5 moles of monatomic gas, 4 moles of linear triatomic gas without vibrational modes and 3 moles of nonlinear polyatomic gas with 2 modes of vibration excited are mixed together in a vessel. The heat required to raise the temperature by 2 kelvin is  $xR$ joules where R is universal gas constant. Find the value of  $\frac{x}{13}$.

As the gases are in a vessel the heat required will be the change in internal energy of the mixture of gases.

for monatomic gas the degrees of freedom are 3 

for linear triatomic gas without vibrational modes the degrees of freedom are 5

for nonlinear polyatomic gas with two vibration modes the degrees of freedom are 10 ( 2 degrees of freedom for each vibration mode)

$$\begin{gathered} \Delta Q=[\left(5\right)\left(\frac{3}{2}R\right)+\left(4\right)\left(\frac{5}{2}R\right)+3\left(\frac{10}{2}R\right)]2 \\ =65R \\ \therefore x=65 \end{gathered}$$