Question

A 2.35 L container of \(\mathrm{H}_{2}(\mathrm{g})\) at \(762 \mathrm{mmHg}\) and \(24^{\circ} \mathrm{C}\) is connected to a 3.17 L container of \(\mathrm{He}(\mathrm{g})\) at \(728 \mathrm{mmHg}\) and \(24^{\circ} \mathrm{C}\). After mixing, what is the total gas pressure, in millimeters of mercury, with the temperature remaining at \(24^{\circ} \mathrm{C} ?\)

Short answer

The total gas pressure after mixing of the hydrogen and helium gases is the sum of the individual pressures calculated following Boyle's Law.

Step-by-step solution

Convert pressure of H2 gas after mixing

Find the pressure of the hydrogen gas once mixed. According to Boyle's Law, the pressure and volume of a gas have an inverse relationship when temperature is held constant. Hence, \(P_{1}V_{1} = P_{2}V_{2}\). Given \(P_{1} = 762 \, mmHg\) and \(V_{1} = 2.35 \, L\), while \(V_{2}\) is the total volume of the system which is \(V_{1} + V_{2} = 2.35 \, L + 3.17 \, L = 5.52 \, L\). Solve for \(P_{2}\) to get the pressure of H2 gas after mixing.

Convert pressure of He gas after mixing

Similar to Step 1, determine the pressure of the helium gas once mixed. With \(P_{1} = 728 \, mmHg\) and \(V_{1} = 3.17 \, L\), and the same \(V_{2} = 5.52 \, L\), we can solve for \(P_{2}\) which will give us the pressure of He gas after mixing.

Add the two pressures

The total pressure of the system will be the sum of the pressures of H2 and He after mixing. Calculate the sum of the results obtained in Step 1 and Step 2 to get the total gas pressure of the system.

Key concepts

Boyle's Law

Boyle's Law is a fundamental gas law stating that the pressure of a gas is inversely proportional to its volume when temperature is constant. This means if you increase the volume of a gas, its pressure decreases and vice versa, provided temperature doesn't change. The relationship is described by the equation: \[ P_1V_1 = P_2V_2 \]where:

• \(P_1\) and \(V_1\) are the initial pressure and volume, respectively
• \(P_2\) and \(V_2\) refer to the final pressure and volume

In the given exercise, Boyle's Law is used to find the new pressures of hydrogen and helium gases after they are mixed into a larger volume. By using this equation, we are able to determine the pressure that each gas exerts when the total volume changes from 2.35 L and 3.17 L each, to a combined 5.52 L.

Partial Pressure

Partial pressure refers to the pressure that a single gas in a mixture of gases would exert if it occupied the whole volume on its own at the same temperature. In a mixture of non-reacting gases, each gas contributes to the total pressure proportionally to its amount. This is known as Dalton's Law of Partial Pressures.Dalton's Law can be expressed as: \[ P_{total} = P_1 + P_2 + ... + P_n \] where each \(P_i\) is the partial pressure of each gas in the mixture.In the original exercise, once we determine the partial pressures of hydrogen and helium after mixing, their combined partial pressures give us the total pressure of the gas mixture. This principle allows us to understand how individual gas components contribute to the overall pressure in a mixed system.

Mixture of Gases

When dealing with a mixture of gases like hydrogen and helium, it's essential to understand how these gases behave when combined. Gases in a mixture don't interact or interfere with each other but follow the gas laws independently. This means each gas in the mixture behaves as if it were alone in the container. In the exercise provided, two gases are combined in a total volume thus requiring analysis of each gas separately using Boyle's Law, followed by a summation of their partial pressures.

• Each gas’s initial conditions are assessed using the individual pressure-volume pairs.
• Boyle's Law helps us find out how their pressures change upon combining.

By analyzing each gas separately, and then summing their partial pressures, we efficiently determine the overall pressure of the gas mixture, respecting each component's contribution as described by Dalton's Law.