Question

The average molar mass of a mixture of methane and ethene present in the ratio a:b is found to be 20 $gmo{l}^{-}$ . If the ratio were reversed, what would be the molar mass of the mixture.

Answer 1

Hint: Average molecular mass can be written as ${\displaystyle \frac{n_1{m}_1+n_2{m}_2+n_3{m}_3.....}{n_1+n_2+n_3....}}$ where ‘n’ is the no of moles of compound and ‘m’ is the molecular mass of compound.

Complete step by step answer:
In the question it is given that the mixture of methane and ethane are present in the ratio a:b and the average molar mass of the mixture is given to be 20 $g/mol$ .
Molar mass of Methane $(C{H}_4)=(12\times 1)+(1\times 4)=16g/mol$
Molar mass of ethene $(C_2{H}_4)=(12\times 1)+(1\times 4)=28g/mol$
Average molar mass of the mixture = 20 $g/mol$
We know,
Number of moles = Weight (g) / Molecular mass
$\Rightarrow$ Weight (g) = Number of moles * Molecular mass
There are ‘a’ mole of methane and ‘b’ moles of ethene in the mixture.
$$\begin{array}{l}\Rightarrow 16a+28b=20a+20b\\ \Rightarrow 4a=8b\end{array}$$
So, solving the equation we get: ${\displaystyle \frac{a}{b}}={\displaystyle \frac{8}{4}}={\displaystyle \frac{2}{1}}$
So, $$a:b=2:1$$
If the ratio is reversed, i.e. Methane and ethene present in the mixture in the ratio of b:a,
Then,
$\Rightarrow$ ${\displaystyle \frac{16b+28a}{a+b}}={\displaystyle \frac{16+28{\displaystyle \frac{a}{b}}}{1+{\displaystyle \frac{a}{b}}}}$
$\Rightarrow$ ${\displaystyle \frac{16+28\left({\displaystyle \frac{2}{1}}\right)}{1+{\displaystyle \frac{2}{1}}}}$
$\Rightarrow$ ${\displaystyle \frac{(16+56)}{3}}$
$\therefore$ ${\displaystyle \frac{72}{3}}=24$

So, if the ratio is reversed, the molar mass of the mixture will be 24g/mol.


Note: Molar mass is a physical quantity defined as the mass of $6.022\times {10}^{23}$ atoms of an element. Also the higher the molar mass the higher the density of a gas and vice versa. Average molecular weight is defined for mixture of atoms or molecules. The molar ratios are very important for quantitative chemistry calculations.