# Entropy of aluminum at low temperatures

Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 3.14.

A curve fit to experimental measurements of the heat capacity of one mole of alumin(i)um at low temperatures is

$\displaystyle C_{V}=aT+bT^{3} \ \ \ \ \ (1)$

where ${a=0.00135\mbox{ J K}^{-2}}$ and ${b=2.48\times10^{-5}\mbox{ J K}^{-3}}$.

We can use this to work out the entropy from the formula

 $\displaystyle S_{f}-S\left(0\right)$ $\displaystyle =$ $\displaystyle \int_{0}^{T_{f}}\frac{C_{V}\left(T\right)}{T}dT\ \ \ \ \ (2)$ $\displaystyle$ $\displaystyle =$ $\displaystyle aT_{f}+\frac{b}{3}T_{f}^{3} \ \ \ \ \ (3)$

If we take ${S\left(0\right)=0}$, we can evaluate the formula at a few temperatures.

At ${T=1\mbox{ K},}$ we have

$\displaystyle S\left(1\right)=1.358\times10^{-3}\mbox{ J K}^{-1} \ \ \ \ \ (4)$

In dimensionless form, we have

$\displaystyle \frac{S\left(1\right)}{k}=9.84\times10^{19} \ \ \ \ \ (5)$

At ${T=10\mbox{ K}}$

 $\displaystyle S\left(10\right)$ $\displaystyle =$ $\displaystyle 0.0218\mbox{ J K}^{-1}\ \ \ \ \ (6)$ $\displaystyle \frac{S\left(10\right)}{k}$ $\displaystyle =$ $\displaystyle 1.58\times10^{21} \ \ \ \ \ (7)$