A-level Chemistry/AQA/Module 5/Thermodynamics/Specific Heat

Measuring the heat capacity of solids and gases was one of the first clues that we live in a quantum world. Classically, the heat capacity was constant at 1/2Nk per degree of freedom. This would put at a gas at 3/2Nk. Where k is Boltzmann's constant.

The specific heat capacity of a substance is defined as heat capacity per unit mass.

Experimental Values

When people started looking at heat capacity around the turn of century, they discovered some interesting things about how it changed with temperature. These problems took many years to figure out, including many attempts to create models.

Heat capacity of a Diatomic Gas

Heat Capacity of a Quantum Oscillator

If we take a single quantum oscillator with energy

$\epsilon =\hbar \omega \left(n+{\frac {1}{2}}\right)$

Then by summing over n with a Boltzmann distribution for energy. We get a value fot the internal energy of the oscillator.

$U$	$=\sum _{n}^{\infty }e_{n}p_{n}$
	$=\sum _{n}\hbar \omega \left(n+{\frac {1}{2}}\right)e^{\hbar \omega \left(n+{\frac {1}{2}}\right)/kT}$
	$={\frac {\hbar \omega }{2}}+\sum _{n}\hbar \omega ne^{\hbar \omega \left(n+{\frac {1}{2}}\right)/kT}$
	$={\frac {\hbar \omega }{2}}+{\frac {\hbar \omega }{e^{\hbar \omega /{kT}+1}}}$

When we differentiate this with respect to T, we get the Heat Capacity.

$C={\frac {dU}{dT}}$

When we look at the high a low temperature limits for this we see that at high temperatures we get the classical result: C = k_B. At the low temperature limit we get that

$C=k_{B}\left({\frac {\hbar \omega }{k_{b}T}}\right)^{2}e^{\frac {\hbar \omega }{k_{b}T}}$

The Harmonic crystal approximation

One reason why solids are easy to study is that the atoms in a solid are held very tightly to their equilibrium positions. So tightly, in fact, that all the atoms may be assumed to execute coupled simple harmonic motion about their mean positions.

Einstein's model for a solid

Einstein's model pretty much said that a solid was comprised of 3N quantum oscillators, it provided quite a good model at higher temperatures, but near T = 0K, it didn't agree with experimental values.Let derive the mathematical expression for that.

First Einstein assumed that in solids, the electrons are in the form of waves that don't react with each other (in the real case this does not happen but it can be a good approximating in some cases). Making this assumption, according to Einstein, the probability f(v) that an oscillator has frequency v is given by $f(v)={\frac {1}{e^{\frac {hv}{kT}}-1}}$.also internal energy of a solid is E=3Nhvf(v).So Einstein specific heat is $c_{v}={\frac {dE}{dT}}_{v}=3R{\frac {hv}{kT}}*{\frac {hv}{kT}}*{\frac {e^{\frac {hv}{kT}}}{\left(e^{\frac {hv}{kT}}-1\right)^{2}}}$

Debye's Model

Debye's model was a large improvement over Einstein's, it was based on the fact that the individual atoms do not oscillate independently. He said that waves travel through the material, these phonons were the important part. The maths is a bit more complicated in Debye's model, but is essentially the same as a boson gas, for example, it's analogous to photons.