Moles of \(CO _{2}=\frac{11}{44}\)

\(=0.25\)

Moles of \(N _{2}\) are,

Moles of \(N _{2}=\frac{14}{28}\)

\(=0.50\)

The value of \(\gamma\) for the mixture of gases is given by the relation,

\(\frac{n_{1}+n_{2}}{\gamma_{\operatorname{mix}}-1}=\frac{n_{1}}{\gamma_{1}-1}+\frac{n_{2}}{\gamma_{2}-1}\)

Where, \(n_{1}\) and \(n_{2}\) are the number of moles of the two gases and \(\gamma_{\text {mix }}\) is the specific heat ratio for mixture. Therefore,

\(\frac{0.25+0.50}{\gamma_{\text {mix }}-1}=\frac{0.25}{1.29-1}+\frac{0.50}{1.4-1}\)

\(\frac{0.75}{\gamma_{\operatorname{mix}}-1}=0.862+1.25\)

\(2.112 \gamma_{\operatorname{mix}}=0.75+2.112\)

\(\gamma_{\operatorname{mix}} \approx 1.4\)

\(=\frac{7}{5}\)