Understanding Equipartition & Statistical Mechanics: Part a & b

[A.Brown]

Homework Statement

Part a: A monoatomic ideal gas is confined to move in two dimensions. What is C_v for this gas?

Part b: Consider a system composed of N independent harmonic oscillators in two dimensions. What is C_v for the system?

Homework Equations

C_v = ∂U/∂T
U = (number of degrees of freedom)(1/2 N_Ak_BT = (number of degrees of freedom)(1/2)RT

The Attempt at a Solution

I'm not very confident about this:

Part a: Since the gas is monatomic, it only has translational degrees of freedom, and it is given that these are restricted to two.

(2)(1/2)RT = RT and C_v = R

Part b: Since it's a harmonic oscillator, there's a degree of freedom for U and K for each dimension, so 4 degrees of freedom, and C_v = 2R? I'm not sure what to do with the N though, since it's an average, it shouldn't affect the C_v? Or am I misunderstanding?

 

[anathema]

For part a, your answer is correct. Since the gas is confined to move in two dimensions and it is monatomic, it only has two translational degrees of freedom. Therefore, Cv = R.

For part b, you are correct that each harmonic oscillator has two degrees of freedom (for U and K in each dimension), so for N oscillators, there are 2N degrees of freedom. Therefore, Cv = (2N)(1/2)R = NR. The N does not affect the Cv as it is a constant and does not change with temperature.