### Ideal Gas Law Multiple Gases Mixed Together (Ex5)

You may have a mixture of gases together. It will have a total pressure, but that total pressure will be the sum of all the partial pressures (this is called Dalton's Law). Here's a practice example where multiple gases and vessels are mixed together:

A 1L volume bulb of methane at 10 kPa is connected to a 3L volume bulb of helium gas at 20 kPa. Find the total pressure afterwards, and also find the mole fractions of hydrogen and methane after the mixing. Use the ideal gas law.

When you have a "change in situation" you can equate two ideal gas laws at once.

$\frac{{{P_1}{V_1}}}{{nRT}} = \frac{{{P_2}{V_2}}}{{nRT}}$

How did I get this above? You need to consider one of the species at a time for the equation above. The volume changes because you're connecting two volumes at once, which means the pressure will also change because pressure and volume are inversely related. BUT, considering one species, the number of moles n will NOT change. R will always be constant. Temperature WON'T change here because it's just inert gases, there's no reaction no burning, etc. Cancelling out:

${P_1}{V_1} = {P_2}{V_2}$

And you use this equation twice: once for methane, and once for hydrogen. These will get you the partial pressures, which can then be summed to provide the total pressure after the mixing. So for V2, add 1L and 3L to get 4L.

For methane:

$(10kPa)(1L) = {P_2}(4L)$

${P_2} = 2.5kPa$

For hydrogen:

$(20kPa)(3L) = {P_2}(4L)$

${P_2} = 15kPa$

And sum together to get the total pressure:

${P_{total}} = 2.5kPa + 15kPa = 17.5kPa$

That's done. To get the mole fractions, you can just use the partial pressures over the total pressure:

${x_{C{H_4}}} = \frac{{2.5kPa}}{{17.5kPa}} = 0.143$

${x_{{H_2}}} = \frac{{15kPa}}{{17.5kPa}} = 0.875$

And to check your mole fractions are correct, you know that all of the mole fractions will sum up to 1.

Please see our other worked-out examples of the ideal gas law below:

Ideal Gas Law with Molecules (Ex1)

Ideal Gas Law Mole Fractions (Ex2)

Ideal Gas Law Two Gases Combined (Ex3)

Ideal Gas Law Partial Pressures and Mole Fractions (Ex4)