# n moles of an ideal gas with constant volume heat capacity $C_{V}$undergo an isobaric expansion by certain volume. The ratio of the work donein the process, to the heat supplied is : Option 1) $\frac{nR}{C_{V}+nR}$ Option 2) $\frac{nR}{C_{V}-nR}$ Option 3) $\frac{4nR}{C_{V}-nR}$ Option 4) $\frac{4nR}{C_{V}+nR}$

First law in isobaric process -

$\Delta U= n\, C_{v}\Delta T$

$= n\frac{R}{\gamma -1}\Delta T$

- wherein

$\Delta Q= \Delta U+W$

$= n\frac{\gamma \: R}{\gamma -1}\cdot \Delta T$

$\Delta Q= nC_{p}\: \Delta T$

$C_V=n\frac{f}{2}R$

$W=nR\Delta T$

$\Delta Q=nC_P\Delta T=(\frac{f}{2}+1)nR\Delta T$

$\frac{W}{\Delta Q}=\frac{nR\Delta T}{nC_P\Delta T}=\frac{1}{\frac{f}{2}+1}=\frac{2}{2+f}$

$\frac{W}{\Delta Q}=\frac{2}{2+\frac{2C_V}{nR}}=\frac{2nR}{2(nR+C_V)}$

$\frac{W}{\Delta Q}=\frac{nR}{(nR+C_V)}$

Option 1)

$\frac{nR}{C_{V}+nR}$

Option 2)

$\frac{nR}{C_{V}-nR}$

Option 3)

$\frac{4nR}{C_{V}-nR}$

Option 4)

$\frac{4nR}{C_{V}+nR}$

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