Question

One mole of ideal monoatomic gas ($\gamma $  = 5/3) is mixed with one mole of diatomic gas ($\gamma $  = 7/5). What is $\gamma $ for the mixture? ( $\gamma $ denotes the ratio of specific heat at constant pressure, to that at constant volume.)
(a) 3/2 
(b) 23/15 
(c) 35/23 
(d) 4/3

Answer 1

Hint: For the given mixture, the specific heat for a mixture of two gases is given by\[{{\gamma }_{mix}}=~\dfrac{\left[ {{\mathbf{n}}_{\mathbf{1}}}\text{ }{{\gamma }_{\mathbf{1}}}\text{ }+\text{ }{{\mathbf{n}}_{2}}\text{ }{{\gamma }_{2}} \right]}{\left[ \mathbf{n1}\text{ }+\text{ }\mathbf{n2} \right]}\text{, where n refers to the number of moles of each gas }\]
Now, apply this formula for the given mixture.

Step-by-Step Solution:
Let us first understand the concept of the ratio of specific heats before moving on to the particulars of this question.
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 (C_P) to heat capacity at constant volume (C_V). It is sometimes also known as the isentropic expansion factor and is denoted by $\gamma $ for an ideal gas or κ, the isentropic exponent for a real gas. The symbol gamma is used by aerospace and chemical engineers.
\[\gamma \text{ }=\text{ }\dfrac{{{C}_{P}}}{{{C}_{V}}}=\dfrac{{{c}_{P}}}{{{c}_{V}}}\]
where C is the heat capacity and c the specific heat capacity (heat capacity per unit mass) of a gas. The suffixes P and V refer to constant pressure and constant volume conditions respectively.
The change in internal energy and enthalpy of mixing ideal gases is zero. According to Gibbs' Theorem, the individual contribution of each species in an ideal gas mixture to the extensive thermodynamic properties of the mixture is the same as that of the pure species at the same temperature and at the partial pressure of the species in the mixture.
Thus, the resulting formula the specific heat for a mixture of two gases is given by\[{{\gamma }_{mix}}=~\dfrac{\left[ {{\mathbf{n}}_{\mathbf{1}}}\text{ }{{\gamma }_{\mathbf{1}}}\text{ }+\text{ }{{\mathbf{n}}_{2}}\text{ }{{\gamma }_{2}} \right]}{\left[ \mathbf{n1}\text{ }+\text{ }\mathbf{n2} \right]}\text{, where n refers to the number of moles of each gas }\]
Now, applying this formula for the given gases
\[\begin{align}
  & {{\gamma }_{mix}}=~\dfrac{\left[ \text{1}\text{. 5/3 }+\text{ 1}\text{.7/5} \right]}{1+1}\text{, where n refers to the number of moles of each gas } \\
 & {{\gamma }_{mix}}\approx 1.5 \\
\end{align}\]
Therefore, the required answer is a)

Note: Remember that the Ratio of Specific Heat is dimensionless and the value is the same in the SI and the Imperial system of units. Another way of obtaining the specific heats of a gaseous mixture would be calculating the sum of the product of mole fraction times the specific heat of that gas component.