**Gaseous
Mixture**

Gaseous Mixture is a mixture composed of gases. Mixture of gases are common in many applications. If a substance is in gaseous state and there are at least two different types of molecules in it, it’s a mixture. The most common example of a gas mixture is air.

If two non-reactive gases are
enclosed in a vessel of volume (V). In the mixture n₁ moles of one gas are
mixed with n₂ moles of another gas. If N_{A} is Avogadro’s number then,

Number of molecules of first gas (N₁)
= n₁N_{A}

And number of molecules of second gas
(N₂) = n₂N_{A}

Total mole fraction (n) = (n₁ + n₂)

If M₁ is the molecular weight of first gas and M₂ that of second gas.

Then molecular weight of mixture \(\left( M \right)=\frac{n{{M}_{1}}+n{{M}_{2}}}{{{n}_{1}}+{{n}_{2}}}\).

As there is no loss of internal
energy, the internal energy should be same before mixing and after mixing. Let
specific heat of the mixture at constant volume be (C_{V})_{max}
and adiabatic constant of mixture be γ.

\(({{n}_{1}}+{{n}_{2}}){{\left( {{C}_{v}} \right)}_{\max }}T={{n}_{1}}{{\left( {{C}_{v}} \right)}_{1}}T+{{n}_{2}}{{\left( {{C}_{v}} \right)}_{2}}T\) … (1)

\({{\left( {{C}_{v}} \right)}_{\max }}=\frac{{{n}_{1}}{{\left( {{C}_{v}} \right)}_{1}}+{{n}_{2}}{{\left( {{C}_{v}} \right)}_{2}}}{{{n}_{1}}+{{n}_{2}}}\) … (2)

\({{\left( {{C}_{v}} \right)}_{\max }}=\frac{R}{\left( \gamma -1 \right)}\), \({{\left( {{C}_{v}} \right)}_{1}}=\frac{R}{\left( {{\gamma }_{1}}-1 \right)}\) and \({{\left( {{C}_{v}} \right)}_{2}}=\frac{R}{\left( {{\gamma }_{2}}-1 \right)}\).

Now, substitute (C_{V})_{max},
(C_{V})₁ and (C_{V})₂ in equation (1), we get:

\(({{n}_{1}}+{{n}_{2}})\left( \frac{R}{\left( \gamma -1 \right)} \right)T={{n}_{1}}\left( \frac{R}{\left( {{\gamma }_{1}}-1 \right)} \right)T+{{n}_{2}}\left( \frac{R}{\left( {{\gamma }_{2}}-1 \right)} \right)T\).

\(\frac{({{n}_{1}}+{{n}_{2}})}{\left( \gamma -1 \right)}=\left( \frac{{{n}_{1}}}{{{\gamma }_{1}}-1} \right)+\left( \frac{{{n}_{2}}}{{{\gamma }_{2}}-1} \right)\) … (3)

Now, from equation (3), we can calculate γ of mixture at constant pressure.

\({{\left( {{C}_{P}} \right)}_{\max }}=\frac{{{n}_{1}}{{\left( {{C}_{P}} \right)}_{1}}+{{n}_{2}}{{\left( {{C}_{P}} \right)}_{2}}}{{{n}_{1}}+{{n}_{2}}}\).