Thermodynamical Potential: Assignment | Kinetic Theory & Thermodynamics - Physics PDF Download

Q.1. The Helmholtz free energy of a system given by, F = A+ BT(1-lnT) -CT lnV where A, B,C are constatnt. Obtain expression for pressure, entropy, internal energy, enthalpy and gibbs energy?

dU = TdS- PdV , H = U+ PV , F = U-TS , G = H- TS

dF = -PdV - SdT

Hence,

and

Also U = ( F+ TS )

U = A + BT (1 - ln T) - CT ln V + TB ln T + TC lnV = (A + BT )

H = (U + PV ) = (A+ BT)+CT

G = H - TS = (A+ BT + CT)- TB ln T - TC lnV = A + BT (1- ln T) + CT (1- lnV )

Q.2. Prove that for n -moles of an ideal gas

(a) F =

(b) G =

where a₁ ,a₂ ,b₁ and b₂ are constant parameter.

(a) F = U- TS , dF = dU- d (TS )

F =

S = S (T ,V)

dS =

TdS =

dS =

S =

(b) G = ( H - TS ) = U + PV - TS

dG = (dU + PdV + VdP) - d (TS ) = dU + nRdT - d (TS )

and S = S (T , P)

S =

Hence, G =

Q.3. If equation of state is given P =

(a) Find the

(b) If volume is expanded from V₁ to V₂ at very high temperature then what will be change in internal energy?

(a)

P =

=

(b)  for very high temperature

Q.4. The entropy function of a system is given by S(E) = aE(E₀ - E) where a and  E₀ are positive constants.

(a) For what is condition that energy E temperature is negative .

(b) Plot temperature with energy of the system .

(a) From first and second law of thermodynamics 

 =  ⇒ T =

For negative temperature

(b) T =

Q.5. Calculate the Variation of C_P with pressure at constant temperature for a substance whose equation of state is given by relation V =

 =

Q.6. If α = , β =  then prove that for an ideal gas prove that

(a)

(b)

(c)

(a) α = , β =

  

V =

dU =

dU =

(b) H = U+ PV , dH = dU + PdV +VdP = TdS - PdV + PdV +VdPdH = TdS +VdP

S = S (T , P)

dS =

(c) F = (U- TS )

dF = (TdS - PdV) - (TdS + SdT) = -PdV - SdT

and, V = V (T , P)

Q.7. The specific Gibbs energy of an ideal gas is given as-

G =

where B is a function of T only obtain the equation of state of gas?

we know that 

G = , dG =

comparing on both sides, we have

Q.8. The equation of state of a gas is given by V =

where R is the gas constant and b is another constant parameter. Show that the specific heat at constant pressure C_P and the specific heat at constant volume C_V for this gas is related by

It is given V =

P =

=

=

Q.9. Helmholtz free energy is given by αVT⁴ where α > 0 .

(a) Find the value of internal energy

(b) For the adiabatic process prove that VT³ = K where K is constant parameter.

(a)    U = F+ TS

U =αVT⁴ + T.4αVT³ = 3αVT⁴ 

(b) for adiabatic change dQ = 0 ⇒ TdS = 0, S = constant

 = 4αVT³ = constant so VT³ = K

Q.10. For a particular thermodynamic system the entropy S is related to the internal energy U and volume V by

S =

 ⇒ T =

G = U  = 0