Question

A gas mixture contains \[1g{\text{ }}{H_2}\] ​and \[1g{\text{ }}{H_e}\] temperature of the gas mixture is increased from \[0^\circ \] to \[100^\circ C\] an isobaric process. Then find the given heat of the gas mixture. $[{\gamma _{{H_2}}} = \dfrac{5}{3},{\gamma _{He}} = \dfrac{7}{5},R = 2cal/mol - K]$
\[(A)124cal\]
\[(B)\;327cal\]
\[(C)\;218cal\]
\[(D)\;475cal\]

Answer 1

Hint: By using the heat capacity ratio we will find the constant volume \[\left( {{C_V}} \right)\]. To find the value of constant pressure \[\left( {{C_P}} \right)\], we will use the mayor’s relation. Through the formula of enthalpy change, we will find the heat of the gas mixture.

Formulae used:
$\gamma = \dfrac{{{C_P}}}{{{C_V}}}$
${C_P} - {C_V} = R$ (Mayor’s relation)
$H = n{C_P}\Delta T$

Complete step by step answer:
Specific heat capacity can be given by
$\gamma = \dfrac{{{C_P}}}{{{C_V}}}$
$\gamma - 1 = \dfrac{{{C_P} - {C_V}}}{{{C_V}}} = \dfrac{R}{{{C_V}}}$
$\Rightarrow {C_V} = \dfrac{R}{{\gamma - 1}} = \dfrac{R}{{\dfrac{{67}}{{45}} - 1}}$
$\Rightarrow {C_V} = \dfrac{{45R}}{{22}}$
According to the mayor’s relation,
$\Rightarrow {C_P} = R + {C_V} = \dfrac{{67}}{{22}}R$
As per the data given in the question, the enthalpy change can be given by
$\Rightarrow \Delta T = 100K$
$\Rightarrow H = n {C_P} \Delta T = 0.75 \times \dfrac{{67}}{{22}} \times 2 \times 100 = 475cal$

Hence, the right option is in option (D).

Additional information:
Enthalpy is the sum of the internal energy and the product of pressure and volume given by the equation:
$H = E + PV$ . Where, \[E\] = internal energy
\[P\] = pressure
\[V\] = volume
\[H\]= Enthalpy
When a process occurs at constant pressure, the heat evolved (either released or absorbed) is equal to the change in enthalpy.
The heat capacity ratio, also known as the adiabatic index. It is the ratio of the heat capacity at constant pressure \[\left( {{C_P}} \right)\] to heat capacity at constant volume \[\left( {{C_V}} \right)\]. It is sometimes also known as the isentropic expansion factor and is denoted by (gamma) for an ideal gas .
$\gamma = \dfrac{{{C_P}}}{{{C_V}}}$

Note: Mayor’s formula states that the difference between the specific heat of a gas at constant pressure and its specific heat at constant volume is equal to the gas constant divided by the molecular weight of the gas.