One mole of an ideal gas $(\gamma = 1.4)$ is adiabatically compressed so that its temperature rises from $27\,^oC$ to $35\,^oC$ . The change in the internal energy of the gas is .... $J$. (given $R = 8.3\,J/mole-K$ )
Diffcult
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By definition.
$\Delta \mathrm{U}=\mathrm{C}_{\mathrm{v}} \Delta \mathrm{T}$
(for $1$ mole of ideal gas)
and from Mayer's formula,
$\mathrm{C}_{\mathrm{p}}-\mathrm{C}_{\mathrm{v}}=\mathrm{R}$
$\frac{C_{p}}{C_{v}}-1=\frac{R}{C_{v}} \Rightarrow \gamma-1=\frac{R}{C_{v}}$ or $C_{v}=\frac{R}{\gamma-1}$
$\therefore \Delta \mathrm{U}=\frac{\mathrm{R} . \Delta \mathrm{T}}{\gamma-1}=\frac{8.3 \times(35-27)}{1.4-1}$
$=\frac{8.3 \times 8}{0.4}=166 \mathrm{J}$
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