Equation of State of Ideal Gases

Equation of state of ideal gases

Objective For a constant amount of gas (air) investigate the correlation of 1. Volume and pressure at constant temperature (Boyles law)

2. Volume and temperature at constant pressure (Gay-Lussac’s law)

3. Pressure and temperature at constant volume Charles’s law

Introduction The gas laws are thermodynamic relationships that express the behavior of a quantity of gas in terms of the pressure P , volume V , and temperature T . The kinetic energy of gases expresses the behavior of a ”perfect” or ideal gas to be PV = nRT , which is commonly referred to as the ideal gas law. However, the general relationships among P , V and T contained in this equation had been expressed earlier in the classical gas laws for real gases. These gas laws were Based on the empirical observations of early investigators, in particular the English scientist Robert Boyle and the French scientists Jacques Charles and Joseph Gay-Lussac.

Theory I Boyle’s law: Around 1660, Robert Boyle had established an empirical relationship between the pressure and the volume of a gas, which is known as Boyle’s law: At constant temperature, the volume occupied by a given mass of gas is inversely proportional to its pressure. In mathematical notation, we write 1 V ∝ , (1) P In equation form, we may write k ∆V = . (2) P Where k is the constant of proportionality for a given temperature. The pressure of a conﬁned gas is commonly measured by means of a manometer. The gas pressure for such an open-tube manometer is the sum of the atmospheric pressure Pa and the pressure of the height diﬀerence of mercury

P = Pa + ρm × g × hm. (3)

1 but Pa = 76 cm Hg = ρm × g × 76 then equation (3) becomes:

P = ρm × g × (hm + 76). (4)

3 2 where ρm = 13.6 gm/cm and g = 980cm/s The volume of the enclosed gas is the volume of the measuring tube segment marked in brown added to the volume calculated from the length of the column of air:

V = Vl + 1.01 ml (5)

or V = πr2l + 1.01 ml µ ¶ 1.14 2 = π l + 1.01 ml 2 = 1.02l + 1.01 ∼= l + 1 = l0 (6)

Substituting with P and V in eq. (2):

0 PV = ρm × g × (hm + 76) × l = k. (7)

Let k k0 = ρm g then Boyle’s law is k0 h = − 76 (8) m l0

Where hm represents the change in pressure, and since the cross sec- tional area of the tube is constant,then l0 represents V .

II Charle’s and Gay-Lussac’s laws : In 1787, the French physicist Jacques Charles reported the result of a series of experiments which is known as Charles’ law : At constant volume, the pressure exerted by a given mass of gas is proportional to its absolute temperature. In mathematical notation,

P ∝ T, (9)

or P = k T. (10)

2 where k is a constant of proportionality. If we substitute for the value of P from eq(4 ), we get

ρm × g × (hm + 76) = k T. (11)

or k hm + 76 = × T. (12) ρm g then 0 hm = k T − 76. (13) Notice that the gas laws are expressed in terms of absolute temperature (Kelvin). Where TK = Tc + 273

In 1802, Gay-Lussac, using a somewhat diﬀerent experimental ap- proach, essentially restarted Charles’ law in a form more commonly used today (also known as Gay-Lussac’s law): At constant pressure, the volume of a given mass of gas is proportional to its absolute temperature. That is, V ∝ T, (14) or V = k00 T. (15) Substituting for the value of V from eq.(6)

l0 = k00 T. (16)

Equipments Gas laws apparatus, lab thermometer (−10... + 100c), distilled water.

Procedure • PART (I)

1. During the experiment, the temperature in the measuring tube must be kept constant. Keep the heater oﬀ and record reading of the thermometer in the air tube. 2. raise or lower the open end of the tube until the mercury levels in both sides are equal, then record the length of the enclosed air 0 column l , where the diﬀerence of mercury levels hm = 0. Repeat this step several times

3 3. Raise the mercury level in the open tube to decrease the air column, then record l, hm. 1 4. Plot hm on the y-axis, and l on the x-axis, then calculate the slope. • PART(II) 1. Set the thermostat to 30 c◦ wait until the thermometer read 30 c◦. 2. Raise or lower the open end of the tube until the mercury levels in both sides are equal, and then record the length of the enclosed 0 0 air column l = l0, and the diﬀerence of mercury levels hm = 0. 3. During the experiment the volume must be constant; this means l0 must be constant. Increase the thermostat by 5 c◦ wait for temperature constancy in the measuring tube and the increment of l0

4 4. Raise the open end tube to return the length of air column to 0 0 l = l0, then record hm and TK . 5. ) repeat steps 3, 4 several times.

6. Plot hm on the y-axis, and TK on the x-axis, calculate the slope. • PART(III)

1. Set the thermostat to 30 c◦ wait until the thermometer read 30 c◦. 2. Raise or lower the open end of the tube until the mercury levels in both sides are equal, and then record the length of the enclosed 0 0 air column l = l0, and the diﬀerence of mercury levels hm = 0. 3. During the experiment the pressure must be constant; this means ◦ hm = 0. Increase the thermostat by 5 c wait for temperature constancy in the measuring tube and the increment of l0. 4. Lower the open end tube to return to equal levels of mercury 0 hm = 0, then record l and TK . 5. Repeat steps 3, 4 several times 0 6. Plot l on the y-axis, and TK on the x-axis, calculate the slope.

Results Data table (I ): T0 = ...... l(cm)

l0(cm)

hm(cm)

Data table (II ): 0 l0 = ...... , l0 = ......

hm(cm)

5 Data table (II ): T0 = ......

l(cm)

l0(cm)