Question

An ideal gas whose adiabatic exponent equals to is expanded according to the law $P=2V.$ The initial volume of the gas is equal to $V_0=1$ unit. As a result of expansion the volume increases 4 times. (Take $R=$ units)

Column I		Column II
i.	Work done by the gas	p.	25 units
ii.	Increment in internal energy of the gas	q.	45 units
iii.	Heat supplied to the gas	r.	75 units
iv.	Molar heat capacity of the gas in the process	s.	15 units
		t.	55 units

Now, match the given columns and select the correct option from the codes given below.
Codes

• A. i - (t) , ii - (s) , iii - (q) , iv - (r)
• B. i - (s) , ii - (r) , iii - (q) , iv - (p)
• C. i - (q) , ii - (r) , iii - (s) , iv - (t)
• D. i - (p) , ii - (q) , iii - (r) , iv - (s)

Answer 1

Answer: (B)

$W=\int PdV=\underset{V_0}{\overset{4V_0}{\int }}2VdV={\left(V^2\right)}_{V_0}^{4V_0}=15V_0^2=15$ units
From $PV=nRT,2V^2=nRT$
$\Rightarrow 2\left(V_2^2-V_1^2\right)=nR(\Delta T)nR\Delta T=30V_0^2$
$\Delta U=n{C}_V\Delta T=\frac{nR}{\gamma -1}\Delta T=\frac{30V_0^2}{\gamma -1}=\frac{30(1{)}^2}{\frac{7}{5}-1}=\frac{30}{2}(5)$
$=75$ units
$Q=W+\Delta U=15+30=45\text{ units }$
Molar heat capacity:
$C=C_V+\frac{R}{1-x}=\frac{5}{2}R+\frac{R}{1-(-1)}=\frac{5}{2}R+\frac{R}{2}=3R$
$=3\times \frac{25}{3}=25$ units