Question

A box contains 100 atoms in a configuration that has 50atoms in each half of the box. Suppose that you could count the different microstates associated with this configuration at the rate of 100billion states per second, using a supercomputer. Without written calculation, guess how much computing time you would need: a day, a year, or much more than a year.

Short answer

The time for counting the different microstates associated with this configuration is more than a year.

Step-by-step solution

The given data

a) The number of atoms in the box is N =100

b) The number of atoms in each half of the box is n =50

c) The rate of the configuration is$P=100\mathrm{billion}\mathrm{states}/\mathrm{s}\mathrm{or}100\times {10}^9/s$

Understanding the concept of microstates configuration

We can first find the multiplicity of the given configuration. Then dividing it by the count rate, we can find the time for counting the given microstates.

Formulae:

The multiplicity of the microstates,$W=\frac{N!}{n_1!n_2!}$ …(i)

The rate of multiplicity of the configuration, $P=\frac{W}{t}$ …(ii)

Calculation of the time for counting the microstates

Using the given values in equation (i), we can get the multiplicity of the microstate configuration as follows:

$\begin{array}{l}W=\frac{100!}{\left(50!\left)\right|50!\right)}\\ =1.01\times {10}^{29}\end{array}$

Now, using the rate equation (ii) of the microstates associated with the configuration, we can get the time taken for counting the configuration as follows:

$$\begin{gathered} t=\frac{W}{P} \\ =\frac{1.01\times {10}^{29}}{100\times {10}^9/s} \\ =1.01\times {10}^{18}s \\ =32\mathrm{billion}\mathrm{years} \end{gathered}$$

Hence, the time taken is more than one year.