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)

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