Question

Two moles of helium are mixed with n moles of hydrogen. If $\dfrac{{{C_P}}}{{{C_V}}} = \dfrac{3}{2}$ for the mixture, then the value of n is
(A) 3/2
(B) 2
(C) 1
(D) 3

Answer 1

We know that the adiabatic index $\gamma = \dfrac{{{C_P}}}{{{C_V}}}$. Therefore adiabatic index for mixture ${\gamma _{mix}} = \dfrac{{{C_{Pmix}}}}{{{C_{Vmix}}}}$. Also, heat capacity of mixture is,${C_{mix}} = \dfrac{{{n_1}{C_1} + {n_2}{C_2}}}{{{n_1} + {n_2}}}$, where ${n_1}$ and ${n_2}$ are the number of moles. Using this we can find the heat capacity of the mixture at constant pressure and volume.

Complete step by step answer:
We know that the adiabatic index is given as $\gamma = \dfrac{{{C_P}}}{{{C_V}}}$.
The value of ${C_P}$ and ${C_V}$ for a monatomic gas (here helium) is $\dfrac{{5R}}{2}$ and $\dfrac{{3R}}{2}$ respectively.
The value of ${C_P}$ and ${C_V}$ for diatomic gas (here hydrogen) is $\dfrac{{7R}}{2}$ and $\dfrac{{5R}}{2}$ respectively.
Heat capacity of mixture is,${C_{mix}} = \dfrac{{{n_1}{C_1} + {n_2}{C_2}}}{{{n_1} + {n_2}}}$, where${n_1}$ and ${n_2}$ are the number of moles.
Given, the number of moles of helium is 2.
Let the number of moles of hydrogen be n.
Obtaining the heat capacity at constant pressure for mixture.
${C_P}^{mix} = \dfrac{{{n_1}{C_{{P_1}}} + {n_1}{C_{{P_2}}}}}{{{n_1} + {n_2}}}, \\\ \Rightarrow{C_P}^{mix} = \dfrac{{2\left( {5R/2} \right) + n\left( {7R/2} \right)}}{{2 + n}} = \dfrac{{R(10 + 7n)}}{{2\left( {2 + n} \right)}} \\\$
Obtaining the heat capacity at constant volume for mixture.
${C_V}^{mix} = \dfrac{{{n_1}{C_{{V_1}}} + {n_1}{C_{{V_2}}}}}{{{n_1} + {n_2}}}, \\\ \Rightarrow{C_V}^{mix} = \dfrac{{2\left( {3R/2} \right) + n\left( {5R/2} \right)}}{{2 + n}} = \dfrac{{R(6 + 5n)}}{{2\left( {2 + n} \right)}} \\\$
Calculating adiabatic index of the mixture,
${\gamma _{mix}} = \dfrac{{{C_{Pmix}}}}{{{C_{Vmix}}}}$ $= \dfrac{3}{2} = \dfrac{{\dfrac{{R(10 + 7n)}}{{2(2 + n)}}}}{{\dfrac{{R(6 + 5n)}}{{2(2 + n)}}}} = \dfrac{{10 + 7n}}{{6 + 5n}}$
$\Rightarrow 18 + 15n = 20 + 14n \\\ \therefore n = 2 \\\$
Hence,option (B) is the correct answer.

Note: We should know the value of heat capacity at constant pressure and volume for a monatomic and a diatomic gas to solve such type of questions.In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure (CP) to heat capacity at constant volume (CV). It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas.