You allow a gas to expand freely from volume V to volume 2V. Later you allow the gas to expand freely from volume 2V to volume 3V. Is the net entropy change for these two expansions greater than, less than, or equal to the entropy change that would occur if you allowed the gas to expand freely from volume V directly to volume 3V? Explain.

Question

You allow a gas to expand freely from volume V to volume 2V. Later you allow the gas to expand freely from volume 2V to volume 3V. Is the net entropy change for these two expansions greater than, less than, or equal to the entropy change that would occur if you allowed the gas to expand freely from volume V directly to volume 3V? Explain.

Answer 1

Assuming that no heat enters or leaves the gas, and that it expands by pushing against an external gas, or a moving wall or piston, then there is no entropy change in any case.

If there is a "free expansion" into a larger volume with no heat addition, and no work done against an outside gas or piston, then internal energy and temperature remain the same while entropy increases. The entropy gain will depend upon the ratio of initial to final volumes, and it will not matter if the expansion was done in two stages.

Answer 2

To determine the net entropy change for the two expansions, we need to consider the entropy change for each individual expansion and then add them together.

For the first expansion, from volume V to volume 2V, the change in entropy can be calculated using the formula:

∆S1 = nR ln(V2/V1)

Here, n represents the number of moles of the gas, R is the ideal gas constant, and V1 and V2 are the initial and final volumes, respectively.

For the second expansion, from volume 2V to volume 3V, the change in entropy can be calculated similarly:

∆S2 = nR ln(V3/V2)

Now, to determine the entropy change from volume V directly to volume 3V, we can add the individual changes in entropy:

∆S_total = ∆S1 + ∆S2

Now, let's compare the net entropy change with the entropy change that would occur if you allowed the gas to expand directly from volume V to volume 3V.

If we substitute V3 = 3V and simplify the equation, we get:

∆S_total = nR ln(2) + nR ln(3/2)

To determine if the net entropy change is greater than, less than, or equal to the entropy change for the direct expansion, we need to compare ∆S_total with the entropy change for the direct expansion, let's call it ∆S_direct.

∆S_direct = nR ln(3)

If we compare the two expressions, we can see that ln(3) > ln(2) + ln(3/2). Therefore, ∆S_total is greater than ∆S_direct.

In conclusion, the net entropy change for the two expansions (from V to 2V, and then from 2V to 3V) is greater than the entropy change that would occur if the gas expanded directly from volume V to volume 3V.

Assuming that no heat enters or leaves the gas, and that it expands by pushing against an external gas, or a moving wall or piston, then there is no entropy change in any case.

sgfs\d

To determine the net entropy change for the two expansions, we need to consider the entropy change for each individual expansion and then add them together.