1. 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.

To determine the net entropy change for these two expansions, let's consider each expansion separately and then calculate the total entropy change.

Expansion 1:
Initially, the gas expands from volume V to volume 2V. Entropy change for this expansion (ΔS1) can be calculated using the equation:

ΔS1 = nR ln(Vf1/Vi)

where n is the number of moles of the gas, R is the ideal gas constant, Vf1 is the final volume (2V), and Vi is the initial volume (V).

Expansion 2:
Next, the gas expands from volume 2V to volume 3V. Entropy change for this expansion (ΔS2) can be calculated using the same equation:

ΔS2 = nR ln(Vf2/Vf1)

where Vf2 is the final volume (3V) and Vf1 is the volume at the end of the first expansion (2V).

Total entropy change:
The net entropy change for the two expansions is given by the sum of the individual entropy changes:

ΔS_total = ΔS1 + ΔS2

Now, let's compare this total entropy change with the entropy change that would occur if the gas expanded directly from volume V to volume 3V.

Direct expansion:
The entropy change (ΔS_direct) for the direct expansion from volume V to volume 3V can be calculated using the same equation:

ΔS_direct = nR ln(V_direct/Vi)

where V_direct is the final volume (3V) and Vi is the initial volume (V).

To determine if the net entropy change for the two expansions is greater than, less than, or equal to the entropy change for the direct expansion, we compare ΔS_total with ΔS_direct.

If ΔS_total > ΔS_direct, then the net entropy change for the two expansions is greater than the direct expansion.

If ΔS_total < ΔS_direct, then the net entropy change for the two expansions is less than the direct expansion.

If ΔS_total = ΔS_direct, then the net entropy change for the two expansions is equal to the direct expansion.

By comparing the calculated entropy changes, you can determine the relationship between the net entropy change and the entropy change for the direct expansion.

To determine the net entropy change for the two expansions compared to the entropy change if the gas expanded directly from V to 3V, we need to look at the entropy changes for each individual expansion.

Entropy is a measure of the distribution of energy in a system, or the level of disorder. In a thermodynamically reversible process, the change in entropy (ΔS) is given by the equation:

ΔS = ∫(dQ/T)

where dQ represents an infinitesimally small amount of heat transferred into or out of the system, and T is the temperature at which the heat transfer occurs.

In the case of an ideal gas expansion, if the process is adiabatic (i.e., no heat is transferred), the change in entropy can be expressed using another equation:

ΔS = n * R * ln(V₂/V₁)

where n is the number of moles of gas and R is the ideal gas constant.

Now let's analyze each expansion separately:

1. Expansion from V to 2V:
The entropy change for this expansion is given by ΔS₁ = n * R * ln(2V/V) = n * R * ln(2). Since ln(2) is a positive value, ΔS₁ is also positive.

2. Expansion from 2V to 3V:
Similarly, the entropy change for this expansion is ΔS₂ = n * R * ln(3V/2V) = n * R * ln(3/2). Again, ln(3/2) is a positive value, so ΔS₂ is positive as well.

To find the net entropy change for both expansions, we add ΔS₁ and ΔS₂:

Net entropy change = ΔS₁ + ΔS₂ = n * R * ln(2) + n * R * ln(3/2) = n * R * [ln(2) + ln(3/2)].

Now let's calculate the entropy change for the direct expansion from V to 3V:

Entropy change for direct expansion = n * R * ln(3V/V) = n * R * ln(3).

To compare these two quantities, we can simplify the expressions:

Net entropy change = n * R * [ln(2) + ln(3/2)] = n * R * ln(2*3/2) = n * R * ln(3) = Entropy change for direct expansion.

Therefore, the net entropy change for the two separate expansions is equal to the entropy change that would occur if the gas expanded directly from V to 3V.