Degrees of Freedom and Molar Specific Heats — Question 8

Question 8

A gas restricted to a plane has two translational energy terms

Consider a classical monatomic ideal gas confined strictly to a two-dimensional plane of area $A$. There is no out-of-plane motion or internal excitation. Use the two-dimensional equation of state $\Pi A=Nk_BT=nRT$, where $\Pi$ is force per unit length. The work is $\delta W=\Pi\,dA$.

1. Derive $U$, molar $C_A$, and the mean square speed in the plane.

2. Derive the molar heat capacity $C_\Pi$ at fixed two-dimensional pressure and its ratio to $C_A$. Explain why $\Pi$ is not a pressure measured in pascals.

Question 8 - Solution

A gas restricted to a plane has two translational energy terms

(a) Method and calculation

$$f=2,\quad U=Nk_BT=nRT,\quad C_A=R.$$$$\tfrac12m\langle v_x^2+v_y^2\rangle=k_BT\ \Rightarrow\langle v^2\rangle=2k_BT/m.$$

(b) Interpretation and further calculation

$$\Pi\,dA=nR\,dT\quad(\Pi\text{ fixed}),\quad\delta Q=dU+\Pi\,dA=2nR\,dT.$$$$C_\Pi=2R,\quad C_\Pi/C_A=2,\quad[\Pi]=\mathrm{J/m^2}=\mathrm{N/m}.$$The geometry changes both the mode count and the mechanical pressure variable.

Physical check

The two-dimensional model must not be combined with the three-dimensional factor $3/2$.