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 NA is Avogadro’s number then,
Number of molecules of first gas (N₁) = n₁NA
And number of molecules of second gas (N₂) = n₂NA
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 (CV)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 (CV)max, (CV)₁ and (CV)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.
\({{\left( {{C}_{P}} \right)}_{\max }}=\frac{{{n}_{1}}{{\left( {{C}_{P}} \right)}_{1}}+{{n}_{2}}{{\left( {{C}_{P}} \right)}_{2}}}{{{n}_{1}}+{{n}_{2}}}\).