Question

Equal weight of ${H_2}$ & $C{H_4}$ are mixed in a container at ${25^ \circ }C$. Calculate fraction of total pressure exerted by ${H_2}$.

Answer 1

Hint: Dalton's law of partial pressure of the gas will help to find the answer. The equation of this law is
\[{p_{{H_2}}} = {p_{total}} \cdot {x_{{H_2}}}{\text{ }}\]

Complete step by step solution:
We will use Dalton's law of partial pressure to measure the ratio of pressure of hydrogen gas to the total pressure.
Molecular weight of ${H_2}$ = 2( Atomic weight of H) = 2(1) = 2$gmmo{l^{ - 1}}$
Molecular weight of $C{H_4}$ = Atomic weight of C + 4(Atomic weight of H) = 12+4(1) = 16$gmmo{l^{ - 1}}$
Now, for the ease of calculation, we assume that the mixture of gas contains 16 grams of methane and hydrogen gas each.
So, we know that
\[{\text{Number of moles = }}\dfrac{{{\text{Weight}}}}{{{\text{Molecular weight}}}}\]
For ${H_2}$ ,
\[{\text{Number of moles = }}\dfrac{{16}}{2} = 8\]
For $C{H_4}$,
\[{\text{Number of moles = }}\dfrac{{16}}{{16}} = 1\]
Now, we have obtained that number of moles of both hydrogen and methane in the mixture. We will calculate the mole fraction of hydrogen gas in the mixture according to the below given formula.
\[{\text{Mole fraction = }}\dfrac{{{\text{Moles of gas}}}}{{{\text{Total number of moles in mixture}}}}\]
For Hydrogen gas, we can write the above equation as
\[{\text{Mole fraction = }}\dfrac{8}{{8 + 1}} = \dfrac{8}{9}\]
Now, we will use Dalton's law of partial pressure to find the given ratio.
The law states that the partial pressure of a gas is the product of the total pressure of the mixture with the mole fraction of the gas in the mixture. So, the equation can be written as
\[{p_{{H_2}}} = {p_{total}} \cdot {x_{{H_2}}}{\text{ }}.....{\text{(1)}}\]
Here, ${p_{{H_2}}}$ = partial pressure of hydrogen gas
${p_{total}}$ is the total pressure of the mixture and ${x_{{H_2}}}$ is the mole fraction of hydrogen gas in the mixture.
We need to find the ratio of pressure exerted by hydrogen gas to the total pressure. So, this ratio can be expressed as $\dfrac{{{p_{{H_2}}}}}{{{p_{total}}}}$ .
We can rewrite the equation (1) as
\[\dfrac{{{p_{{H_2}}}}}{{{p_{total}}}} = {x_{{H_2}}}\]
We already found that the mole fraction of hydrogen gas ${x_{{H_2}}}$ = $\dfrac{8}{9}$ .

So, \[\dfrac{{{p_{{H_2}}}}}{{{p_{total}}}} = \dfrac{8}{9}\].

Note: Remember that we need to write the total number of moles in the formula to find the mole fraction of a gas. So, we need to add the moles of all the components present in the mixture in the denominator of the formula.