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An ideal gas of mass $m$ and temperature $T_1$ undergoes a reversible isothermal process from an initial pressure $P_1$ to final pressure $P_2$. The hear loss during the process is $Q$. The entropy change $\Delta S$ of the gas is

  1. $mR \: \ln \bigg( \dfrac{P_2}{P_1} \bigg) \\$
  2. $mR \: \ln \bigg( \dfrac{P_1}{P_2} \bigg) \\$
  3. $mR \: \ln \bigg( \dfrac{P_2}{P_1} \bigg) - \dfrac{Q}{T_1} \\$
  4. $\text{zero}$

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