Figure shows the heat capacity of a Bose gas as a function of temperature. In this problem you will calculate the shape of this unusual graph.
(a) Write down an expression for the total energy of a gas of N bosons confined to a volume V, in terms of an integral (analogous to equation 7.122).
(b) For T < Tc you can set µ = 0. Evaluate the integral numerically in this case, then differentiate the result with respect to T to obtain the heat capacity. Compare to Figure.
(c) Explain why the heat capacity must approach Nk in the high-T limit.
(d) For T > Tc you can evaluate the integral using the values of µ calculated in Problem. Do this to obtain the energy as a function of temperature, then numerically differentiate the result to obtain the heat. capacity. Plot the heat capacity, and check that your graph agrees with Figure.
Figure: Heat capacity of an ideal Bose gas in a three-dimensional box.
If You have a computer system that can do numerical integrals, It’s not particularly difficult to evaluate μ for T > Tc.
(a) As usual when solving a problem on a computer, it’s best to start by putting everything m terms of dimensionless variables. So define t == T /Tc , c = μ/kTc, and x = ϵ/kTc. Express the integral that defines μ, equation , in terms of these variables. You should obtain the equation
(b) According to Figure 7.33, the correct value of c when T = 2Tc is approximately –0.8. Plug in these values and check that the equation above is approximately satisfied.
(c) Now vary μ, holding T fixed, to find the precise value of μ for T = 2Tc Repeat for values of T/Tc ranging from 1.2 up to 3.0, in increments of 0.2. Plot a graph of μ as a function of temperature.
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