$Q$ amount of heat is given to $0.5\ mole$ of an ide al mono-atomic gas by a process $TV^n$ constant. Following graph shows variation of temperature with $Q$ . Find value of $n$.
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$\Delta \mathrm{Q}=\Delta \mathrm{U}+\Delta \mathrm{W}$
$\Delta \mathrm{Q}=\frac{\mathrm{f}}{2} \mathrm{n} \mathrm{R} \Delta \mathrm{T}+\frac{\mathrm{n} \mathrm{R} \Delta \mathrm{T}}{1-\mathrm{n}_{0}}$
$15 \times 10^{3}=\frac{3}{2} \times \frac{1}{2} \times \frac{25}{3} \times 200+\frac{0.5 \times 25 / 3 \times 200}{1-\mathrm{n}_{0}}$
$150=\frac{25}{2}+\frac{25}{3\left(1-n_{0}\right)}$
$6=\frac{1}{2}+\frac{1}{3\left(1-n_{0}\right)}$
$5.5=\frac{1}{3\left(1-n_{0}\right)}$
$16.5-16.5 \mathrm{n}_{0}=1$
$-16.5 \mathrm{n}_{0}=-15.5$
$\mathrm{T} \mathrm{V}_{0}^{\mathrm{n}-1}=\mathrm{constant}$
$\mathrm{n}_{0}-1=\mathrm{n}$
$\frac{{155}}{{165}} - 1 = n$
$\frac{-10}{165}=n$
$n=\left(\frac{-2}{33}\right)$
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