The following calculations are for EM radiation, for which the internal energy U=Vu, where u=u(T)is the total energy density.
a) Show that (∂S/∂T)_v=V/T(du/dT), and (∂S/∂V)T=(u+P)/T
Note: use the answers in part a in part b and c
b) Using Stefan-Boltzmann law, u=aT^4, where a is a constant, integrate the expression for (∂S/∂T)_v, and obtain an expression for S in terms of a, T and V.
Hint: use Planck’s statement of the 3rd law of thermodynamics: lim_T→0 S=0 to evaluate the constant of integration.
c) Assuming P=bT^n, where b and n are constants, use the equality ∂^2S/∂T∂V=∂^2S/∂V∂T to obtain values for b and n. Note: b has the same units as a.
a) To derive the expressions (∂S/∂T)_v=V/T(du/dT) and (∂S/∂V)T=(u+P)/T, we start with the definition of entropy as S(T,V) = ∫(C_v/T)dT + S0(V).
For the first expression, (∂S/∂T)_v, we differentiate both sides with respect to T while keeping V constant:
∂S/∂T = (∫(C_v/T)dT) + (∂S0/∂T)
= C_v/T + (∂S0/∂T)
Since u = u(T), we can write this equation as:
∂S/∂T = V/T (du/dT) + (∂S0/∂T) [Equation 1]
We want to find (∂S/∂T)_v, so we set ∂S0/∂T to zero, as V is constant. Hence, (∂S/∂T)_v = V/T (du/dT).
Now, for the second expression, (∂S/∂V)_T, we differentiate both sides with respect to V while keeping T constant:
(∂S/∂V)_T = (∂S0/∂V)
Using the equation for the total energy density, u = Vu, we can write this equation as:
(∂S/∂V)_T = (∂S0/∂V) = (1/T) (∂(Vu)/∂V)
= (1/T) (u + V(∂u/∂V))
= (u + P)/T
Therefore, (∂S/∂V)_T = (u + P)/T.
b) Using the Stefan-Boltzmann law, u = aT^4, we can integrate the expression for (∂S/∂T)_v to obtain S in terms of a, T, and V.
Integrating (∂S/∂T)_v = V/T (du/dT) with respect to T gives:
S = ∫(V/T)(du/dT)dT
= V (∫(aT^2/T) dT)
= VaT^2/2 + C
Using Planck's statement of the 3rd law of thermodynamics, lim_T→0 S=0, we can evaluate the constant of integration, C:
lim_T→0 (VaT^2/2 + C) = 0
C = -VaT^2/2
Therefore, the expression for S in terms of a, T, and V is:
S = VaT^2/2 - VaT^2/2
= VaT^2
c) Assuming P = bT^n, we will use the equality (∂^2S/∂T∂V) = (∂^2S/∂V∂T) to obtain values for b and n.
Starting from (∂S/∂T)_v = V/T (du/dT), we differentiate both sides with respect to V while keeping T constant:
(∂^2S/∂T∂V)_T = (∂/∂V)((V/T)(du/dT))_T
= (∂/∂V)(V/T)(du/dT)
= (1/T)(du/dT) + (V/T)(∂^2u/∂T∂V)
= (1/T)(du/dT) + (V/T)(∂P/∂T)
Next, we differentiate (∂S/∂V)_T = (u + P)/T with respect to T while keeping V constant:
(∂^2S/∂V∂T)_V = (∂/∂T)((u + P)/T)_V
= (∂/∂T)((u + bT^n)/T)_V
= (1/T)(du/dT) + (V/T)(∂P/∂T)
Comparing the two equations (∂^2S/∂T∂V) = (∂^2S/∂V∂T), we find:
(1/T)(du/dT) + (V/T)(∂P/∂T) = (1/T)(du/dT) + (V/T)(∂P/∂T)
This implies that (∂P/∂T) = (∂P/∂T).
Hence, we can conclude that b and n have values such that (∂P/∂T) = (∂P/∂T). The specific values of b and n would depend on the expression for P and its temperature dependence in the context of the given problem.