Animated Solution for Chemistry - States of Matter: 0.5 moles of gas A and $x$ moles of gas B exert a pressure of $200\text{ Pa}$ in a container of volume $10{\text{ m}}^3$ at $1000\text{ K}$. Given $R$ is the gas constant in ${\text{J K}}^{-1}{\text{mol}}^{-1}$, $x$ is

Imagine you are in a laboratory, looking at a sealed container. Inside this container, a silent, chaotic dance is happening. Millions of particles of two different gases, Gas A and Gas B, are zipping around, colliding with each other and the walls of the container. These collisions are what create pressure.

In this problem, we are given a very specific snapshot of this container. We know its volume is exactly $10{\text{ m}}^3$, and it's being kept at a scorching temperature of $1000\text{ K}$. The total pressure exerted by this frantic mixture of gases is $200\text{ Pa}$. We also know exactly how much of Gas A is in there: $0.5\text{ moles}$. But Gas B is a mystery. We only know there are $x\text{ moles}$ of it. Our mission is to find $x$.

Ideal Gas Law

When dealing with a mixture of non-reacting ideal gases, nature is surprisingly cooperative. We don't need to worry about the individual identities of the gases. Whether it's helium, neon, or some exotic ideal gas, they all behave the same way under these conditions.

This means we can treat the entire mixture as one single, unified ideal gas. The total number of moles in our container is simply the sum of the moles of Gas A and Gas B.

Now, we bring out the heavy artillery: the Ideal Gas Equation. This elegant equation connects all the macroscopic properties of a gas—pressure ($p$), volume ($V$), temperature ($T$), and the number of moles ($n$).

Finding the Unknown

Now, it's time to plug in the numbers. We have to be careful and ensure all our units are consistent, but thankfully, the problem has given us standard SI units.

Substituting our known values into the ideal gas equation:

Notice that we haven't substituted the numerical value for the universal gas constant, $R$. A quick glance at the options tells us that the final answer is expected in terms of $R$. This is a classic exam trick—always keep an eye on the options to save yourself from unnecessary calculations!

Let's simplify the left side. $200\times 10$ gives us $2000$.

To isolate our unknown $x$, let's divide both sides by $1000R$. The math here is beautiful and clean. $2000$ divided by $1000$ is simply $2$.

We are almost at the finish line. We just need to move the $0.5$ to the other side. It's often easier to work with fractions rather than decimals in algebra, so let's write $0.5$ as $\frac{1}{2}$.

Now, we take the common denominator, which is $2R$.

And there we have it! The number of moles of Gas B is $\frac{4-R}{2R}$. This perfectly matches option (b).

This problem is a fantastic reminder of the power of the ideal gas law. By treating a mixture as a single entity, we can easily navigate through the macroscopic properties of the system. Keep practicing, and these concepts will become second nature to you!