How to derive partial gas equation?

Question

I was given a question as follows:

If the following gasses are to mix, what is the partial pressure of the Ne gas?

The correct answer is 0.1. I cannot figure out how my professor derived this equation. From the ideal gas law, all I can arrive to is this:

$P_x = \frac{P_tV_t}{V_x}$

This gives me an answer of 4.4, which is obviously incorrect.

I've determined the equation he used is as follows:

$P_x = \frac{P_xV_x}{V_t}$

which gives the correct answer of 0.1. How did he arrive to this answer?

Answer 1

Prepare for an amazing feat of algebra! I sincerely hope there is a shorter way to the endpoint. EDIT - there is! See the end of my answer.

To calculate partial pressure, $P_{\text{He}}$, we need the total pressure $P_t$ and the fraction of the gas that is He $X_{\text{He}}$: $$P_{\text{He}}=X_{\text{He}} P_t$$

You have $PV=nRT$. $T$ is not given, so assume it is constant. $n$ is unknown, but the system is closed, so $n_x$ for each gas is constant. Thus we have:

$$PV=\text{ constant}$$

Total Pressure

It is tempting to write variations of $$P_1 V_1 = P_2 V_2$$

However, for each gas, both $P$ and $V$ are changing, so we need to consider their products: $$(PV)_1 = (PV)_2$$

Thus, as trb456 suggests, $$P_t V_t = (PV)_t \implies P_t =\frac{\sum{(PV)_i}}{\sum{V_i}}$$

Fraction of He

The fraction of the mixture that is helium is determined by the ratio $\dfrac{n_{\text{He}} }{n_t}$. Since $n=PV/RT$, and $R$ and $T$ are constant, we can write: $$X_{\text{He}}=\frac{n_{\text{He}}}{\sum{n_i}}=\frac{(PV)_{\text{He}}}{\sum{(PV)_i}}$$

At last!

$$P_{\text{He}}=X_{\text{He}} P_t = \left( \frac{(PV)_{\text{He}}}{\sum{(PV)_i}}\right) \left( \frac{\sum{(PV)_i}}{\sum{V_i}}\right)$$ $$P_{\text{He}_2}=\frac{(PV)_{\text{He}_1}}{V_t}$$

Or, as it struck me as I finished:

Helium expands to fill the total volume. Now we can use $P_1 V_1 = P_2 V_2$ and completely ignore the other gasses (there is a lot empty space for the He atoms to fit in).

$$P_{\text{He}_2}=\frac{P_{\text{He}_1}V_{\text{He}_1}}{V_t}$$

This equation is very similar to your second equation, except the subscripts are added to denote initial and final pressure of He.

Answer 2

The total pressure of the system is:

$\frac{1.0\times0.5+1.0\times0.4+2.0\times0.2}{1.0+1.0+2.0}=1.3/4=0.325$

i.e. just the sum of the $PV$s divided by the sum of the $V$s. Then $0.325$ is the sum of the partial pressures by the Law of Partial Pressures. So this at least suggest that the individual partial pressures are near $0.1$.

Can you figure out how to parcel out the individual pressures from here?