PH-102 (Modern Physics)Tutorial Sheet 1
(Microstates, Macrostates, microscopic description of system)
1Show that there are 10 macrostates A, B, ……J corresponding to the distribution of energy 6E0 to a collection of five distinguishable particles which can absorb energy in units of E0. Determine the probability P(E) of the particle to have energy E and plot P(E) as a function of E. Does the curve resemble any known function?
2Consider a box containing 5 particles a1, a2, a3, a4, a5. The box is divided into two equal parts, the right and the left. Write all the different ways in which we can have 3 particles in the left-part and 2 particles in the right-part of the box.
3Consider a small volume ‘v’ in a box of volume V. The box contains N identical but distinguishable particles. What is the probability that n particles are in the volume v and (N-n) particles are in the remaining volume (V-v)? Maximize this probability and prove that n = (v/V)N when the probability is maximum. Use the Stirling's approximation formula to show that the probability distribution has a Gaussian shape.
4A box is separated by a partition which divides its volume in the ratio 3:1. The larger portion of the box contains 1000 molecules of the Ne gas and the smaller one has 100 molecules of the He gas. The partition is punctured to make a small hole and let the equilibrium be attained. Assuming that all the particles are distinguishable, find the (a) mean number of molecules of each category on either side of the partition. And (b) probability of the larger portion having 1000 molecules of the Ne gas and the smaller portion 100 molecules of the He gas, that is, the same distribution as in the initial system.
5A system A consists of 4 distinguishable particles of spin zero. The energy of the system is 3 energy units. A second system B consists of 2 distinguishable particles of spin zero. The energy of this second system is also 3 units. Particle energy levels of both the systems are non-degenerate and have energies 0,1,2…..energy units. Find the number of microstates accessible to each system as well as to the composite system (A+B). (Ans. 20, 4, 80)
6A system consists of four independent but distinguishable particles. Each particle can be in one of the two states of energy 0 and ε. Find the probability that the energy of the system is 2 ε.
7A simple harmonic oscillator is in thermal contact with a heat reservoir at a temperature T, which is such that kT/ħω <1. Find the ratio of the probability of the oscillator being in the first excited state to the probability of its being in the ground state. Assuming that only the ground state and the first excited states are occupied, find the mean energy of the oscillator as a function of temperature T. (Ans. Exp(-ħω/kT), (ħω/2)(1+3exp(-ħω/kT))(1+exp(-ħω/kT))-1)
8Consider the model magnet of N dipoles, each of which may exist in either of two states (orientations). For the case N = 4 identify explicitly, and count, the different microstates associated with each possible energy macrostates (identified by each possible value of the system energy E).
9Review the analysis of the model magnet, according to which the weight function (N,E), giving the number of microstates of energy E, satisfies, for large N,
where
and x=n/N, with n (which is fixed by E) the number of dipoles in excited states. Sketch (ln Ω)/N as function of n/N, and think about why it has this form.
10One mole of ideal gas is maintained by a partition in one half of a container of total volume 2V , which is thermodynamically isolated. The partition is removed allowing the gas to expand freely to fill the entire volume. Calculate the entropy change by appeal to statistical mechanics arguments.
[Help: you may assume that the number of states available to a single molecule of gas is proportional to the volume V it occupies, from which you should deduce that the number of microstates of N molecules in volume V is proportional to VN .]
11Calculate the probability of finding 0.499 x 1023 particles in the right-part of a box and 0.501 x 1023 particles in the left part of the box. The total number of particles is 1023, the volume in the right and left part being the same.
12Consider a one dimensional harmonic oscillator of mass m whose total energy is E = p2/2m + ½ mω2x2. Find the volume in the phase space of the oscillator with the restriction that the energy lies between U and (U+ΔU).
13Consider a single particle enclosed in a cube of side L. Calculate the number of microstates available to this system if the total energy of the system is between U and (U+ΔU).