Essential Physics

Section 3 review

Statistical mechanics explains how the average behavior of trillions of microscopic particles creates macroscopic properties such as temperature, density, and pressure. This explanation is known as the kinetic theory of matter. One important prediction of kinetic theory is that, in thermal equilibrium, each degree of freedom has a mean thermal energy of ½k_BT. The kinetic theory also shows that particles in a gas assume an equilibrium range of speeds described by the Maxwellian distribution. The average thermal speed of a gas particle can be predicted by using the Maxwellian distribution and depends on temperature, mass, and Boltzmann’s constant.

Another result of kinetic theory is the ideal gas law, which relates pressure, volume, temperature, and quantity for a gas. The ideal gas law can be derived by calculating the average force exerted on the walls of a container by the impact of N particles per second. The specific heat of an ideal gas is derived by summing the kinetic energies of all constituent atoms. In contrast to atoms in a gas, atoms in a solid interact strongly with each other. The bonds between neighboring atoms constitute additional degrees of freedom because they can store potential energy (akin to “springs”). Kinetic theory predicts that the specific heat of a monatomic solid is twice that of an ideal gas. Read the text aloud

Read the text aloud

statistical mechanics, Maxwellian distribution

E = \frac{3}{2} k_{B} T

P V = N k_{B} T

gas: c_{v} = \frac{3}{2} \frac{N_{A} k_{B}}{m_{m o l}}

solid: c_{p} = \frac{3 N_{A} k_{B}}{m_{m o l}}

Review problems and questions

Calculate the mean thermal speed of an atom in xenon gas at room temperature (21ºC). The molecular mass of xenon is 131.2 g/mol.

Answer: 7.5 m/s
A temperature of 21°C corresponds to 294 K. The mass of a xenon atom is

m = \frac{131.2 g/mol}{6.022 \times 10^{23} atoms/mol} = 2.179 \times 10^{- 22} g/atom

Use this mass for each xenon atom to calculate the thermal speed for the gas:

v_{t h} = \sqrt{\frac{3 k_{B} T}{m}} = \sqrt{\frac{3 (1.38 \times 10^{- 23} J/K) (294 K)}{2.179 \times 10^{- 22} g}} = 7.5 m/s

How many particles are in one cubic meter of air at a pressure of one atmosphere (101,325 Pa) and a temperature of 20ºC?

Answer: 2.51×10²⁵ atoms
The temperature needs to be converted to kelvins: 20°C corresponds to 293 K. Rearrange terms from the ideal gas law to solve for the number of atoms, N:

N = \frac{P V}{k_{B} T} = \frac{(101, 325 Pa) (1 m^{3})}{(1.38 \times 10^{- 23} J/K) (293 K)} = 2.51 \times 10^{25} atoms

The specific heat of an unknown gas is measured to be 620 J/(kg ºC). The gas is most likely to be which of the following?
1. helium (4 g/mol)
2. neon (20.1 g/mol
3. argon (40 g/mol)
4. xenon (137 g/mol)

Answer: The gas is neon (b).
Start with the equation for a gas

c_{v} = \frac{3 N_{A} k_{B}}{2 m_{m o l}}

and rewrite it to solve for m_mol:

m_{m o l} = \frac{3 N_{A} k_{B}}{2 c_{v}} = \frac{3 (6.022 \times 10^{23} atoms/mol) (1.38 \times 10^{- 23} J/K)}{2 (620 {J kg}^{- 1} ° C^{- 1})} = 0.0201 kg/mol = 20.1 g/mol

That corresponds to neon.

Calculate the theoretical specific heat of silver. The atomic mass of silver is 107.9 g/mol. How does this compare with the actual measured value? Research possible explanations for any discrepancy.

Answer: 231 J kg⁻¹ °C⁻¹
Use the equation for the specific heat of a solid:

c_{v} = \frac{3 N_{A} k_{B}}{m_{m o l}} = \frac{3 (6.022 \times 10^{23} atoms/mol) (1.38 \times 10^{- 23} J/K)}{0.1079 kg/mol} = 231 {J kg}^{- 1} ° C^{- 1}

This is 1% lower than the measured value of 233 J kg⁻¹ °C⁻¹. This is pretty good! It is off a little bit because the derivation of the above equation in the text utilized several approximations.


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