The average molar mass of a mixture of methane (CH_4) and ethane …

Question

The average molar mass of a mixture of methane ($CH_4$) and ethane ($C_2H_4$) present in the ratio of a : b is found to be 20.0g $mol^{-1}$. If the ratio were reversed, what would be the molar mass of the mixture?

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Answer

Molar mass of $CH_4 = 16g mol^{-1}$
Molar mass of $C_2H_4 = 28g mol^{-1}$
When they are present in the a : b, their average molar mass
$=\frac{\text{a}\times16+\text{b}\times28}{\text{a}+\text{b}}=20\text{g mol}^{-1}$ (Given)
i.e. 16a + 286 = 20 (a + b) or 4a + 7b = 5 (a + b)
$\text{or }\text{a}=2\text{b}\text{ or }\frac{\text{a}}{\text{b}}=\frac{2}{1}=2:1$
If the ratio is reversed, now the ratio a : b = 1 : 2
$\therefore$ Average molar mass $=\frac{1\times16+2\times28}{1+2}=24\text{g mol}^{-1}$

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