Ideal gas law? implied volume?

Question

Working on a practice exam. The question is

A volume of gas contains
0.25 moles of $\ce{CO2}$
0.30 moles of $\ce{N2}$
0.15 moles of $\ce{O2}$

What is the partial pressure of $\ce{O2}$ if the total pressure was $690\ \mathrm{mmHg}$ and the temperature was $25.0\ \mathrm{^\circ C}$

I know the formula is $p_a=\dfrac{n_aRT}{V}$

So far I know that

$n_a = 0.25\ \mathrm{mol}$

$R = 0.0821\ \mathrm{l\ atm\ K^{−1}\ mol^{−1}}$

$T = 298\ \mathrm K$

But I can't seem to figure out volume. I was reading somewhere that volume is directly proportional to The amount of moles, but maybe it has something to do with the total pressure. I'm a little lost.

The only thing I can think of would be getting volume via the Ideal Gas Law as well. Would that work?

Answer 1

You don't need the Ideal Gas Law. You can use it (and I'll show you how). The question asks for partial pressure. You want Dalton's Law of Partial Pressures.

Dalton's Law

Dalton's Law states that the total pressure in a system is equal to the sum of the partial pressures of the components:

$$P_T=P_A + P_B + P_C + ...$$

At constant volume and temperature (which you have, even if you don't know the volume), the ideal gas law can be rearranged so that:

$$\frac{P}{n}=\frac{RT}{V}$$ $$\frac{P_T}{n_T}=\frac{P_A}{n_A}$$ $$P_A=\frac{n_A}{n_T}P_T$$

Ideal Gas Law

You can find the volume of the system because you have the total moles $n_T$ and the total pressure $P_T$:

$$V=\frac{n_T RT}{P_T}$$

You can then replace the number of moles with $n_A$ and solve for $P_A$;

$$P_A=\frac{n_A RT}{V}$$

This approach is basically equivalent to the previous except:

1. Using the ideal gas law you have to do two calculation (volume and partial pressure), while you only need to do one with Dalton's Law
2. Fewer math steps means less rounding, which means less error and higher precision.
3. If you use Dalton's Law, you don't have to worry about the units of temperature or pressure. Temperature is not in the equation, and the pressures are a ratio. You can use whatever pressure unit you want without converting.
4. You don't need to worry about $R$ and its units in the Dalton's Law approach.

Solve problems as ratios whenever you have the chance.