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Three moles of an ideal gas at $300 \mathrm{~K}$ are isothermally expanded to five times its volume and heated at this constant volume so that the pressure is raised to its initial value before expansion. In the whole process $83.14 \mathrm{~kJ}$ heat is required. Calculate the ratio $\mathrm{\left(C_P / C_V\right)}$ of the gas.

$\mathrm{\left[\log _{\mathrm{e}} 5=1.61\right. and \left.R=8.31 \mathrm{~J} / \mathrm{mol} \mathrm{K}^{-1}\right]}$

Option: 1
$1.2$

Option: 2
$1.3$

Option: 3
$1.4$

Option: 4
$1.5$

Answers (1)

According to first law of thermodynamics,

$\mathrm{ Q=\Delta U+W }$

For an isothermal change,

$\mathrm{ T=\text { constant, } U=\text { constant, } \Delta U=0 }$

and $\mathrm{ W=n R T \log _{\mathrm{e}}\left|\frac{V_f}{V_i}\right| }$

$\mathrm{ i.e. W=3 \times 8.31 \times 300 \times \log _e(5) }$

$\mathrm{ =12.03 \mathrm{~kJ} }$

$\mathrm{ \therefore Q_{\text {isothermal }}=0+12.03=12.03 \mathrm{~kJ} }$ .........(i)

For isochoric change as $\mathrm{ V= }$ constant,

$\mathrm{W=\int P d V=0 }$

$\mathrm{\Delta U=n C_v \Delta T=3 C_v \Delta T \quad(\because \quad n=3)}$

Applying gas equation between points $\mathrm{A \: and \: C,}$

$\mathrm{\frac{P V}{300}=\frac{P(5 \mathrm{~V})}{T_C} \quad i.e. T_C=1500 \mathrm{~K} }$

$\mathrm{so \: that \Delta T=T_C-T_B=1500-300=1200 }$

$\mathrm{\therefore \quad \Delta U=3 C_v \times 1200=3.6 C_v \mathrm{~kJ}}$

Note: $\mathrm{T_0 \: find \: T_C}$ you can apply gas equation between points $\mathrm{B\: and \: C}$ also.

and hence,
$\mathrm{ (\Delta Q)_{\text {isochoric }}=3.6 C_v+0=3.6 C_v \mathrm{~kJ} }$ .........(ii)

According to given problem,

$\mathrm{ \Delta Q_{\text {isothermal }}+\Delta Q_{\text {isochoric }}=83.14 \mathrm{~kJ} }$

Using equation (i) and (ii), we get

$\mathrm{ 12.03+3.6 C_v=83.14 }$

$\mathrm{ \text { or } C_v=(71.11 / 3.6)=19.75 \mathrm{~J} }$

$\mathrm{ \text { Thus } C_P=C_V+R=19.75+8.3=28.05 \mathrm{~J} / \mathrm{mol}-\mathrm{K} }$

$\mathrm{\therefore \gamma=\frac{C_P}{C_V}=\frac{28.05}{19.75}=1.40}$

Posted by

Kshitij

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