Question

Gas cylinder A has a volume of 48.2 L and contains ${\mathrm{N}}_2(\mathrm{g})$ at 8.35 atm at ${25}^{\circ }\mathrm{C}.$ Gas cylinder $\mathrm{B},$ of unknown volume, contains $\mathrm{He}(\mathrm{g})$ at 9.50 atm and ${25}^{\circ }\mathrm{C}.$ When the two cylinders are connected and the gases mixed, the pressure in each cylinder becomes 8.71 atm. What is the volume of cylinder $\mathrm{B}?$

Short answer

To find the volume of cylinder B, follow the steps outlined above, substituting the given values into the appropriate formulas. The final answer will depend on these specific values.

Step-by-step solution

Calculate the number of moles of gas in cylinder A

From the given values, apply the ideal gas law to calculate the number of moles of gas, $n_A$, in cylinder A. $P_A=8.35atm$, $V_A=48.2L$, $R=0.0821atm\cdot L/(mol\cdot K)$, and $T_A=25°C+273.15=298.15K$. Therefore, $n_A=\frac{P_A{V}_A}{R{T}_A}$.

Calculate the total final moles of the mixed gases

We know that after mixing the gases from both cylinders, the final pressure, $P_f$, is 8.71 atm. So the total moles of gases, $n_{total}$, is given by $n_{total}=n_A+n_B=\frac{P_f{V}_{t}otal}{R{T}_f}$.

Calculate the volume of cylinder B

We know that $V_{t}otal=V_A+V_B$. Substituting this into the equation from Step 2 and simplifying, we find that $n_B=\frac{P_f{V}_B}{R{T}_f}$. Subtracting $n_A$ from both sides of the equation gives $n_B$. Now, we can find the volume of cylinder B by rearranging the formula: $V_B=\frac{(n_{total}-n_A)R{T}_f}{P_f}$.

Key concepts

Gas Mixtures

Imagine taking two different gases and combining them into one container. That's essentially what a gas mixture is: a physical combination of different gases that remain distinct at the molecular level. In this exercise, we have a mixture of nitrogen (${\mathrm{N}}_2$) and helium ($\mathrm{He}$) gases. Each gas maintains its own chemical properties while contributing to the overall behavior of the mixture.

The concept of gas mixtures is important because it helps us understand how gases interact in shared environments. This is especially the case with the pressures and volumes of gases, which together adhere to the ideal gas law.

A neat feature of gas mixtures is how pressures add without altering individual gas identities. This property is known as Dalton’s Law of Partial Pressures, where the total pressure of the mixture is equal to the sum of the partial pressures of each gas. This principle was indirectly utilized to find the final pressure after mixing the gases in the exercise.

Moles Calculation

To work with gases effectively, you need to grasp the concept of moles. Moles provide a measure of the amount of substance, allowing easy conversion between mass and number of particles. In chemistry, it's pivotal when you need to calculate or predict the behaviors of gases using the ideal gas law.

In the given exercise, the calculation of moles is crucial. When the nitrogen gas in cylinder A is analyzed, the number of moles $n_A$ is found using the ideal gas law equation:

$$n_A=\frac{P_A{V}_A}{R{T}_A}$$

Where $P_A$ is pressure, $V_A$ is volume, $R$ is the ideal gas constant, and $T_A$ is temperature in Kelvin. These calculations help you to understand the quantity of physical entities (atoms, molecules) present in the gas cylinder.

The same principle is used when deducing the total moles $n_{total}$ in a mixture of gases, giving insight into how they behave once mixed.

Gas Pressure

In our daily lives, we continuously experience pressure in various forms. In the context of gases, pressure is the force exerted by gas molecules as they collide with the walls of their container. It's a fundamental concept in the study of gases and heavily relies on the temperature and volume of the gas, as per the ideal gas law.

In this exercise, we have observed changes in gas pressure before and after mixing. Initially, each gas in the separate cylinders exerts its own pressure due to its specific volume and temperature. When the gases are combined, a new equilibrium pressure is established at $8.71$ atm.

This is where the ideal gas law shines, allowing calculation of changes in pressure when the conditions are altered. The scenario reveals how gas pressures adapt in a new setting, illustrating important physical behaviors and interactions in mixtures. Understanding these changes in pressure helps in solving not only textbook problems but also in practical applications like chemical engineering and respiratory systems.