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| In the general case, the change in entropy of an ideal gas when transitioning from state $1$ to state $2$ is described by the formula: | | In the general case, the change in entropy of an ideal gas when transitioning from state $1$ to state $2$ is described by the formula: |
| $$\Delta S = \frac{m}{M} C_V \ln\left(\frac{T_2}{T_1}\right) + \frac{m}{M} R \ln\left(\frac{V_2}{V_1}\right)$$ | | $$\Delta S = \frac{m}{M} C_V \ln\left(\frac{T_2}{T_1}\right) + \frac{m}{M} R \ln\left(\frac{V_2}{V_1}\right)$$ |
| where $\frac{m}{M}$ is the amount of substance (moles), $C_V$ is the molar heat capacity at constant volume, and $R$ is the universal gas constant. | | where $\frac{m}{M}$ is the amount of substance (moles), $C_V$ is the molar heat capacity at constant volume, and $R$ is the universal gas constant. |
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| Let us consider both cases: | | Let us consider both cases: |
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| <b>b) In an isothermal process</b> | | <b>b) In an isothermal process</b> |
| According to the given conditions, the process is isothermal, hence the temperature is constant ($T_1 = T_2$). The first term in the entropy equation (which depends on temperature) becomes zero. | | According to the given conditions, the process is isothermal, hence the temperature is constant ($T_1 = T_2$). The first term in the entropy equation (which depends on temperature) becomes zero. |
| The change in entropy is determined solely by the change in volume from $V_1 = V$ to $V_2 = 2V$: | | The change in entropy is determined solely by the change in volume from $V_1 = V$ to $V_2 = 2V$: |
| $$\Delta S_b = \frac{m}{M} R \ln\left(\frac{2V}{V}\right) = \frac{m}{M} R \ln 2$$ | | $$\Delta S_b = \frac{m}{M} R \ln\left(\frac{2V}{V}\right) = \frac{m}{M} R \ln 2$$ |
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| <b>a) Upon expansion in a vacuum</b> | | <b>a) Upon expansion in a vacuum</b> |
| The expansion of a gas into a vacuum is an adiabatic process with no work done ($Q = 0$, $A = 0$). According to the first law of thermodynamics, the change in internal energy is $\Delta U = Q - A = 0$. | | The expansion of a gas into a vacuum is an adiabatic process with no work done ($Q = 0$, $A = 0$). According to the first law of thermodynamics, the change in internal energy is $\Delta U = Q - A = 0$. |
| Since the internal energy of an ideal gas depends only on its temperature, a constant internal energy implies that the gas temperature remains unchanged ($T_1 = T_2$). | | Since the internal energy of an ideal gas depends only on its temperature, a constant internal energy implies that the gas temperature remains unchanged ($T_1 = T_2$). |
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| Thus, the initial and final states of the gas upon expansion in a vacuum are exactly the same as in the isothermal expansion. Since entropy is a <b>state function</b>, its change does not depend on the path the system took to get from the initial state to the final one. | | Thus, the initial and final states of the gas upon expansion in a vacuum are exactly the same as in the isothermal expansion. Since entropy is a <b>state function</b>, its change does not depend on the path the system took to get from the initial state to the final one. |
| Therefore, the entropy increment in a vacuum will be exactly the same: | | Therefore, the entropy increment in a vacuum will be exactly the same: |
| $$\Delta S_a = \Delta S_b = \frac{m}{M} R \ln 2$$ | | $$\Delta S_a = \Delta S_b = \frac{m}{M} R \ln 2$$ |
