(Average) Kinetic Energy of Molecules

Leoragon

What? So average kinetic energy is 3/2, and the total mechanical energy is 5/2? What happens when the molecule is triatomic? Is it the same?

the_emi_guy

K^2
Ok,
Thanks for the correction on this.

Leoragon,
Its (3/2)kT and (5/2)kT.

I believe that when the molecules get bigger it depends on their structure, for example whether they are linear or non-linear.

K^2

With triatomic it depends on whether all three are different, and if not, how they are arranged. For CO2, H2O, and similarly structured molecules, you get something very close to 7/2, because rotational modes are still frozen out, but you pick up an extra vibrational mode.

You can look up these values easily. The quantity most directly associated with average mechanical energy of molecules in a gas is specific heat capacity at constant volume. So when average mechanical energy per molecule is 3/2 k_bT, the C_V = 3/2 R per mol of gas. R is the ideal gas constant, of course. You can also measure heat capacity of gas at constant pressure, allowing volume to expand as gas heats up. C_P = C_V + nR. Finally, the quantity that's most commonly used and measured is the heat capacity ratio, γ = C_P/C_V. For light monatomic gases, γ=5/3=1.67 almost perfectly. For diatomic, it's closer to γ=7/5=1.4, but here you'll start seeing significant temperature dependence. It's slightly higher at lower temperatures. For triatomic gasses with structure of H2O and CO2, it's close to γ=9/7=1.29. You can compare these to values on this page. The simple theoretical prediction works out very close at about 100°C, where it's hot enough to excite vibrational modes, but cold enough to leave rotational more or less alone.

As the molecule gets more complicated, this approach gets worse. At some point, you need to honestly consider QM and see what the probabilities are of exciting states at given temperature.