Temperature and Kinetic Theory of Gases

Luis Anchordoqui Atomic Theory of Matter (cont’d)

On a microscopic scale, the arrangements of molecules in solids, liquids, and gases are quite different.

solids gases

liquids Luis Anchordoqui Temperature and Thermometers

Temperature is a measure of how hot or cold something is. Most materials expand when heated.

Luis Anchordoqui Temperature and Thermometers (cont’d) Thermometers are instruments to measure temperature they take advantage of some property of matter that changes with temperature.

Early thermometers designed by Galileo made use of the expansion of a gas

Luis Anchordoqui Temperature and Thermometers (cont’d)

Common thermometers used today include the liquid-in-glass type and the bimetallic strip.

Luis Anchordoqui Temperature and Thermometers (cont’d)

Temperature is generally measured using either the Fahrenheit or the Celsius scale The freezing point of water is 0°C = 32°F the boiling point is 100°C = 212°F 5 T = ― ( T - 32º) C 9 F

9 T = ― ( T + 32º) F 5 C

Luis Anchordoqui Thermal Expansion Linear expansion occurs when an object is heated.

L = L 0 (1+ α Δ T)

the coefficient of linear expansion.

Luis Anchordoqui Thermal Expansion (cont’d)

Volume expansion is similar except that it is relevant for liquids and gases as well as solids:

Δ V = β V0 Δ T

the coefficient of volume expansion.

For uniform solids, β ≈ 3 α

Luis Anchordoqui Thermal Expansion (cont’d) Coefficients of expansion, near 20ºC

Luis Anchordoqui Thermal Equilibrium: Zeroth Law of Thermodynamics

Two objects placed in thermal contact will eventually come to the same temperature. When they do we say they are in thermal equilibrium

The zeroth law of thermodynamics says that if 2 objects are each in equilibrium with a 3 object they are also in thermal equilibrium with each other.

Luis Anchordoqui The Gas Laws and Absolute Temperature The relationship between the volume, pressure, temperature, and mass of a gas is called an equation of state.

We will deal here with gases that are not too dense.

Boyle’s Law: the volume of a given amount of gas is inversely proportional to the pressure as long as the temperature is constant.

Luis Anchordoqui The Gas Laws and Absolute Temperature

The volume is linearly proportional to the temperature, as long as the temperature is somewhat above the condensation point and the pressure is constant:

Extrapolating, the volume becomes zero at −273.15°C; this temperature is called absolute zero.

Luis Anchordoqui The Gas Laws and Absolute Temperature (cont’d) The concept of absolute zero allows us to define a third temperature scale – the absolute, or Kelvin, scale.

This scale starts with 0 K at absolute zero but otherwise is the same as the Celsius scale.

Therefore, the freezing point of water is 273.15 K, and the boiling point is 373.15 K.

Finally, when the volume is constant, the pressure is directly proportional to the temperature:

Luis Anchordoqui The temperature of various places and phenomena

Luis Anchordoqui The Ideal Gas Law We can combine the three relations just derived into a single relation:

What about the amount of gas present?

If the temperature and pressure are constant the volume is proportional to the amount of gas:

Luis Anchordoqui Ideal Gas Law (cont’d)

Atomic and molecular masses are measured in unified atomic mass units: u This unit is defined so that the carbon-12 atom has a mass of exactly 12.0 u Express in kilograms

-27 1 u = 1.6605 x 10 kg

Luis Anchordoqui The Ideal Gas Law (cont’d) A mole (mol) is defined as the number of grams of a substance that is numerically equal to the molecular mass of the substance

1 mol H2 has a mass of 2 g 1 mol Ne has a mass of 20 g

1 mol CO2 has a mass of 44 g

The number of moles in a certain mass of material:

mass (grams) n (mol) = Molecular mass (g/mol)

Luis Anchordoqui The Ideal Gas Law (cont’d)

We can now write the ideal gas law:

PV = n RT the number of moles the universal gas constant

R = 8.315 J/ ( mol · K ) = 0.0821 ( L · atm) / ( mol · K ) = 1.99 calories / ( mol · K )

Luis Anchordoqui Ideal Gas Law in Terms of Molecules: Avogadro’s Number

Since the gas constant is universal, the number of molecules in one mole is the same for all gases. That number is called Avogadro’s number:

23 NA = 6.02 x 10

The number of molecules in a gas is the number of moles times Avogadro’s number:

N = n NA

Luis Anchordoqui Ideal Gas Law in Terms of Molecules: Avogadro’s Number

Therefore we can write: PV = N k T

Boltzmann’s constant

R 8.315 J / mol · K K = = = 1.38 X 10-23 J / K N 6.02 X 10 23 / mol A

Luis Anchordoqui A small hot air balloon has a volume of 150 m3 and is open at the bottom.

The air inside the balloon is at an average temperature of 46º C, while the air next to the balloon has a temperature of 24º C, and a pressure, on average, of 1 atm.

The balloon is tethered to prevented from rising, and the tension in the tether is 10 N. Use 0.029 kg/mol for the molar mass of air.

(Neglect the gravitational force on the fabric of the balloon.)

What is the pressure on average inside the balloon?

P = 1.01 atm Luis Anchordoqui Kinetic Theory and the Molecular Interpretation of Temperature Assumptions of kinetic theory:

• large number of molecules, moving in random directions with a variety of speeds

• molecules are far apart, on average

• molecules obey laws of classical mechanics and interact only when colliding

• collisions are perfectly elastic

Luis Anchordoqui Kinetic Theory and the Molecular Interpretation of Temperature (cont’d)

The force exerted on the wall by the collision of one molecule is

Δ (mv) 2mv mv² F = = x = x Δ t 2l / vx l

Then the force due to all N molecules colliding with that wall is m F = ― N ¯v² l x

Luis Anchordoqui Kinetic Theory and the Molecular Interpretation of Temperature (cont’d)

The averages of the squares of the speeds in all three directions are equal:

m ¯v² F = ― N ― l 3 So the pressure is:

F 1 Nmv¯ ² 1 Nmv¯ ² P = ― = = A 3 Al 3 V

Luis Anchordoqui Kinetic Theory and the Molecular Interpretation of Temperature (cont’d)

Rewriting, 2 ¯ PV = 3 N ( ½ mv²)

so 2 ¯ ―3 ( ½ mv²) = kT

¯ 3 KE = ½ mv² = ―2 kT

The average translational kinetic energy of the molecules in an ideal gas is directly proportional to the temperature of the gas. Luis Anchordoqui Kinetic Theory and the Molecular Interpretation of Temperature (cont’d)

We can invert this to find the average speed of molecules in a gas as a function of temperature:

Luis Anchordoqui The escape speed for gas molecules in the atmosphere of Jupiter is 60 km/s and the surface temperature of Jupiter is typically -150º C. Calculate the rms speeds for (a) H 2 (b) O 2 (c) CO 2 at this temperature. (d) Are H 2 , O 2 , and CO 2 likely to be found in the atmosphere of Jupiter?

v rms (H 2 ) = 1.24 km/s

v rms (O 2 ) = 310 m/s

v rms (CO 2 ) = 264 m/s v 20% = 12 km/s

Then H 2 , O 2 , and CO 2 should be found in the atmosphere of Jupiter Luis Anchordoqui The escape speed for gas molecules in the atmosphere of Mars is 5.0 km/s and the surface temperature of Mars is typically 0º C. Calculate the rms speeds for (a) H 2 (b) O 2 (c) CO 2 at this temperature. (d) Are H2 , O 2 , and CO 2 likely to be found in the atmosphere of Mars?

v rms (H 2 ) = 1.8 km/s v rms (O 2 ) = 461 m/s v rms (CO 2 ) = 393 m/s v 20% = 1 km/s

Then O , and CO should be found in the atmosphere of 2 2 Mars, the molecules of H will escape. 2 Luis Anchordoqui Mean Free Path

The average speed of molecules in a gas at normal pressure is several hundred meters per second, yet if somebody across the room from you opens a perfume bottle, you do not detect the odor for several minutes.

The reason for the time delay is that the perfume molecules do not travel directly towards you, but instead travel a zigzag path due to collisions with the air molecules.

The average distance λ travelled by a molecule between collisions is called its mean free path

Luis Anchordoqui Mean Free Path (cont'd)

Consider one gas molecule of radius r moving with speed v through a 1 region of stationary molecules.

The moving molecule will collide with another molecule of radius r if the centers of the two molecules come within a 2 distance d = r + r from each other 1 2

If all the molecules are of the same type

d = molecular diameter

Luis Anchordoqui Mean Free Path (cont'd)

As the molecules moves it will collide with any molecule whose center is in a circle of radius d

After a time t the molecule moves a distance vt and collides with every molecule in the cylindrical volume π d² vt

The number of molecules in this volume is n π d² vt V

number density: number of molecules per unit volume n = N/V V After each collision the direction of the molecule changes so the path actually zigs and zags

The total path leangth divided by the number of collisions is the mean free path

vt 1 λ = = n π d² vt n π d² V V Luis Anchordoqui Mean Free Path (cont'd) Our previous calculation of the mean free path assumes that all but one of the gas molecules are stationary, which is not a realistic situation When motion of all the molecules is taken into account the mean free path becomes 1 λ = 2 √ nV π d²

The average time between collisions is called collision time τ

The reciprocal of the collision time is equal to the average number of collisions per second

Then if vav is the average speed

the average distance travelled between collisions is v λ = av τ Luis Anchordoqui Mean Free Path of CO Molecules in Air The local poison control center wants to know more about carbonmonoxide and how it spreads through a room. You are asked (a)to calculate the mean free path of a carbon monoxide molecule and (b) to estimate the mean time between collisions. The molar mass of carbon monoxide is 28 g/mol. Assume that the CO molecule is traveling in air at 300 K and 1 atm, and that the diameter of both CO molecules and air molecules are -10 3.75 x 10 m.

-8 λ= 6.53 x 10 m

-10 τ =1.26 x 10 s

Luis Anchordoqui Distribution of Molecular Speeds These two graphs show the distribution of speeds of molecules in a gas, as derived by Maxwell.

The most probable speed, vp , is not quite the same as the rms speed.

As expected, the curves shift to the right with temperature.

Luis Anchordoqui Real Gases and Changes of Phase The curves here represent the behavior of the gas at different temperatures. The cooler it gets, the farther the gas is from ideal.

In curve D, the gas becomes liquid; it begins condensing at (b) and is entirely liquid at (a).

The point (c) is called the critical point.

Luis Anchordoqui Real Gases and Changes of Phase (cont’d)

Below the critical temperature, the gas can liquefy if the pressure is sufficient; above it, no amount of pressure will suffice.

Critical temperatures and pressures

Luis Anchordoqui Real Gases and Changes of Phase (cont’d) A PT diagram is called a phase diagram; it shows all three phases of matter. The solid-liquid transition is melting or freezing; the liquid- vapor one is boiling or condensing; and the solid-vapor one is sublimation.

Phase diagram of water

Luis Anchordoqui Real Gases and Changes of Phase (cont’d) The triple point is the only point where all three phases can coexist in equilibrium.

Phase diagram of carbon dioxide

Luis Anchordoqui