Statistical interpretation of entropy It is possible to develop a statistical in

Question

Statistical interpretation of entropy

It is possible to develop a statistical interpretation of entropy and entropy changes. For example, suppose there was just one molecule in a box. Now suppose that the volume of the box is doubled. What is the chance that the molecule in the box will remain in the one-half it originally occupied rather than spreading out to occupy the entire space? The answer is 1/2.

1. Suppose that there are three molecules in the box and the box is tripled in volume. What is the probability that the three molecules will remain in the space they originally occupied, rather than spreading out to occupy the entire space? Enter the fractional probability as a decimal.

Probability that three will remain in original space =  

2. Now extend the same idea to a larger system. Suppose you have a 73 liter vessel containing an ideal gas. Now imagine a smaller 39 liter volume within the vesssel. If the vessel contains only 24molecules, what is the probability that all 24 would be found within the imagined 39 liter smaller volume rather than spread out through the entire vessel? Enter your answer in exponential notation.

Probability that the 24 molecules will remain in the imagined smaller space =  

3. Suppose that the ideal gas described in the previous question was actually confined to the smaller imagined volume (39 liters). Now suppose the gas is allowed to expand isothermally to fill the entire container (73 liters). Calculate the entropy change for this expansion. Enter your answer in exponential notation.

Entropy change for this expansion =  JK-1

Explanation / Answer

1)

In general, the probability that N molecules are all in the same half of a box is p = (1/2)^N, and the probability that all

N molecules of a gas are in 1/Mth of the available volume of a space is p = (1/M)^N.

The probability that they are three in the same half of the box is therefore given by the product of the individual probabilities, so

probability p = (1/2)^3   or 1/2 x 1/2 x 1/2

                    = 0.125

Probability that three will remain in original space = 0.125

2)

the probability that all 24 molecules would be found in a particular 39/73th of the volume of a vessel. From the previous expression, that probability would be

p = (39 / 73)^24 = 2.92 x 10^-7

Probability that the 24 molecules will remain in the imagined smaller space = 2.92 x 10^-7

3)

dS= Change in entropy;

n = Number of molecules / Avogadro's number;

   = 24 / 6.023 x 10^23

   = 3.98 x 10^-23

R is the gas constant ( in this case we use 8.3145 JK-1 mol-1);

Vf is the space that the gas is allowed expand into, so in this case 73 liters;

Vi is what the gas was originally confined to 39 liters.

S = n*R*ln(Vfl/Vi)

     = 3.98 x 10^-23 x 8.3145 x ln (73 / 39)

     = 2.07 x 10^-22 JK-1

Entropy change for this expansion = 2.07 x 10^-22 JK-1

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