$C _{ V _{\operatorname{mix}}}=\frac{ n _1 C _{ V _1}+ n _2 C _{ V _2}}{ n _1+ n _2}=\frac{13 R }{6}$

$\Longrightarrow \frac{ n _1 C _{ V _1}+2 n _1 C _{ V _2}}{ n _1+2 n _1}=\frac{13 R }{6} \quad\left[\because \frac{ n _1}{ n _2}=\frac{1}{2}\right]$
$\Rightarrow C _{ V _1}+2 C _{ V _2}=\frac{13 R }{2}$
Possible values are
$C _{ V _1}=\frac{3 R }{2}, C _{ V _2}=\frac{5 R }{2}$
$\therefore$ Gases are monatomic (like $He$ ) and diatomic (like $N _2$ )