It is possible to develop a statistical interpretation of entropy and entropy ch

Question

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.

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 = 3.7E-2

Now extend the same idea to a larger system. Suppose you have a 68 liter vessel containing an ideal gas. Now imagine a smaller 30 liter volume within the vesssel. If the vessel contains only 21 molecules, what is the probability that all 21 would be found within the imagined 30 liter smaller volume rather than spread out through the entire vessel? Enter your answer in exponential notation.

Probability that the 21 molecules will remain in the imagined smaller space = ______________

Suppose that the ideal gas described in the previous question was actually confined to the smaller imagined volume (30 liters). Now suppose the gas is allowed to expand isothermally to fill the entire container (68 liters). Calculate the entropy change for this expansion. Enter your answer in exponential notation.

Entropy change for this expansion =_____________ JK-1

Explanation / Answer

For the first question, the probability that the first molecule is in the original half of the box is 1/2, and the probability that the second molecule is in that half is also 1/2. We will assume that the molecules do not interact (i.e., attract or repel one another), so their positions are independent of one another. The probability that they are *both* in the same half of the box is therefore given by the product of the individual probabilities, so p = 1/2 * 1/2 = 1/4.

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.


If we write the entropy as a function of temperature and volume, then the total differential of S(T,V) is given by:

dS = ?S/?T_V dT + ?S/?V_T dV

for a constant-temperature process, dT = 0, so:

dS = ?S/?V_T dV

Using a Maxwell's relation (see source), we can equate ?S/?V_T = ?P/?T_V, so:

dS = ?P/?T_V dV

For an ideal gas, P = N*k*T/V

where N is the number of molecules and k is Boltzmann's constant.

Differentiating this with respect to temperature at constant volume gives:

?P/?T_V = N*k/V

so

dS = (N*k/V) dV

Integrating this gives:

?S = N*k*ln(V_final/V_initial)

In the last question, V_final = 68L and V_initial = 30L, so

?S = N*k*ln(68/30) = N*k*ln(2.26)

N here is 21 molecules, and k = R/(Avogadro's number) = 1.38*10^-23 J/(K*molecule) so:

?S = 21*1.38*10^-23 J/K * 2ln(2.26) = 4.72*10^-22 J/K

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