Question

Four mole of hydrogen, two mole of helium and one mole of water vapor form an ideal gas mixture. What is the molar specific heat at constant pressure of mixtures?
A.\[\dfrac{{16}}{7}R\]
B.\[\dfrac{7}{{16}}R\]
C.\[R\]
D.\[\dfrac{{23}}{7}R\]

Answer 1

Hint: The molar specific heat at constant volume and the molar specific heat at constant pressure were differed by ideal gas constant. The molar specific heat at constant volume can be calculated from the average degrees of freedom. When average degrees of freedom were divided by two gives molar specific heat at constant volume.

Complete answer:
Given that four mole of hydrogen, two mole of helium and one mole of water vapor form an ideal gas mixture. Hydrogen is a diatomic molecule; helium is a monatomic molecule and water vapor is a triatomic molecule.
The degrees of freedom for a monatomic gas are \[3\] and a diatomic gas has \[5\]degrees of freedom, and a triatomic molecule has \[6\] degrees of freedom.
The average degrees of freedom will be obtained by dividing the degrees of freedom of each molecule in a mixture with the total number of moles.
Thus, the average degrees of freedom is \[\dfrac{{3\left( 2 \right) + 5\left( 4 \right) + 6\left( 1 \right)}}{{2 + 4 + 1}} = \dfrac{{32}}{7}\]
The molar specific heat at constant volume is \[{C_v} = \dfrac{{\dfrac{{32}}{7}}}{2}R = \dfrac{{16}}{7}R\]
The molar specific heat at constant pressure will be \[{C_p} = {C_v} + R = \dfrac{{16}}{7}R + R = \dfrac{{23}}{7}R\]
The molar specific heat at constant pressure of mixtures is \[\dfrac{{23}}{7}R\]

Option D is the correct one.

Note:
Hydrogen exists as a diatomic molecule and thus it has \[3\]degrees of freedom, helium is a diatomic gas and has \[5\]degrees of freedom. Helium has a completely filled configuration and does not form molecules easily. and water is a triatomic as three atoms are present in it and has \[6\] degree of freedom.