For an ideal gas the internal energy depends on temperature...

For an ideal gas the internal energy depends on temperature only. We can consider the process in question to be one of simultaneous free expansion. Then the total energy $U=U_{1}+U_{2}$. Since $U_{1}=C_{V} T_{1}, U_{2}=C_{V} T_{2}, U=2 C_{V} \frac{T_{1}+T_{2}}{2}$ and $\left(T_{1}+T_{2}\right) / 2$ is the final temperature. The entropy change is obtained by considering isochoric processes because in effect, the gas remains confined to its vessel. $$ \Delta S=\int_{T_{1}}^{\left(T_{1}+T_{2} / 2\right.} \frac{C_{V} d T}{T}-\int_{\left(T_{1}+T_{2}\right) / 2}^{T_{2}} C_{V} \frac{d T}{T}=C_{V} \ln \frac{\left(T_{1}+T_{2}\right)^{2}}{4 T_{1} T_{2}} $$ Since $\left(T_{1}+T_{2}\right)^{2}=\left(T_{1}-T_{2}\right)^{2}+4 T_{1} T_{2}, \Delta S>0$

For an ideal gas the internal energy depends on temperature only. We can consider the process in question to be one of simultaneous free expansion. Then the total energy $U=U_{1}+U_{2}$. Since
$U_{1}=C_{V} T_{1}, U_{2}=C_{V} T_{2}, U=2 C_{V} \frac{T_{1}+T_{2}}{2}$ and $\left(T_{1}+T_{2}\right) / 2$ is the final temperature. The
entropy change is obtained by considering isochoric processes because in effect, the gas remains confined to its vessel.
$$
\Delta S=\int_{T_{1}}^{\left(T_{1}+T_{2} / 2\right.} \frac{C_{V} d T}{T}-\int_{\left(T_{1}+T_{2}\right) / 2}^{T_{2}} C_{V} \frac{d T}{T}=C_{V} \ln \frac{\left(T_{1}+T_{2}\right)^{2}}{4 T_{1} T_{2}}
$$
Since $\left(T_{1}+T_{2}\right)^{2}=\left(T_{1}-T_{2}\right)^{2}+4 T_{1} T_{2}, \Delta S>0$

Key Concepts

Ideal Gas Internal Energy

In an ideal gas, the internal energy is a function solely of temperature and does not depend on volume or pressure. This principle simplifies analyses of thermodynamic processes because any change in the internal energy of the gas results only from a change in its temperature.

Free Expansion

Free expansion is a process where a gas expands without performing work on the surroundings and without exchanging heat with the environment. In such a process, particularly for an ideal gas, the internal energy remains unchanged if the temperature stays constant, which highlights the unique characteristics of ideal gas behavior.

Isochoric Process

An isochoric process is one in which the volume remains constant. In thermodynamic calculations, particularly in the determination of entropy changes, considering a process as isochoric allows one to directly relate the change in temperature to changes in entropy using the relation dS = CV dT/T, since the volume-related work is zero.

Energy Conservation in Composite Systems

In systems composed of multiple parts or compartments, the conservation of energy principle dictates that the total internal energy is the sum of the internal energies of the individual parts. For an ideal gas, since each part’s internal energy depends only on its temperature, energy conservation can be used to find the final equilibrium temperature after the gas components mix or expand.

Entropy and the Second Law of Thermodynamics

Entropy is a state function that quantifies the degree of disorder or randomness in a system. When calculating the entropy change during a process, even if the process itself is irreversible, one can use a reversible path as a reference. The result that the change in entropy (?S) is greater than zero is consistent with the Second Law of Thermodynamics, which states that the entropy of an isolated system tends to increase in spontaneous processes.

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