Question

A flask contains three times as many moles as `H_2` as it does `O_2` gas. If hydrogen and oxygen are the only present gases, what is the total pressure flask if the partial pressure due to oxygen is `P`?

Answer 1

`P_"total" = 4 xx P`

Explanation:

According to Dalton's Law of Partial Pressures, the total pressure of a gaseous mixture will be equal to the individual partial pressure of each component of the mixture.

Simply put, to get the total pressure of a gaseous mixture, you add up the partial pressure you'd get for each component of that mixture if it occupied the volume of the mixture alone.

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Mathematically, this can written as

`color(blue)(|bar(ul(color(white)(a/a)P_"total" = sum_i P_icolor(white)(a/a)|)))" "`, where

`P_"total"` - the total pressure of the gaseous mixture
`P_i` - the partial pressure of component `i` of said mixture

In your case, the mixture is said to contain hydrogen gas, `"H"_2`, and oxygen gas, `"O"_2`, in a `3:1` mole ratio.

This means that if you take `n_(O_2)` to be the number of moles of oxygen gas present in the mixture, you can say that

`n_(H_2) = 3 xx n_(O_2)" "color(purple)("(*)")`

Now, let's assume that the mixture is being held in a container that has a volume of `V` liters. If you take `P` to be the partial pressure of oxygen gas, you can use the ideal gas law equation

`color(blue)(|bar(ul(color(white)(a/a)PV = nRT color(white)(a/a)|)))`

to express `V` in terms of `P` and `n_(O_2)`. You will thus have

`P * V = n_(O_2) * RT implies V = n_(O_2)/P * RT" "color(red)("(*)")`

Now, you can write the same thing for `P^'`, the partial pressure of hydrogen gas in the same volume `V`

`P^' * V = n_(H_2) * RT implies V = n_(H_2)/P^' * RT`

Use equation `color(red)("(*)")` to write

`n_(O_2)/P * color(red)(cancel(color(black)(RT))) = n_(H_2)/P^' * color(red)(cancel(color(black)(RT)))`

Rearrange to find `P^'`, the partial pressure of hydrogen gas

`P^' = n_(H_2)/n_(O_2) * P`

Use equation `color(purple)("(*)")` to write

`P^' = (3 xx color(red)(cancel(color(black)(n_(O_2)))))/color(red)(cancel(color(black)(n_(O_2)))) * P`

`P^' = 3 xx P`

Finally, plug this into the equation that describes Dalton's Law of Partial Pressures to get the total pressure of the mixture, `P_"total"`

`P_"total" = P^' + P`

`P_"total" = 3 xx P + P = color(green)(|bar(ul(color(white)(a/a)4 xx Pcolor(white)(a/a)|)))`