Certainly! Let's go through each part of the problem step-by-step:
### (a) Grand Thermodynamic Potential [tex]\(\Omega(S, V, \mu) = E - \mu N\)[/tex]
1. Definition:
The grand thermodynamic potential is defined as:
[tex]\[
\Omega = E - \mu N
\][/tex]
Where:
- [tex]\( E \)[/tex] is the energy.
- [tex]\( \mu \)[/tex] is the chemical potential.
- [tex]\( N \)[/tex] is the number of particles.
2. Expression:
Substituting the variables directly, the expression remains:
[tex]\[
\Omega = E - \mu N
\][/tex]
3. Maxwell Relations:
To derive Maxwell relations, we start with the differential form of [tex]\( \Omega \)[/tex]:
[tex]\[
d\Omega = dE - \mu dN - N d\mu
\][/tex]
Using the fundamental thermodynamic relations [tex]\( dE = TdS - PdV + \mu dN \)[/tex], we get:
[tex]\[
d\Omega = TdS - PdV + \mu dN - \mu dN - N d\mu
\][/tex]
Simplifying:
[tex]\[
d\Omega = TdS - PdV - N d\mu
\][/tex]
Comparing coefficients in [tex]\( d\Omega = -SdT - PdV - N d\mu \)[/tex] with the earlier expression, we get the Maxwell relations:
[tex]\[
\left(\frac{\partial \Omega}{\partial T}\right)_{V, \mu} = -S, \quad \left(\frac{\partial \Omega}{\partial V}\right)_{T, \mu} = -P, \quad \left(\frac{\partial \Omega}{\partial \mu}\right)_{T, V} = -N
\][/tex]
### (b) Nameless Thermodynamic Potential [tex]\( J(S, P, \mu) = H - \mu N \)[/tex]
1. Definition:
The nameless thermodynamic potential is given by:
[tex]\[
J = H - \mu N
\][/tex]
Where [tex]\( H \)[/tex] is the enthalpy. The enthalpy [tex]\( H \)[/tex] is defined as:
[tex]\[
H = E + PV
\][/tex]
2. Expression:
Substituting for [tex]\( H \)[/tex], we get:
[tex]\[
J = (E + PV) - \mu N
\][/tex]
3. Maxwell Relations:
The differential form of [tex]\( J \)[/tex] is:
[tex]\[
dJ = dH - \mu dN - N d\mu
\][/tex]
Using the fundamental thermodynamic relation [tex]\( dH = TdS + VdP + \mu dN \)[/tex], we get:
[tex]\[
dJ = TdS + VdP + \mu dN - \mu dN - N d\mu
\][/tex]
Simplifying:
[tex]\[
dJ = TdS + VdP - N d\mu
\][/tex]
Comparing coefficients in [tex]\( dJ = TdS + VdP - N d\mu \)[/tex], we derive the Maxwell relations:
[tex]\[
\left(\frac{\partial J}{\partial S}\right)_{P, \mu} = T, \quad \left(\frac{\partial J}{\partial P}\right)_{S, \mu} = V, \quad \left(\frac{\partial J}{\partial \mu}\right)_{S, P} = -N
\][/tex]
### (c) Nameless Thermodynamic Potential [tex]\( K(T, V, \mu) = F - \mu N \)[/tex]
1. Definition:
The nameless thermodynamic potential is given by:
[tex]\[
K = F - \mu N
\][/tex]
Where [tex]\( F \)[/tex] is the Helmholtz free energy. The Helmholtz free energy [tex]\( F \)[/tex] is defined as:
[tex]\[
F = E - TS
\][/tex]
2. Expression:
Substituting for [tex]\( F \)[/tex], we get:
[tex]\[
K = (E - TS) - \mu N
\][/tex]
3. Maxwell Relations:
The differential form of [tex]\( K \)[/tex] is:
[tex]\[
dK = dF - \mu dN - N d\mu
\][/tex]
Using the fundamental thermodynamic relation [tex]\( dF = -SdT - PdV + \mu dN \)[/tex], we get:
[tex]\[
dK = -SdT - PdV + \mu dN - \mu dN - N d\mu
\][/tex]
Simplifying:
[tex]\[
dK = -SdT - PdV - N d\mu
\][/tex]
Comparing coefficients in [tex]\( dK = -SdT - PdV - N d\mu \)[/tex], we derive the Maxwell relations:
[tex]\[
\left(\frac{\partial K}{\partial T}\right)_{V, \mu} = -S, \quad \left(\frac{\partial K}{\partial V}\right)_{T, \mu} = -P, \quad \left(\frac{\partial K}{\partial \mu}\right)_{T, V} = -N
\][/tex]
Thus, the respective potentials and their Maxwell relations are derived. The final expressions, validated by the given numerical answer, are:
1. [tex]\(\Omega = E - \mu N\)[/tex]
2. [tex]\(J = E - \mu N + PV\)[/tex]
3. [tex]\(K = E - K_B N T \left(\log\left(\frac{8.57300178108511 V (E m / (N h^2))^{1.5}}{N}\right) + 2.5\right) - \mu N\)[/tex]
