Application of Maxwell Relation | Kinetic Theory & Thermodynamics - Physics PDF Download

First T - dS equation

Let T and V are independent variable, such that S = S(T,V)

dS =

TdS =  put

TdS =

Second T - dS equation

Let T and P are independent variable, such that S = S(T, P).

TdS =

From Maxwell relation

TdS =

Third

T - dS

Equation

Let P, V are independent variable, such that S = S(P, V).

dS =

TdS = =

=

The First Energy Equation

Let T and V are independent variable, and U = U (T,V)

From first law of thermodynamics.

dU = TdS - PdV

Using Maxwell relation,

dU =

Second Energy Equation

Using Maxwell relation

This is popularly known as second energy equation

Application of second energy equation:

If U is function of independent variable of T and P.

Example 2: From relation, dU = TdS - PdV,  derive Maxwell relation,

dU = TdS - PdV

Hence, U is exact differential

Example 3: A real gas which obey van der Waal’s equation of state are kept in container which has temperature T0 and volume V0 . If volume of container changes to V such that temperature of gas become T , then what is change in entropy?

Assume C_v is specific heat of constant volume

For van der Waal’s gas

From first T - dS equation

dS =

where, S₀ is integration constant

Example 4: For van der Wall gases, prove that , where U is internal energy

From first energy equation

 put the value of  in equation (i)

Example 5: Prove that

(a)

(b)

(a) We know that,

Using Maxwell relation,

One can get,

(b)

Use Maxwell relation,

Example 6: If α_p is thermal expansivity at constant pressure and K_T is isothermal compressibility, then prove that

(a)

(b)

(c)

From Maxwell relation

(a)

(b)

(c)

Using Maxwell relation,

 

Example 7: Prove that

(a)

(b)

(c) For the van der Waal’s gas, prove that

(a)

=

S = S(T,V)

 and

Put the value of in equation (A)

=  

From Maxwell relation

From Maxwell relation  So

 Put the value of

(b) For van der Waal’s gas

 → differentiate w.r.t. to T

Differentiate (B) with respect to V

Substituting the value  in equation

 =

 =  =

Example 8: From

Prove, C_p - C_v = TEα²V, where E is bulk modulus of elasticity and α is coefficient of volume expansion.

Let C_p - C_{v =}

_{P = P(T,V)}

_{For constant pressure dP = 0}

_{Cp - Cv = TVEα²}

Example 9: Prove that

dH = TdS + VdP and put TdS =  dP in equation

Example 10: Over a certain range of pressure and temperature the Equation of a Certain substance is given by the relation V =

(a) Find the change in enthalpy at constant temperature if pressure change from P₁ to P₂

(b) Find the change of entropy of this substance in isothermal process

(a) dH =

for Isothermal process dT = 0

dH = , V =

dH = H₂ - H₁ =

(b) From second TdS equation TdS =

For Isothermal process dT = 0

dS =

=

Example 11: For Vander Waal gas

(a) Prove that  for Iso-Entropic process.

(b) If pressure and Volume changes from P₁,V₁ to P₂,V₂ at constant temperature then find Change in enthalpy

(a) From first TdS equation, TdS =

P =  ⇒

At constant entropy dS = 0 ⇒

 or

By integration,  = constant

(b) From

H =U + PV Hence enthalpy is point function

So, H₁ =U₁ + P₁V₁ and H₂ = U₂ + P₂V₂

H₂ - H₁ = U₂ - U₁ + P₂V₂ - P₁V

Example 12: If Helmholtz free energy for radiation is given by F =

(a) What is radiation pressure?

(b) If S is entropy of the system, prove that specific heat at constant volume is given by C_v= 3S

(a) dF = -SdT - PdV

(b) S =

C_v =  =  =

C_v = 3S

Example 13: The internal energy E of a system is given by E = , where b is constant and other symbols have their usual meaning.

(a) Find the temperature of the system

(b) Find Pressure of the system

From first law of thermodynamics

TdS = dU + PdV ⇒dU = TdS - PdV

As, U = E =

(a) T =

(b) P =

Example 14: Consider an Ideal gas where entropy is given by S = where n = number of moles, R = universal gas constant, U = internal energy V = volume and σ = constant

(a) Calculate specific heat at constant pressure and volume

(b) Prove that internal energy is given by U = 5/2PV

(a) From first law of thermodynamics

TdS = dU - PdV, dS =  ⇒

∴

C_v =  ⇒ C_p = C_v + R ⇒C_p = 7/2nR

(b) U =  ⇒

PV = nRT ⇒ V =

⇒ U= 5/2PV

Example 15: Using the equation of state, PV = nRT and the specific heat per mole,  for monatomic ideal gas

(a) Find Entropy of given system.

(b) Find free energy of given system

dU = , P = , C_v =

TdS = dU + PdV

dS =  or

S = 3/2  where, S₀ is constant

(c) F = U - TS

=  where F0 = T.S₀ is again constant

Example 16: From electromagnetic theory, Maxwell found that the pressure P from an isotropic radiation equal to 1/3 the energy density i.e., P = , where V is volume of the cavity, then using the first energy equation, prove that Energy density u is proportional to T⁴.

, where u = =

 = u and u =

 = 4u ⇒

u ∝T⁴ ⇒ u = α T⁴ , where α is a constant.