PC2352 Examples 5&6

10.

Questions on probability and microstates

Arrangements are covered here.

(a)

Four dice are rolled. What is the probability of obtaining four sixes? A six, a five, a four and a three? Two sixes and two fives?

(b)

A monkey types eight letters at random on a keyboard of 26 symbols. What is the probability that all are ``a''? That all of the first eight letters of the alphabet are present? Five ``a''s and three ``b''s? Three ``a''s, three ``b''s and two ``c''s?

(c)

A child distributes eight identical marbles between three different boxes. How many arrangements are there? (Hint: consider all the possible arrangements of eight marbles and two partitions in a line, but remember that the partitions are identical, as are the marbles.)

(d)

One joule of heat is added reversibly to 1 kg of water at 300 K. What is the increase in entropy of the water? By what factor does the number of accessible microstates increase?

Hint: first calculate the increase in entropy (the isothermal expression is a very reasonable approximation) then use the connection between entropy and microstates.

11.

(a)

Suppose we define the macrostate of a lattice of $N$ spin-$1/2$ atoms by specifying the number of up-spins $n$. What is the statistical weight $\Omega (N,n)$ of this macrostate? What is the maximum value $\Omega_{max}$ of $\Omega (N,n)$ as a function of $n$?

(b)

Plot $\Omega (N,n)/\Omega_{max}$ versus $\frac{n}{N}$ for $N = 10$ and $N = 100$ on the same graph. Comment on the form of your graph.

(Most calculators will fail to cope with factorials above 69, as the power of ten exceeds 100. For $N = 100$ you can use use the most accurate form of Stirling's approximation, $m! \approx \sqrt{2\pi m} m^{m} \exp(-m)$, or you can stick to $n$ in the range 35 to 65 since $\Omega (N, n)/\Omega _{\rm max}$ is very small outside that range. Either way you will still have to do some cancellation by hand and enter large and small terms alternately!)

(c)

Plot the function $\exp ( - 2 s^2 /N )$, where $s = n - N/2$, and show that it is a good approximation to $\Omega (N, n)/\Omega _{\rm max}$ when $N = 10$ and an excellent one when $N = 100$.

See here.

12.

The entropy of some gas can be written as $S=\frac 4 3 a^{1/4} V^{1/4}E^{3/4}$ where $a$ is a constant. Find the pressure, temperature and chemical potential, and show that $E/V=a T^4$ and $P=\frac 1 3 E/V$. What kind of gas is it?

$P$, $T$ and $\mu$ can be found from suitable derivatives of the entropy.

13.

Using Stirling's approximation, $\ln N!\approx N\ln N-N$, show that the entropy of a lattice of $N$ spin-1/2 atoms is

$$S = k_B \left[ N\ln N - n\ln n - (N - n)\ln (N - n) \right] ,$$

where $n$ is the number of up-spins in the $z$-direction.

If the sample is in a magnetic field $B$ which is in the $z$-direction, and the spins have magnetic moment $\mu$, show that the internal energy is $E=(N-2n)\mu B$.

Using the statistical mechanical definition of temperature and

$$\left( \frac{\partial S}{\partial E} \right)_{\scriptscripts... ... \frac{\partial S}{\partial n} \right)_{\scriptscriptstyle N}$$

show that the internal energy is given in terms of the temperature by

$$E = - N\mu B {\rm tanh} \left( \frac{\mu B}{k_BT} \right) .$$

(Hint - first find $T$ as a function of $n$, then invert to get $n(T)$.)

This question is covered here.