A gaseous mixture containing equal masses of He and CH4 is at a total pressure of 2 atm. Calculate the partial pressure of He.

Steps to Solve

1. Identify the masses of the gases

Assume we have equal masses of helium and methane. Let the mass of each be $m$ grams. So, mass of helium, $m_{He} = m$ and mass of methane, $m_{CH_4} = m$.

2. Calculate the number of moles of each gas

Use the molar masses to convert mass to moles. The molar mass of helium is $4 , \text{g/mol}$ and for methane, it is $16 , \text{g/mol}$.

Number of moles of helium: $$ n_{He} = \frac{m}{4} $$

Number of moles of methane: $$ n_{CH_4} = \frac{m}{16} $$

3. Calculate the total number of moles in the mixture

Total moles can be calculated by adding the moles of helium and methane: $$ n_{total} = n_{He} + n_{CH_4} $$

Substituting the values: $$ n_{total} = \frac{m}{4} + \frac{m}{16} $$

4. Simplify the total number of moles

To simplify, find a common denominator (which is 16): $$ n_{total} = \frac{4m}{16} + \frac{m}{16} = \frac{5m}{16} $$

5. Calculate the mole fraction of helium

The mole fraction of helium ($X_{He}$) is given by: $$ X_{He} = \frac{n_{He}}{n_{total}} $$

Substituting in the values: $$ X_{He} = \frac{\frac{m}{4}}{\frac{5m}{16}} = \frac{16}{20} = \frac{4}{5} $$

6. Use Dalton's Law to find partial pressure

The partial pressure of helium ($P_{He}$) can be obtained using Dalton's Law: $$ P_{He} = X_{He} \times P_{total} $$

Substituting the values: $$ P_{He} = \frac{4}{5} \times 2 , \text{atm} $$

7. Calculate the partial pressure

Calculating gives: $$ P_{He} = \frac{8}{5} , \text{atm} = 1.6 , \text{atm} $$