Question

2 Equipartition The laws of statistical mechanics lead to a surprising, simple, and useful result — the Equipartition Theorem. In thermal equilibrium, the average energy of every degree of freedom is the same: hEi = 1 /2 kBT. A degree of freedom is a way in which the system can move or store energy. (In this expression and what follows, h· · ·i means the average of the quantity in brackets.) One consequence of this is the physicists’ form of the ideal gas law: P V = N kBT. In this version of the law, N is the number of particles and kB is Boltzmann’s constant, kB = 1.38 · 10^−23 J/K. This can be transformed into the chemists’ version of the ideal gas law as follows: P V = N kBT = N/ NA (NAkB) T = νRT. NA = 6.02 · 10^23 is Avogadro’s number, ν is the number of moles particles in the system, and R = 8.31 J/mol · K is the ideal gas constant. The ideal gas constant is useful for describing the energy in a huge collection of particles, where as Boltzmann’s constant is useful for describing the energy of individual atoms and molecules. In this problem, we will explore the implications of the equipartition theorem. Note that kBT ≈ 4 · 10^−21 J ≈ 25 meV at room temperature.

Kinetic Energy Particles are generally free to move along three independent directions (x, y, and z, if you like). Such a particle has three degrees of freedom. What is the total kinetic energy of a particle that moves freely in three dimensions? (Express the energy in terms of vx, vy, and vz.)

Equipartition The equipartition theorem states that each of the three degrees of freedom from the previous question will have an average energy of 1/2 kBT. What is the average kinetic energy of the particle, in terms of the temperature T?

RMS Speed The average velocity of a particle that is undergoing many collisions will be zero: It is just as likely to be moving in any direction. However, its average speed may not be zero. We can use the equipartition theorem to determine the average of the square of the speed, v^2 , because it is related to the average kinetic energy: (E) = 1/2 mv^2 = 1/2 m v ^2 The square root of this quantity gives the “root mean square speed” — the rms speed: vrms =sqrt(v^2) Use the equipartition theorem to derive an expression for the rms speed of a molecule of mass m that is in thermal equilibrium with a system at temperature T.

Molecular Speed What is the rms speed of a nitrogen molecule (N2) at room temperature? Can you drive that fast?

Slow Down At what temperature would the average speed of a nitrogen molecule be 100 mph?

Thermal Noise The equipartition theorem applies to every degree of freedom in a system — every way in which it can store energy. This includes circuit elements! The energy stored in a capacitor and an inductor is U = 1/2 CV^ 2 + 1/2 LI^2 Use the equipartition theorem to find the rms voltage and rms current in a circuit that contains an inductor L = 40 mH and a capacitor C = 10 µF.

Diatomic Molecules Consider a diatomic molecule like N2. We can model this molecule as two masses attached by a spring. How many degrees of freedom does such a molecule have? Describe all of the ways the molecule can move, including rotation, as well as the potential energy stored in the spring. Draw a diagram, too.

Internal Energy What is the average thermal energy of a diatomic molecule?

Specific Heat Do you expect the specific heat of a diatomic molecule to be larger or smaller than a monatomic molecule? Explain your reasoning.

Answer 1