One mole of a monoatomic ideal gas undergoes an adiabatic expansion in which its volume becomes eight times its initial value. If the initial temperature of the gas is $100$ K and the universal gas constant R = $8.0 J m o l^{- 1} K^{- 1}$ , the decrease in its internal energy, in Joule, is ________.

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Solution

$v_{i} = v$
$v_{F} = 8 v$
For an adiabatic process $γ = \frac{5}{3}$ for monoatomic process

$T_{1} V_{1}^{γ - 1} = T_{2} V_{2}^{γ - 1}$
$100 (v)^{\frac{2}{3}} = T_{2} (8 v)^{\frac{2}{3}}$
$T_{2} = 25 k$

$△ U = n c_{v} △ T = 1 [\frac{F R}{2}] [100 - 25] = 12 \times 75 = 900 J o u l e$

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