physics assignment

Question

i need answers for these questions.

 

Thanks

Nazia

 

 

i) State Boltzmann’s two principles of statistical mechanics. (2 marks)

(ii) In the context of statistical mechanics, explain what is meant by a phase cell, and explain how such cells may be used in the specification of the configuration of a classical gas. (3 marks)



(b) (i) State Boltzmann’s distribution law for a gas in equilibrium. (2 marks)

(ii) If the probability of any given molecule occupying any given phase cell is known, what additional information is required in order to work out the equilibrium distribution of molecular speeds in a pure gas? (2 marks)



(c) A sealed container of volume 0.10 m^3 holds a sample of 3.0 × 10^24 atoms of helium gas in equilibrium. The distribution of speeds of the helium atoms shows a peak at 1100 m s−1.

(i) Calculate the temperature and pressure of the helium gas. (6 marks) (ii) What is the average kinetic energy of the helium atoms? (3 marks) (iii) What is the position of the maximum in the energy distribution? (3 marks) (Take the mass of each helium atom to be 4.0 amu.)



(d) (i) Explain what is meant by the heat capacity at constant volume, CV, and the heat capacity at constant pressure, CP. How are these two properties related for an ideal gas? Why is CP generally greater than CV? (4 marks)

(ii) When 117 J of energy is supplied as heat to 2.00 moles of an ideal gas at constant pressure, the temperature rises by 2.00 K. Calculate the molar heat capacity at constant pressure, CP,m and the molar heat capacity at constant volume, CV,m for the gas. Is the gas monatomic or diatomic? (6 marks)



(e) Explain what isothermal processes and adiabatic processes are, and the differences between them, taking care to explain the role, or otherwise, of the environment to which the system under study is exposed. (4 marks)



(f) A sample of 8.02 × 10−1 moles of nitrogen gas (γ = 1.40) occupies a volume of 2.00 × 10−2 m^3 at a pressure of 1.00 × 105 Pa and temperature of 300 K. The sample is adiabatically compressed to half its original volume. Nitrogen behaves as an ideal gas under these conditions.

(i) What is the change in entropy of the gas? (1 mark)

(ii) Show from the adiabatic condition and the equation of state that TV γ − 1 remains constant, and hence determine the final temperature of the gas. (6 marks)



(g) The gas sample is now returned to its initial state and then isothermally compressed to half its original volume.

(i) Find the change in entropy of the gas. (Hint: see Equation 3.30a in CPM.) (4 marks)

(ii) What is the change in internal energy of the gas? (1 mark)

(iii) What is the amount of heat transferred from the gas to its environment? (2 marks)
(iv) Calculate the amount of work done in compressing the gas. (1 mark)

Answer 1

i -Here are Boltzmann’s two main principles of statistical mechanics,

1. The Principle of Equal A Priori Probabilities

For an isolated system in equilibrium, all accessible microstates are equally probable. That means if the system has Ω possible microstates, each microstate has probability

𝑃_𝑖=1/ Ω

1. Boltzmann’s Entropy Principle.

The entropy S of a system is proportional to the logarithm of the number of accessible microstates:

S=k_Bln⁡Ω,,where

1. S = entropy,
2. k_B​ = Boltzmann constant,
3. Ω= number of microstates.

ii- A phase cell is a small region (or “cell”) in phase space, where phase space is the space of all possible values of positions and momenta of particles. A phase cell has a finite volume, typically taken to be of size

ℎ ^{3 𝑁} for a system of 𝑁 particles, where ℎ is Planck’s constant.

It represents a group of microstates that are indistinguishable at the macroscopic level.

How Phase Cells Are Used for a Classical Gas For a classical gas of 𝑁 N particles:

Each particle has a position 𝑟

( 𝑥 , 𝑦 , 𝑧 ) r=(x,y,z) and momentum 𝑝 ( 𝑝 𝑥 , 𝑝 𝑦 , 𝑝 𝑧 ) so the entire gas is represented by a point in a 6N-dimensional phase space.

Instead of specifying the exact continuous position and momentum of every particle (which would give infinitely many microstates), the phase space is divided into discrete phase cells.

Each cell represents a possible microscopic configuration that is treated as one microstate.

Thus, the configuration of a classical gas is specified by identifying which phase cell the system’s representative point occupies. Counting these cells allows statistical mechanics to compute quantities like entropy and the number of microstates Ω .