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*Answer can only contain numeric values

QUESTION: 1

A system of N localized, non-interacting spin 1/2 ions of magnetic moment in each is kept in an external magnetic field H. If the system is in equilibrium at temperature T, the Helmholtz free energy of the system is given as Calculate the value of α?

Solution:

Partition function

Vibrational partition function in this

where

For the whole system Z(v) = [Z(v)]^{N}

Helmholtz free energy F = –Nk_{B} TlnZ

⇒ α = 2

The correct answer is: 2

*Answer can only contain numeric values

QUESTION: 2

The free energy of a photon gas enclosed in a volume V is given by where a is the constant and T is the temperature of the gas.

The chemical potential of the photon gas is?

Solution:

and

No term depends on N

hence

µ = 0

The correct answer is: 0

*Answer can only contain numeric values

QUESTION: 3

Calculate the standard Gibb’s free energy change for the formation of methane from carbon and hydrogen at 298 K in the units of kJ/mol. Given that ΔH is –74.9 kJ/mol and ΔS = -80.75J{\rm{/K - mol}}.\)

Solution:

Given ΔH = –74.9 kJ/mol

ΔS = –80.75 J/K-mol

Now use the equation

ΔS = ΔH – TΔS

⇒ ΔS = –74.9 kJ/mol +298 K (80.75/K-mol) × (1 kJ/1000J)

⇒ ΔS = – 50.9 kJ/mol

The correct answer is: -50.9

*Answer can only contain numeric values

QUESTION: 4

Calculate the depression of melting point of ice (in K) produced by 1 atm increase of pressure, given that L_{ice} = 80 cal gm^{-1} and specific volume of ice and water at 0°C are 1.091 cm^{3} and 1 cm^{3} respectively

Solution:

1 atm = 10^{5} N/m^{2}

= 106 dynes/cm^{2
}

The correct answer is: 0.007429

*Answer can only contain numeric values

QUESTION: 5

To change the melting point of ice by 1K, what should be the change in the pressure, given that L = 80 cal gm^{-1} and specific volume of ice and water at 0°C are 1 cm^{3} and 1.091 cm^{3} respectively. Give your answer in atm.

Solution:

dT = 1K

dp = 1.3525 × 10^{8} dynes/cm^{2}

⇒dp = 135.25 atm.

The correct answer is: 135.25

*Answer can only contain numeric values

QUESTION: 6

Calculate the value of ΔH for an isothermal process?

Solution:

ΔH = C_{p} ΔT

ΔT = 0

⇒ ΔH = 0

The correct answer is: 0

*Answer can only contain numeric values

QUESTION: 7

If Z be the partition function and N represent number of particles the chemical potential is given as Find the value of α.

Solution:

dF = dU – pdV + µdN

Since F = –kT ln Z

⇒ α = –1

The correct answer is: -1

*Answer can only contain numeric values

QUESTION: 8

A solid melts into liquid. The relation between the pressure p and the temperature T of the phase transition is P = –2T + P_{0} , the entropy charge associated with the phase transition is 1 Joule mol^{-1} K^{-1} . The Claussius-Clapeyron equation for the latent heat latent heat then is equal to?

Solution:

p = –2T + p_{0
}

ΔS = 1

The correct answer is: 0.5

*Answer can only contain numeric values

QUESTION: 9

It the partition function of a harmonic oscillator with frequency at temperature T in then the free energy of 8 such independent oscillators in terms of is

Solution:

The Helmholtz free energy N independent harmonic oscillator in terms of partition function Z is

F = –NkT ln Z

here

N = 8

The correct answer is: 16

*Answer can only contain numeric values

QUESTION: 10

A system of N non-interacting and distinguishable particles of spin1 is in thermodynamic equilibrium. The entropy of the system turns out to be of the form Nk_{B}ln x. Find the value of x?

Solution:

Number of microstates Ω = 3^{N}

Entropy is given by

∴ S = NK_{B}ln 3

⇒ x = 3

The correct answer is: 3

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