Heat and Thermodynamics 5 Question 40
41. 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 $R=8.0 j mol^{-1} K^{-1}$, the decrease in its internal energy in joule, is …….
(2018 Adv.)
Show Answer
Answer:
Correct Answer: 41. -900 J
Solution:
- Given, $n=1, \gamma=\frac{5}{3}$
$T-V$ equation in adiabatic process is
$$ \begin{aligned} & T V^{\gamma-1}=\text { constant } \\ & \therefore \quad T _1 V _1^{\gamma-1}=T _2 V _2^{\gamma-1} \\ & \Rightarrow \quad T _2=T _1\left(\frac{V _1}{V _2}\right)^{\gamma-1} \\ & =100 \times\left(\frac{1}{8}\right)^{\frac{2}{3}} \\ & \Rightarrow \quad T _2=25 K \\ & C _V=\frac{3}{2} R \text { for monoatomic gas } \\ & \therefore \quad \Delta U=n C _V \Delta T=n \times\left(\frac{3 R}{2}\right)\left(T _2-T _1\right) \\ & =1 \times \frac{3}{2} \times 8 \times(25-100) \\ & =-900 J \end{aligned} $$
$\therefore$ Decrease in internal energy $=900 J$
