Animated Solution for Physics - Thermodynamics: A closed container of volume $0.02{\text{ m}}^3$ contains a mixture of neon and argon gases, at a temperature of $27^{\circ }\text{C}$ and pressure of $1\times 10^5{\text{ Nm}}^{-2}$. The total mass of the mixture is $28\text{ g}$. If the molar masses of neon and argon are $20$ and $40{\text{ g mol}}^{-1}$ respectively, find the masses of the individual gases in the container assuming them to be ideal. (Universal gas constant $R=8.314\text{ J/mol-K}$).

The behavior of gases is one of the most fascinating topics in thermodynamics. When you mix two non-reacting gases in a closed container, they don't just sit there; they zip around, colliding with each other and the walls of the container, creating pressure. In this problem, we are going to act like molecular detectives. We have a container with a known volume, temperature, and pressure, and we know the total mass of the gas mixture inside it. Our mission? To figure out exactly how much of that mass belongs to Neon and how much belongs to Argon.

Let's dive into the physics and math behind this beautiful problem!

Imagine you are looking at this closed container. It has a volume of $V=0.02{\text{ m}}^3$. Inside, it's a chaotic dance of Neon and Argon molecules.

The temperature is given as $27^{\circ }\text{C}$. Now, in thermodynamics, we must always work with the absolute temperature scale. So, our first step is to convert this to Kelvin:

The pressure exerted by these bouncing molecules on the walls of the container is $p=1\times 10^5{\text{ N/m}}^2$ (or Pascals).

We are also given the total mass of the mixture, $m_{\text{total}}=28\text{ g}$. But we don't know how this mass is distributed between Neon and Argon.

To solve this, let's introduce a variable. Let the mass of the Neon gas be $m\text{ g}$. Since the total mass is $28\text{ g}$, the mass of the Argon gas must be the remainder:

The ideal gas law, $pV=nRT$, relates the macroscopic properties of a gas (pressure, volume, temperature) to the number of molecules it contains, expressed in moles ($n$).

To use this law, we need to convert our masses into moles. The molar mass of Neon is $M_{\text{Ne}}=20\text{ g/mol}$, and the molar mass of Argon is $M_{\text{Ar}}=40\text{ g/mol}$.

The number of moles of Neon ($n_1$) is its mass divided by its molar mass:

Similarly, the number of moles of Argon ($n_2$) is:

Dalton's Law

Here is where the magic happens. According to Dalton's Law of Partial Pressures, a mixture of non-reacting ideal gases behaves exactly like a single ideal gas. The total pressure exerted by the mixture is the sum of the partial pressures of the individual gases.

This means we can apply the ideal gas equation to the entire mixture by simply adding up the total number of moles!

The total number of moles $n$ is:

Now, we plug this into the ideal gas equation:

Let's substitute all our known values into this master equation. We have:

This might look like a terrifying equation, but let's take a breath and simplify it step by step. First, let's move the volume to the left side:

Notice how we took the common denominator of $40$ inside the bracket. This simplifies the numerator to $m+28$. Now, let's isolate the term with $m$:

Subtracting $28$ from both sides, we get the mass of Neon:

We did it! We found the mass of Neon. Now, finding the mass of Argon is a piece of cake. We just subtract the mass of Neon from the total mass:

The container holds $4.074\text{ g}$ of Neon and $23.926\text{ g}$ of Argon. By trusting the ideal gas law and carefully setting up our variables, we were able to peer inside the container and determine its exact composition.