In a cyclic thermodynamic process, an ideal gas is first compressed at constant volume while its internal energy is increased by $350 J$ . Then it expands adiabatically until it returns to its original pressure. Finally, the gas is returned isobarically to its original state when $60.0 J$ of work is performed on the gas while its internal energy decreases by $200 J$ . If the change in internal energy (in 𝑆𝐼 𝑢𝑛𝑖𝑡) during the adiabatic expansion is $X$ , then find the value of $(X + 500)$ .Assume the gas is ideal with $C_{P} = \frac{20.8 J}{m o l - K}$ and $C_{v} = \frac{12.5 J}{m o l - K}$ .

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Solution

Given, change in internal energy during isochoric compression,
$U_{1 \to 2} = 350 J$

Work done on the gas during isobaric compression,
$W_{3 \to 1} = 60 J$

And internal energy decreases during isobaric compression,
$U_{3 \to 1} = 200 J$

Let initial internal energy of the gas be $U_{1}$ .
So,

$U_{2} = U_{1} + 350 \dots (i)$

$U_{3} = U_{1} + 200 \dots (i i)$

So, change in internal energy during adiabatic expansion,
$U_{2 \to 3} = X = U_{3} - U_{2} = - 150$

So, $X + 500 = (- 150 + 500) = 350$

Final answer: $(350)$

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