n moles of an ideal gas with constant volume heat capacity

Question:

$\mathrm{n}$ moles of an ideal gas with constant volume heat capacity $\mathrm{C}_{\mathrm{V}}$ undergo an isobaric expansion by certain volume. The ratio of the work done in the process, to the heat supplied is:

  1. (1) $\frac{\mathrm{nR}}{\mathrm{C}_{\mathrm{V}}+\mathrm{nR}}$

  2. (2) $\frac{\mathrm{nR}}{\mathrm{C}_{\mathrm{V}}-\mathrm{n} \mathrm{R}}$

  3. (3) $\frac{4 n R}{C_{V}-n R}$

  4. (4) $\frac{4 n R}{C_{\mathrm{V}}+\mathrm{n} \mathrm{R}}$

Correct Option: 1

Solution:

(1) At constant volume

Work done $(\mathrm{W})=\mathrm{n} \mathrm{R} \Delta \mathrm{T}$

and Heat given $Q=C_{v} \Delta T+n R \Delta T$

So, $\therefore \frac{\mathrm{W}}{\mathrm{Q}}=\frac{\mathrm{nR} \Delta \mathrm{T}}{\mathrm{C}_{\mathrm{v}} \Delta \mathrm{T}+\mathrm{nR} \Delta \mathrm{T}}=\frac{\mathrm{nR}}{\mathrm{C}_{\mathrm{V}}+\mathrm{nR}}$

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