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*First TdS Equation:*

Entropy (S) can be defined in terms of any two of P, V, and T. Let us define entropy S as the function of V and T, i.e. S (V, T). Derivative of entropy S is:

dS = (∂S/∂V)TdV + (∂S/∂T)VdT

Multiply equation by T to get:

TdS = T(∂S/∂V)TdV + T(∂S/∂T)VdT

(∂S/∂V)T = (∂P/∂T)V and also at constant volume T(∂S/∂T)V = (∂U/∂T)V = C_{V}, thus equation becomes

TdS = T(∂P/∂T)VdV + C_{V}dT

This equation is first TdS equation.

*Second TdS:*

Let us define entropy S in terms P and T i.e. S (P, T), its derivative is

dS = (∂S/∂p)TdP + (∂S/∂T)PdT

Multiplying equation by T provides

TdS = T(∂S/∂p)TdP + T(∂S/∂T)PdT

From Maxwell's relation (∂S/∂p)T = - (∂V/∂T)P, and also at constant pressure T(∂S/∂T)P = (∂H/∂T)P = CP. Thus Equation becomes TdS = -T(∂V/∂T)PdP + C_{P}dT

This equation is second TdS equation.

*Third TdS:*

Then last option is to define entropy S as the function of P and V, i.e. S(P, V), we have

dS = (∂S/∂p)VdP + (∂S/∂V)PdV

Multiplying equation by T to get

TdS = T(∂S/∂p)VdP + T(∂S/∂V)PdV

In the constant volume process, it can be proved that:

T(∂S/∂p)V = C_{V}(∂T/∂P)V

And also constant pressure process, it can be proved that

T(∂S/∂V)P = C_{P}(∂T/∂V)P

As a result equation becomes:

TdS = C_{V}(∂T/∂P)VdP + C_{P}(∂T/∂V)PdV

This equation is third of TdS equations.

These equations allow one to compute heat flow (TdS) in the reversible process. Also, change in entropy between two states of the system can be estimated from these equations given that equations of state are known. This is due to all partial derivatives in these equations can be attained from equation of state.

*Expansion, Compression and TdS Equations:*

**Compression:**

The way the volume of the material decreases with pressure at constant temperature is explained by isothermal compressibility k

k = -1/V(∂V/∂P)T

Isothermal compressibility is different from another parameter known as adiabatic compressibility k_{ad}.

k_{ad} = -1/V(∂V/∂P)s

Reciprocal of k (i.e. 1/k) is known as isothermal bulk modulus or isothermal incompressibility.

**Expansion:**

The way the volume of the material increases with temperature at constant pressure is explained by coefficient of volume expansion or expansivity β.

β = 1/V(∂V/∂T)P

Unit of expansivity is K^{-1}

Using equations, one can demonstrate that

(∂P/∂T)V = β/k and (∂V/∂T)P = βV and reciprocal of equation provides

(∂T/∂P)V = k/β and (∂T/∂V)P = 1/βV

*TdS equations in terms of k and **β:*

Replacing partial derivatives with these, three TdS equations become

dS = β/kdV + (C_{V}/T)dT

dS = -βVdP + (C_{P}/T)dT

ds = (C_{V}k/Tβ)dP + (C_{P}/TβV)dV

These Equations give change in entropy between two states in terms of P, V, T, k, β and heat capacities as the function of temperature and pressure of specific volume. Implication of these equations is that one doesn't even require equation of state to compute change in entropy.

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