Question

if 2 moles of an ideal monoatomic gas at temperature T is mixed with 4 moles of an ideal diatomic gas at temperature 2T, then find the temperature of the mixture ​

Answer 1

first of all, find molar heat capacity at constant volume,

e.g., $C_{mix}=\frac{n_1C_1+n_2C_2}{n_1+n_2}$

here, n denotes number of mole of gases and C denotes molar heat capacity at constant volume.

$n_1=2,C_1=\frac{3R}{2}$

$n_2=4,C_2=\frac{5R}{2}$

now, $C_{mix}={2\times\frac{3R}{2}+4\times\frac{5R}{2}}{2+4}$

= $\frac{3R+10R}{6}$

= $\frac{13R}{6}$

now, $\Delta U_{mix}=\Delta U_1+\Delta U_2$

or, $(n_1+n_2)C_{mix}T_{mix}=n_1C_1T_1+n_2C_2T_2$

or, $(2+4)\frac{13R}{6}T_{mix}=2.\frac{3R}{2}T+4.\frac{5R}{2}(2T)$

or, $13RT_{mix}=3RT+20RT$

or, $T_{mix}=\frac{23T}{13}$

hence, temperature of mixture is 23T/13