencesnail.com/philosophy

33 A stick, a stone, it's the end of the road It's the rest of a stump, it's a little alone It's a sliver of glass, it is life, it's the sun It is night, it is death, it's a trap, it's a gun

The oak when it blooms, a fox in the brush The nod of the wood, the song of a thrush The wood of the wind, a cliff, a fall A scratch, a lump, it is nothing at all

It's the wind blowing free, it's the end of a slope It's a beam, it's a void, it's a hunch, it's a hope And the riverbank talks of the Waters of March It's the end of the strain, it's the joy in your heart

The foot, the ground, the flesh and the bone The beat of the road, a sling-shot stone A truckload of bricks in the soft morning light The shot of a gun in the dead of the night

A mile, a must, a thrust, a bump It's a girl, it's a rhyme, it's a cold, it's the mumps The plan of the house, the body in bed And the car that got stuck, it's the mud, it's the mud

Afloat, adrift, a flight, a wing A cock, a quail, the promise of spring And the riverbank talks of the Waters of March It's the promise of life, it's the joy in your heart

A point, a grain, a bee, a bite A blink, a buzzard, a sudden stroke of night A pin, a needle, a sting, a pain A snail, a riddle, a wasp, a stain

A snake, a stick, it is John, it is Joe A fish, a flash, a silvery glow And the riverbank talks of the Waters of March It's the promise of life in your heart, in your heart

34 A stick, a stone, the end of the load The rest of a stump, a lonesome road A sliver of glass, a life, the sun A night, a death, the end of the run

And the riverbank talks of the Waters of March It's the end of all strain, it's the joy in your heart

The difficulties of living

Jobim emphasizes life’s struggle and suffering through images of physical adversity and pain. For instance, “a stick, a stone”, “a scratch, a lump”, “it's a cold, it's the mumps,” or “a pin, a needle, a sting, a pain.” The suffering is also mental, with feelings of isolation such as “it’s a little alone” and “a lonesome road.” Furthermore, monotony and exhaustion are indicated by lines like “the beat of the road.” Life’s adversity occurs through physical and mental obstacles, such as “the car that got stuck, it's the mud, it's the mud” or “a riddle.” Lastly, the lyrics’ disorganized structure mimics the chaos of life. The invocation of positive, negative, or neutral images occurs without obvious pattern, suggesting that life is unpredictable.

The inevitability of death

Jobim hints at the fragility and impermanence of life, comparing it to “a sliver of glass.” This analogy suggests beauty, yet delicacy. Numerous lines elude to our eventual end, such as “the end of the road,” “a cliff, a fall” “the end of a slope,” “a void,” or “the end of all strain.” Some of these images indicate a sudden death, while others suggest a more gradual decline. Furthermore, Jobim refers to “the wind blowing free,” which is conventionally symbolic of change and the passage of time, emphasizing our limited lifespans.

Despite our eventual end, Jobim reminds us of the beauty and value of life. Many images depict the vitality of nature, such as “the oak when it blooms, a fox in the brush, the nod of the wood, the song of a thrush” or “a fish, a flash, a silvery glow.” Human birth is also referenced, “it’s a girl.” Lastly, there are optimistic suggestions of a positive future: “the promise of spring,” “it’s a hope,” and “the plan of the house.”

Life and death equivalence

Jobim often equates life and death. For instance, “it is life, it’s the sun” is immediately followed by “It is night, it is death, it's a trap, it's a gun.” Likewise, “a life, the sun” precedes “a night, a death, the end of the run.”

More broadly, Jobim compares creation and destruction. “A truckload of bricks in the soft morning light,” an image of construction and optimism, is contrasted against “the shot of a gun in the dead of the night,” a picture of destruction. The light/darkness contrast reinforces this intention. Overall, these comparisons argue that life/creation requires death/destruction; we should think of these as different parts of the same fundamental process.

The equivalence of life and death is also explored using the traditional metaphor of the seasons, embodied by the titled “Waters of March.” The meaning of this term is quite different in the northern and southern hemispheres, a duality which nicely complements the theme.

In the southern hemisphere, this rain marks the end of the summer and the beginning of the colder season. This is the intention in the original Portuguese version.

However, in the northern hemisphere (and the English version of the lyrics), the melting of snow in March announces the start of spring. Therefore, the Waters of March dually refers to a destructive force bringing winter and death, and a constructive force bringing spring and birth.

These waters offer “the promise of spring,” “the promise of life” and “the joy in your heart,” yet paradoxically also bring “the end of all strain.” While the latter may initially sound positive, we can recognize it as a reference to death.

The suggestion is that the death/destruction associated with winter is required for the birth/creation of the summer.