Another Thermodynamics Question

[drcrabs]

Why does U = (3/2)nRT?

 

[so-crates]

Thats not true in all cases. Its onlytrue for , if I remember correctly, monatomic ideal gases. The reason why its true has to do with the "degrees of freedom" of the gas. Monoatomic gaese have only 3 degrees of freedom(they can only move in the x, y and z directions and cannot rotate) For diatomic gases, it would be U = 5/2 nRT since we added two more degrees of freedom(two planes of rotation)

 

[drcrabs]

Yes i understand. I should have asked why

U=(f/2)nRT
where f = degrees of freedom

 

[siddharth]

This is done by the equi partition theorem.
It states that we add 1/2KT per degree of freedom and 1KT per degree of Vibrational freedom.
An interesting case is when we consider a diatomic gas like Hyrdogen gas.
We expect U to be
$$N_a[ \frac{3}{2}KT + \frac{2}{2}KT + 1KT]$$

from velocities in x,y,z directions, the rotation about x,y and vibrational (1/2mv^2 and 1/2kx^2) respectivley
ie,
$$= \frac{7}{2}RT$$


But experimentally we find that
$$U= \frac{5}{2}RT$$

This is because at room temperature vibration does not seem to contribute

Therefore $$\gamma = \frac{7}{5}$$ at room temperature