Statistical Mechanics: Assignment | Kinetic Theory & Thermodynamics - Physics PDF Download

Q.1. One spin in the thermal contact with a small spin system

Consider a system A consisting of a spin having1/2 particle with magnetic moment μ₀ ,and another system A' consisting of 3 spins 1/2 particle, each having magnetic moments μ₀ . Both systems are located in the same magnetic field B . The systems are placed in contact with each other, in such a way, they are free to exchange energy. Suppose that, when the moments of A , points up (i.e., when A is in its + state), two of the moments of A' points up and one of them points down.

(a) Count the total number of state accessible to the combined system A+ A' , when the moment of A , points up, and points down respectively. Hence, calculate the ratio P_- / P₊ ,  where P_- is the probability that the moment of A points down and P₊ is the probability that it points up. Assume that the total system A+ A' is isolated.

(b) Find the average of magnetic moment of sub system A'.

If A has up state and A' has situation that 2 spin is in up state and 1 is in down state and has total energy of the system ( A +A') as 2μ₀B , then the total number of microstate is given by

 = 3 + 1 = 4 number of microstate .

For sub system A p₊ = 3/4 and p_- = 1/4

(a)  For subsystem A' ,

Thus (M) =

Solution 3: Total number of microstate

If moment of A points up, then A' has  number of microstate 

(b) If moment of A points down, then A' has n + 1 up and N - (n - 1) spin points down,

so number of microstate is

(c)  =

For n >> 1 and n' >> 1 , the ratio P_- /P₊ larger or smaller then unity if n >n'

Q.2. In a thermodynamic system in equilibrium, each molecule can exist in three possible states with two of them has probabilities 1/ 2, 1/ 3 respectively. Find entropy per molecule.

S =

S =

S =

S =

=

S =

Q.3. Consider a system of N particles obeying classical statistics, each of which can have an energy 0 or E . The system is in thermal contact with a reservoir maintained at a temperature T . Let k denote the Boltzmann constant. Find heat capacity C of the system .

U =

E₁ = 0, E₂ = E for one particle U =

So for N Particle U =

Q.4. Two classical particles are distributed among N (>2) sites on a ring. Each site can accommodate only one particle. If two particles occupy two nearest neighbour sites, then the energy of the system is increased by ∈.

(a) Find partition function

(b) Find The average Helmholtz free energy of the system at temperature T is

Since two particle two nearest neighbour sites, which energy of system increased by ∈, and remining ( N - 3) particle hs zero energy, then particle function is given
z = 2e^-β∈ + ( N - 3) e^{-β .0} = ( N - 3) + 2e^-β∈

(a) Z = N - 3 +

(b) F =

Q.5. Consider a system of two Ising spins S₁ and S₂ taking values  ± 1with interaction energy given by ε = JεS₁S₂ . When the system is in thermal equilibrium at temperature T ,

(a) Find partition function of system

(b) Find Helmholtz free energy

(c) Internal energy

The interaction energy is given by E = JεS₁S₂ where S₁ and S₂ taking values ± 1 .

Possible values of the Energy of the system are 

 

 

(a)

(b)

(c)

Q.6. Consider an ideal gas of N molecules which is in equilibrium, within a container of volume V₀ . Denoted by n is the number of molecules located within any subvolume V of this container. The probability p that a given molecule is located within this subvolume V is then given by p =V /V₀ .

(a) Find the probability that out of N molecules, n the number of molecules located

within any subvolume V 

(b) What is mean number (n) of molecules within V ? Express your answer in terms of N,V₀ and V

(c) Find the standard derivation Δn, where n is the number of molecules located within volume V .

(d) Calculate , expressing your answer in terms of N,V₀ and V.

(e) What is the value of , when 0 V <<V₀ ?

(f) What value should the standard deviation Δn assume when V →V₀ ?

 (a)

where  so

(b) (n) =

(c)

(d)

(e)

(f)

Q.7. The quantum states available to a given physical system is are group in g_i equally likely states with common energy ε_i .If  p_i is probability associated with energy ε_i such that  then prove that entropy is given by S=

p_i =  where Z is partition function 

Q.8. A rigid and thermally isolated tank is divided into two compartments of equal volumeV, separated by a thin membrane. One compartment contains one mole of an ideal gas A and the other compartment contains one mole of a different ideal gas B . The two gases are in thermal equilibrium at a temperature T . If the membrane ruptures, the two gases mix. Assume that the gases are chemically inert. The Find the change in the total entropy of the gases on mixing is

For A , number of microstate after mixing is 2

For A , number of microstate before mixing is 

⇒ ΔS A = R ln 2 - R ln1 = R ln 2

Similarly, for B ΔS_B =  Rln 2 ⇒ ΔS =ΔS_A + ΔS_B = 2 ln 2

The number of microstates of a gas of N particles in a volume V and of internal energy U , is given by

(where a and b are positive constants). Its pressure P , volume V and temperature U , are related by

Then prove that

S =

Q.9. A particle is confined to the region x ≥ 0 by a potential which increases linearly as V(x) = u₀x . The find ratio between average potential energy V to average kinetic energy T

Partition function Z =

=  =

 =  =

Hence the system is one dimensional so  = 2

Q.10. Gas molecules of mass m are confined in a cylinder of radius R and height L

(with R >> L ) kept vertically in the Earth’s gravitational field.

(a) Find The average energy of the gas at high temperatures (such that mgl<<k_BT ) .

(b) Find The average energy of the gas at low temperatures (such that mgl>>k_BT )

Z =

Z =

Z =

For high temperature Z =

Z =  ⇒

mgl<<k_BT ⇒

For low temperature 

⇒ Z =  =

⇒ (E) =