Thermodynamics and Thermochemistry 1 Question 27
30. A monoatomic ideal gas undergoes a process in which the ratio of $p$ to $V$ at any instant is constant and equals to 1 . What is the molar heat capacity of the gas?
(2006, 3M)
(a) $\frac{4 R}{2}$
(b) $\frac{3 R}{2}$
(c) $\frac{5 R}{2}$
(d) 0
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Solution:
- Given, $\frac{p}{V}=1 \Rightarrow p=V$
Also from first law : $d q=C_{V} d T+p d V$
For one mole of an ideal gas : $p V=R T$
$$ \begin{array}{lrl} \Rightarrow & p d V+V d p & =R d T \ \text { From (i) } & p d V & =V d p \end{array} $$
Substituting in Eq. (ii) gives
$$ \begin{array}{rlrl} 2 p d V & =R d T \quad \Rightarrow \quad p d V=\frac{R}{2} d T \ \Rightarrow \quad & & d q & =C_{V} d T+\frac{R}{2} d T \ \Rightarrow \quad & \int \frac{d q}{d T} & =C_{V}+\frac{R}{2}=\frac{3}{2} R+\frac{R}{2}=2 R \end{array} $$
