The specific heat capacities of an ideal gas at the constant pressure and at constant volume are \(620…
The specific heat capacities of an ideal gas at the constant pressure and at constant volume are \(620 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}\) and \(420 \mathrm{Jkg}^{-1} \mathrm{~K}^{-1}\) respectively. The density of the gas at STP is approximately,
\(2.88 \mathrm{kgm}^{-3}\)
\(4.86 \mathrm{kgm}^{-3}\)
\(3.88 \mathrm{kgm}^{-3}\)
\(1.86 \mathrm{kgm}^{-3}\)
Solution
Given, specific heat capacity of a gas at constant pressure, \(C_p=620 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\) and specific heat capacity of gas at constant volume, \(C_V=420 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\)
\(\because\) Molar specific heat of gas at constant pressure and constant volume are given by
\(\therefore \quad C_p{ }^{\prime}=m C_p=m(620)\)
and \(\quad C_V{ }^{\prime}=m C_v=m(420)\)
As we know that,
\(\because \quad C_p{ }^{\prime}-C_V{ }^{\prime}=R\)
[Here, \(R\) = universal gas constant]
\(\begin{aligned}
m[620-420] & =R \\
\Rightarrow \quad m & =\frac{R}{200} \qquad \ldots (i)
\end{aligned}\)
Now, ideal gas equation,
\(p V=\mu R T\)
where, \(\mu=\) number of moles of gas
or
\(\begin{aligned}
& p V=R T \quad[\because \text { for } 1 \text { mole, } \mu=1] \\
& \therefore \quad p m=\rho R T \quad\left[\because \text { density, } \rho=\frac{m}{V}\right] \\
& 10^5 \times \frac{R}{200}=\rho R(273) \quad \text { [From Eq. (i)] } \\
& \rho=1.85 \mathrm{Kg}-\mathrm{m}^{-3}
\end{aligned}\)
Hence, the option (d) is correct.