Two ideal gases A and B are mixed together at temperature T and pressure P. show that `d=(X_(A)M_(A)+X_(B)M_(B))(P)/(RT),` [d=density of the mixture, `X_(A)`=mole fraction of A, `X_(B)`=mole fraction of B, `M_(A)=`Molar mass of A, `M_(B)`=Molar mass of B].

Explanation

Suppose, the total volume of the gas-mixture is V, and `n_(A) and n_(B)` are the respective number of moles of A and B in the mixture. If `W_(A) and W_(B)` be the masses of A and B respectively, in the mixture, then
`n_(A)=(W_(A))/(M_(A)) and n_(B)=(W_(B))/(M_(B))`
Total mass of A & B in mixture`=W_(A)+W_(B)=n_(A)M_(A)=n_(B)M_(B)`
`therefore`Density of the mixture`=(W_(A)+W_(B))/(V)=(n_(A)M_(A)+n_(B)M_(B))/(V)`
For the gas-mixture,
`PV=(n_(A)+n_(B))RT," "therefore V=(n_(A)+n_B)(RT)/(P)`
So, `d=(n_(A)M_(A)+n_(B)M_(B))/(n_(A)+n_(B))xx(P)/(RT)=((n_(A)M_(A))/(n_(A)+n_(B))+(n_(B)M_(B))/(n_(A)+n_(B)))(P)/(RT)`
`therefore d=(X_(A)M_(A)+X_(B)M_(B))(P)/(RT)[X_(A)=(n_(A))/(n_(A)+n_(B)),X_(B)=(n_(B))/(n_(A)+n_(B))]`.