Question

Three moles of an ideal gas being initially at a temperature $T_0=273K$ were isothermally expanded $n=5.0$ times its initial volume and then isochorically heated so that the pressure in the final state becomes equal to that in the initial state. The total amount of heat transferred to the gas during the process equals $Q=80kJ$. The ratio $\gamma =\frac{C_p}{C_v}$ for this gas is

• A. $1.3$
• B. $1.4$
• C. Data insufficient
• D. $1.5$

Answer 1

The correct option is B $1.4$

Given, number of moles of the gas, $\mu =3$

Initial temperature of the gas, $T_0=273K$

Total heat transferred during the process to the gas, $Q=80kJ$

Let $Q_1$ be the heat transferred during isothermal process and $Q_2$ be the heat transferred during isochoric process
Then total heat transferred to the gas during the process,
$Q=Q_1+Q_2$

$\therefore Q=\mu R{T}_{0}log\frac{n{V}_0}{V_0}+\mu C_v(T_C-T_B)$

As $T_c$ (If volume is made $n=5$ times)

$T_B=T_0$ (Isothermal process)

$\therefore Q=\mu R{T}_{0}logn+\frac{\mu R}{\gamma -1}(n-1)T_0$

$\therefore \gamma =[1+\frac{n-1}{\frac{q}{\mu R{T}_0}-\text{log}n}]$

By substituting the given values,

$\gamma =[1+\frac{5-1}{\frac{80\times {10}^3}{3\times 8.3\times 273}-\text{log}5}]$

We get $\therefore \gamma =1.4$

Final answer:(c)