Calculating Thermal Dilation in Sodium Using Bulk Modulus and Temperature

[daudaudaudau]

Homework Statement

This is problem 5.2 in Kittel's Introduction to solid state physics:
Estimate for 300 K the root mean square thermal dilation for a primitive cell of sodium. Take the bulk modulus as $7\cdot10^{10}$ [unit]. Note that the Debye temperature 158 K is less than 300 K, so that the thermal energy is of the order of $k_BT$.

Homework Equations

Relation between elastic energy density, bulk modulus and volume dilation: $U=\frac{1}{2}B\delta^2$.

The Attempt at a Solution

So obviously I need the elastic energy density. Kittel claims that this is just $\frac{1}{2}k_BT$ because there is only one degree of freedom in this kind of elastic expansion. But I don't understand this. He's supposed to find the elastic energy density, but then he finds the thermal energy density? Are these equal? And why are there suddently LESS degrees of freedom just because the material is expanding? I mean, usually the internal energy is $U=3Nk_BT$, right?

 

[daudaudaudau]

I know this is not a hard question... :-)

 

[daudaudaudau]

No one can tell me how to calculate thermal dilation from temperature and the bulk modulus ? This is not even homework...