Question

If four moles of hydrogen and two moles of helium form a gaseous mixture, then the molar specific heat capacity of the mixture at constant pressure is

• A. $\frac{16R}{7}$
• B. $\frac{7R}{16}$
• C. $R$
• D. $\frac{19R}{6}$

Hint:

For a mixture of ideal gases, the average molar specific heat capacity at constant volume ($C_{V,mix}$) or constant pressure ($C_{P,mix}$) can be calculated as a weighted average of the individual molar specific heats, weighted by their respective number of moles: $C_{V,mix} = \frac{\sum n_i C_{V,i}}{\sum n_i}$ and $C_{P,mix} = \frac{\sum n_i C_{P,i}}{\sum n_i}$. Remember the degrees of freedom ($f$) for different types of ideal gases: \begin{itemize} \item Monoatomic gases (He, Ne, Ar): $f=3$ (translational only) \item Diatomic gases (H$_2$, O$_2$, N$_2$) at moderate temperatures: $f=5$ (3 translational + 2 rotational) \item Non-linear polyatomic gases (NH$_3$, CH$_4$): $f=6$ (3 translational + 3 rotational) \end{itemize} The molar specific heats are related to degrees of freedom by $C_V = \frac{f}{2}R$ and $C_P = \left(\frac{f}{2} + 1\right)R$.

Answer 1

The Correct Option is D

Step 1: Identify the given information and the goal.
Given: \begin{itemize} \item Number of moles of Hydrogen ($n_{H_2}$) = 4 moles \item Number of moles of Helium ($n_{He}$) = 2 moles \end{itemize} We need to find the molar specific heat capacity of the mixture at constant pressure ($C_{P,mix}$). Step 2: Determine the molar specific heat capacities for individual gases.
The molar specific heat capacity at constant pressure ($C_P$) for an ideal gas is related to its degrees of freedom ($f$) by the formula $C_P = \left(\frac{f}{2} + 1\right)R$, where $R$ is the universal gas constant. \begin{itemize} \item For Hydrogen ($H_2$): Hydrogen is a diatomic gas. At normal temperatures, it has 5 degrees of freedom (3 translational and 2 rotational). So, $f_{H_2} = 5$. $C_{P,H_2} = \left(\frac{5}{2} + 1\right)R = \left(\frac{5+2}{2}\right)R = \frac{7}{2}R$ \item For Helium ($He$): Helium is a monoatomic gas. It has 3 degrees of freedom (all translational). So, $f_{He} = 3$. $C_{P,He} = \left(\frac{3}{2} + 1\right)R = \left(\frac{3+2}{2}\right)R = \frac{5}{2}R$ \end{itemize} Step 3: Calculate the molar specific heat capacity of the mixture at constant pressure.
For a gaseous mixture, the molar specific heat capacity at constant pressure ($C_{P,mix}$) is given by the formula: $C_{P,mix} = \frac{n_{H_2} C_{P,H_2} + n_{He} C_{P,He}}{n_{H_2} + n_{He}}$ Substitute the values: $C_{P,mix} = \frac{(4 \text{ moles}) \times \left(\frac{7}{2}R\right) + (2 \text{ moles}) \times \left(\frac{5}{2}R\right)}{4 \text{ moles} + 2 \text{ moles}}$ $C_{P,mix} = \frac{\frac{28}{2}R + \frac{10}{2}R}{6}$ $C_{P,mix} = \frac{14R + 5R}{6}$ $C_{P,mix} = \frac{19R}{6}$ Step 4: Compare the calculated value with the given options.
The calculated molar specific heat capacity of the mixture at constant pressure is $\frac{19R}{6}$, which matches option (4). The final answer is $\boxed{\frac{19R}{6}}$.