Maxwell relation
The four fundamental equations involving the state functions, E, H, P, V, T, A, G and S can be derived as follows:
(i)The entropy change for an infinitesimally small change of the system is given by
dS = dq/T (1)
i.e., Tds = dq (2)
but dq = dE + PdV (3)
therefore TdS = dE + PdV (4)
i.e., dE = TdS – PdV (5)
(ii) The enthalpy change of a system is given by
H = E + PV (6)
i.e., dH = dE + PdV + VdP (7)
from equation 5
dE = dE + PdV + VdP (8)
i.e., dH = TdS – PdV + PdV + VdP (9) 
i.e., dH = TdS – PdVP (10)
(iii) Helmholtz work function is given by
A = E – TS (11)
dA = dE – TdS – SdT (12)
=dE – dq – SdT (13)
But dE – dq = -PdV (14)
i.e., dE – dq = -PdV (15)
Substituting the value of dE – dq in equation (13)
dA = - SdT –PdV (16)
(iv) Gibbs free energy is given by
G = H – TS (17)
But H = E + PV (18)
i.e., G = E + PV – TS (19)
i.e., dG = dE + PdV + VdP – TdS – SdT (20)
from equation (5) TdS = dE + PdV (21)
i.e., dG = TdS + VdP +TdS – SdT (22)
therefore the fundamental equations are
(i) dE = TdS – PdV (23)
(ii) dH = TdS + VdP (24)
(iii) dA = -SdT – PdV (25)
(iv) dG = -SdT + VdP (26)
1. At constant volume i.e., dV = 0 equation (24) becomes
(dE/dS)v = T (27)
At constant entropy dS = 0, then equation (24) becomes
(dE/dV)S = -P (28)
Differentiating equation (27) with respect to V keeping S as a constant and differentiating equation (28) with respect to S keeping V as a constant.
We get
∂square/∂S∂V = (∂T/∂V)s (29)
∂square/∂V∂S = - (∂P/∂S)V (30)
From equation (29) and (30)
(∂T/∂V)S = - (∂P/∂S)V (31)
II. If s is constant then dS = 0 equation (24) becomes
(∂H/∂P)S = T (32)
If P is constant then dS = 0 equation (24) becomes,
(∂H/∂S)P = T (33)
Differentiating equation (32) with respect to S keeping P as a constant and differentiating equation (33) with respect to P keeping S as a constant, we get
∂square H/∂P∂S =(∂V/∂S)P (34)
And ∂squareH/∂S∂P = (∂T/∂P)S (35)
From equation (34) and (35)
(∂T/∂P)S = (∂V/∂S)P (36)
III. At constant T, then dT = 0, equation (35) becomes 
(∂A/∂V)T = -P (37)
At constant V, then dV = 0, equation (35) becomes 
(∂A/∂T)T = -S (38)
Differentiating equation (37) with respect to T keeping v as a constant and differentiating equation (38) with respect to V keeping T as a constant, we get
∂square A/∂V∂T = - (∂P/∂T)V (39)
And ∂square A/ ∂T∂V = - (∂P/∂V)v (40)
From equation (39) and (40)
(∂S/∂V)T = - (∂P/∂T)V (41)

IV. At constant T, then dT = 0, equation (36) becomes
(∂G/∂P)T = V (42)
At constant P, dP = 0 then equation will becomes
(dG/∂P)P = - S (43)
Differentiating equation (42) with respect to T keeping P as Constant and differentiating equation (43) with respect to P keeping T as a constant, we get
∂square G/∂P∂T = (∂V/∂T)P (44)
And ∂square G/∂T∂P = - (∂S/∂P)T (45)
From equation (44) and (45)
-(∂S/∂P)T = (∂V/∂T)P (46)
Therefore the Maxwell’s relations are equations
(∂T/∂V)S = -(∂V/∂S)V
(∂T/∂P)S = (∂V/∂S)P
(∂S/∂V)T = (∂P/∂T)V
- (∂S/∂P)T = (∂V/∂T)P
The above four equations are called Maxwell equations or relationships. All the above equations contain the entropy which is a measure of the spontaneity. The first relations are preferred to as an isotropic or adiabatic and the first last two equations are isothermal relations.