Heat Capacity of Solids: the Einstein and Debye Theories, and the Third Law

From the Dulong-Petit limit to the T-cubed law, and the entropy that vanishes at absolute zero.

Objectives

On completing this experiment you should be able to:

Measure the molar heat capacity of several solids from 2 K to 300 K.
Verify the Dulong-Petit limit and explain why diamond is nowhere near it at room temperature.
Obtain the Debye temperature and the Sommerfeld coefficient from a plot of $\frac{C}{T}$ against T-squared.
Fit the Einstein and Debye models over the whole range and show where the Einstein model fails and why.
Obtain the standard molar entropy by integrating $\frac{C}{T}$ from absolute zero, and verify that the entropy tends to zero.
Identify an unlabelled specimen from its Debye temperature.

Theory

Einstein modelled a solid as 3N independent oscillators of a single frequency, giving C = 3Nk $\frac{x^2 e^x}{e^x}$ - $1)^2$ with $x = \frac{\theta_E}{T}$ . This is correct at high temperature - it gives Dulong-Petit - but it falls off exponentially at low temperature, whereas experiment gives a cube law.

Debye replaced the single frequency by the acoustic spectrum of the solid, cut off so that the total number of modes is 3N. The density of states is then proportional to $\omega^2$ up to $\omega_D = k \theta_D$ / hbar, and

C_V = \frac{9NkT^3}{\theta_D^3} \, \int_{0}^{\theta_D/T} \frac{x^4 e^x}{(e^x-1)^2} \,dx ,

which is 3Nk at high temperature and (12 $\frac{\pi^4}{5)}$ Nk $\left(\frac{T}{\theta_D}\right)^3$ at low temperature. In a metal there is in addition a term gamma T from the conduction electrons, so at low temperature

\frac{C}{T} = \gamma + \left(\frac{12 \pi^4 R}{5 \theta_D^3}\right) T^2 ,

and a plot of $\frac{C}{T}$ against $T^2$ gives both constants at once.

Because C tends to zero at least as fast as T, the integral of $\frac{C}{T}$ from zero converges and the entropy has a definite value at $T = 0$ . The third law asserts that this value is the same for all states of a substance, and Simon's formulation extends this to the statement that the entropy change in any isothermal process tends to zero as T tends to zero.

Key relations

$C_V=9Nk\left(\frac{T}{\theta_D}\right)^{3}\int_0^{\theta_D/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}\mathrm{d}x$
The Debye heat capacity.
$\frac{C}{T}=\gamma+\frac{12\pi^{4}R}{5\theta_D^{3}}T^{2}$
Low-temperature form for a metal.
$S(T)=\int_0^{T}\frac{C_p}{T'}\,\mathrm{d}T'$
Calorimetric entropy, finite only because C tends to zero.

Procedure by part

Part A - Heat capacity by relaxation calorimetry

Stabilise the platform, apply a heat pulse of known energy and measure the temperature rise.

Select the specimen and the platform temperature.
Take the reading. The apparatus reports the molar heat capacity with the addenda already subtracted.
Cover 2 K to 300 K with at least thirty points, closely spaced below 10 K.
Plot C against T for each specimen.

Part B - Calorimetric entropy and the third law

Integrate your own $\frac{C}{T}$ data from the lowest temperature reached, using the cube law to extrapolate the last stretch to absolute zero.

Use the apparatus to obtain closely spaced points below 4 K for the chosen specimen.
Fit $\frac{C}{T}$ against T-squared to obtain gamma and the coefficient of the cube term.
Integrate the fit analytically from 0 K to your lowest measured point, then integrate your data numerically up to 298.15 K.
Compare the total with the tabulated standard molar entropy and account for the difference.

Questions to answer in your report

Fit both the Einstein and the Debye models to your copper data over the whole range. Show where the Einstein model fails, quantify the failure, and explain its physical origin. [10]
State the third law in both the Nernst and the Simon forms. Explain how your entropy integration tests it, and what would go wrong if C did not tend to zero. [8]
Your integrated entropy differs from the tabulated standard entropy. Quantify the discrepancy and discuss which of your assumptions is responsible. [8]
Explain why $C_p$ and $C_V$ differ, derive the expression you used to convert between them, and state which Maxwell relation it rests on. [6]