Translational Kinetic Energy — Question 4

Question 4

Pressure-volume data determine translational energy even for a polyatomic gas

For equilibrium translational motion use $\overline K=\tfrac32k_BT$ per molecule and $U_{\mathrm{tr}}=\tfrac32nRT$. Take $k_B=1.381\times10^{-23}$ J/K and $R=8.314$ J/(mol K). An equilibrium ideal gas occupies .0300 m$^3$ at 200 kPa. Its temperature is not separately measured, and the molecular species may be polyatomic.

1. Find total translational kinetic energy without first finding temperature or amount.

2. A separate measurement gives total internal energy above the chosen nonkinetic reference as 15.0 kJ. Find the energy assigned to nontranslational modes under an additive internal-energy model. Does $\tfrac32pV$ determine all internal energy for every gas?

Question 4 - Solution

Pressure-volume data determine translational energy even for a polyatomic gas

(a) Method and calculation

$$pV=nRT\ \Rightarrow\ U_{\mathrm{tr}}=\tfrac32pV.$$$$U_{\mathrm{tr}}=\tfrac32(200000)(.0300)=9000\,\mathrm J.$$

(b) Interpretation and further calculation

$$U=U_{\mathrm{tr}}+U_{\mathrm{other}}\ \Rightarrow\ U_{\mathrm{other}}=15000-9000=6000\,\mathrm J.$$$$U_{\mathrm{other}}/U=.400.$$Rotational, vibrational or other specified internal contributions are not included in center-of-mass translational kinetic energy.

Physical check

The $pV$ identity determines the translational part for a classical ideal gas; additional internal modes require additional information.