Question

Pressure in a mixture of \(4 \mathrm{~g}\) of \(\mathrm{O}_{2}\) and \(2 \mathrm{~g}\) of \(\mathrm{H}_{2}\) confined in a container of 1 litre capacity at \(0^{\circ} \mathrm{C}\) is: (a) \(25.2\) atm (b) \(35.6 \mathrm{~atm}\) (c) \(15.4 \mathrm{~atm}\) (d) \(48.2 \mathrm{~atm}\)

Short answer

The pressure in the container is \(25.2\, \mathrm{atm}\).

Step-by-step solution

Calculate Moles of Oxygen

First, find the number of moles of oxygen. The molecular weight of \(\mathrm{O}_2\) is \(32 \mathrm{~g/mol}\). Thus, the number of moles of \(\mathrm{O}_2\) is computed as follows: \[\text{Moles of } \mathrm{O}_2 = \frac{4 \mathrm{~g}}{32 \mathrm{~g/mol}} = 0.125 \mathrm{~mol}.\]

Calculate Moles of Hydrogen

Next, calculate the number of moles of hydrogen. The molecular weight of \(\mathrm{H}_2\) is \(2 \mathrm{~g/mol}\). Therefore, the number of moles of \(\mathrm{H}_2\) is calculated as: \[\text{Moles of } \mathrm{H}_2 = \frac{2 \mathrm{~g}}{2 \mathrm{~g/mol}} = 1 \mathrm{~mol}.\]

Total Moles in the Mixture

Sum the moles of both gases to find the total number of moles in the mixture: \[\text{Total Moles} = 0.125 \mathrm{~mol} + 1 \mathrm{~mol} = 1.125 \mathrm{~mol}.\]

Use Ideal Gas Law to Find Pressure

Using the Ideal Gas Law \(PV = nRT\), solve for pressure \(P\). Here, \(n = 1.125 \mathrm{~mol}\), \(R = 0.0821\; \mathrm{L\cdot atm/mol\cdot K}\), \(T = 273\; \mathrm{K}\), and \(V = 1\; \mathrm{L}\). Substituting these into the equation, we have:\[P = \frac{nRT}{V} = \frac{1.125 \times 0.0821 \times 273}{1} = 25.2\; \mathrm{atm}.\]

Key concepts

Moles of Gases

When dealing with chemical reactions and gas calculations, understanding the concept of moles is essential. Moles provide a way to measure the amount of a substance. For gaseous substances, the mole allows us to relate the mass of a gas to its volume and other properties. In the Ideal Gas Law, the mole is a central component in calculations involving volume, temperature, and pressure.

To find the moles of a specific gas, such as oxygen \( \mathrm{O}_2 \), you need to know its molecular weight. The molecular weight informs you how much one mole of that substance weighs in grams. Oxygen’s molecular weight, for example, is 32 \(\mathrm{g/mol}\). If you have 4 grams of oxygen, you divide the mass by the molecular weight to get the number of moles: \[\text{Moles of } \mathrm{O}_2 = \frac{4 \text{ g}}{32 \text{ g/mol}} = 0.125 \text{ mol}.\]

• To determine moles of oxygen \((\mathrm{O}_2)\): Mass of \(\mathrm{O}_2\) divided by Molecular Weight.
• Moles of hydrogen \((\mathrm{H}_2)\) are calculated similarly: 2 grams of \(\mathrm{H}_2\) using its molecular weight of 2 \(\mathrm{g/mol}\) gives 1 mol.

Understanding the moles of each gas in a mixture is crucial for further calculations, such as determining the overall pressure.

Gas Mixtures

Gas mixtures are combinations of two or more gases. To analyze the behavior of a gas mixture, it's important to understand the contribution of each gas to the overall properties of the mixture, such as pressure. The behavior of gas mixtures can often be predicted by assuming that each gas in the mixture behaves independently, as described by Dalton's Law of Partial Pressures.

When gases are combined, the total amount of gas is the sum of the moles of each component gas. For example, if you have 0.125 moles of \( \mathrm{O}_2 \) and 1 mole of \( \mathrm{H}_2 \), the total number of moles in the mixture is \[ \text{Total Moles} = 0.125 \text{ mol} + 1 \text{ mol} = 1.125 \text{ mol}. \]

• Each gas in a mixture exerts its pressure independently.
• The total moles of the mixture is critical for calculating total pressure.

This total mole count becomes a vital parameter in the Ideal Gas Law to find out how these gases act as a mixture in a given volume.

Pressure Calculation

Understanding pressure calculation involves applying the Ideal Gas Law, a fundamental principle for computing the pressure exerted by gases in a mixture. The Ideal Gas Law: \( PV = nRT \) relates the pressure \(P\), volume \(V\), and temperature \(T\) of a gas with its moles \(n\), and the gas constant \(R\).

To find the pressure of a gas mixture:

• Identify \(n\), the total moles of gas ("1.125 mol" in our case).
• Utilize \(R\), with a value of \(0.0821\; \mathrm{L\cdot atm/mol\cdot K}\).
• Remember: room temperature \(T\) converted to Kelvin is 273 K, and the volume \(V\) in this exercise is 1 L.

Plug these values into the formula to compute pressure: \[P = \frac{nRT}{V} = \frac{1.125 \cdot 0.0821 \cdot 273}{1} = 25.2\; \mathrm{atm}.\] This calculation shows how the total moles in combination with the other factors dictate the pressure of the gas confined within the container. Understanding each variable’s role helps predict how changes to any single component affect the resulting pressure, illustrating the interconnectedness of gas properties in physical science.