Question

A vessel of volume \(22.4 \mathrm{~L}\) contains \(2.0 \mathrm{~mol} \mathrm{} \mathrm{H}_{2}(\mathrm{~g})\) and \(1.0 \mathrm{~mol}\) \(\mathrm{N}_{2}\) (g) at \(273.15 \mathrm{~K}\). Calculate (a) their partial pressures and (b) the total pressure.

Short answer

The partial pressure of \( H_{2} \) is \(2.0 \times \frac{0.0821 \times 273.15}{22.4} \text{atm}\), of \( N_{2} \) is \(1.0 \times \frac{0.0821 \times 273.15}{22.4} \text{atm}\), and the total pressure is the sum of the two partial pressures.

Step-by-step solution

Understand the Ideal Gas Law

To solve for the partial pressures, use the Ideal Gas Law, expressed as PV = nRT, where P is pressure, V is volume, n is the amount of gas in moles, R is the ideal gas constant, and T is the temperature in Kelvin. For this exercise, we will need to solve for P for each gas separately.

Calculate the Partial Pressure of Hydrogen \(H_{2}\)

Using the Ideal Gas Law, calculate the partial pressure of \(H_{2}\) by plugging in its values: n = 2.0 mol, V = 22.4 L, T = 273.15 K, and R = 0.0821 L·atm/K·mol. Thus, the partial pressure \( P_{H2} = \frac{nRT}{V} = \frac{2.0\,mol \cdot 0.0821\,L\cdot atm/K\cdot mol \cdot 273.15\,K}{22.4\,L} \).

Calculate the Partial Pressure of Nitrogen \(N_{2}\)

Similarly, for \(N_{2}\), use its values: n = 1.0 mol, V = 22.4 L, T = 273.15 K. Solve for \( P_{N2} = \frac{nRT}{V} = \frac{1.0\,mol \cdot 0.0821\,L\cdot atm/K\cdot mol \cdot 273.15\,K}{22.4\,L} \).

Determine the Total Pressure

The total pressure inside the vessel is the sum of the partial pressures of \(H_{2}\) and \(N_{2}\). Therefore, total pressure \( P_{total} = P_{H2} + P_{N2} \).

Key concepts

Partial Pressure

When multiple gases are present in a container, each gas exerts its own pressure independent of the others. This individual pressure is called the partial pressure. It's crucial to remember that the partial pressures add up to the total pressure inside the container.

To find the partial pressure of a gas, we use a formula derived from the Ideal Gas Law: \( P_{gas} = \frac{nRT}{V} \), where \( P_{gas} \) is the partial pressure of the gas, \( n \) is the number of moles, \( R \) is the ideal gas constant, \( T \) is the temperature in Kelvin, and \( V \) is the volume of the gas. This relationship shows how the amount of gas and the temperature affect the pressure it exerts in a given volume. For example, increasing the number of moles of gas or raising the temperature will increase the partial pressure if the volume remains constant.

In our exercise, the partial pressures of hydrogen (\( H_2 \) gas) and nitrogen (\( N_2 \) gas) are calculated separately using their respective mole amounts and the given temperature and volume. Their sum gives us the total pressure in the vessel.

Gas Constant (R)

The gas constant, often symbolized as \( R \), is a fundamental constant in the Ideal Gas Law. The value of \( R \) depends on the units used for pressure, volume, and temperature. Commonly, in chemistry, we work with \( R = 0.0821 L\cdot atm/K\cdot mol \) for expressivity in terms of liters, atmospheres, Kelvin, and moles.

This constant provides a link between the physical properties of gases and allows us to calculate one property if the others are known. It represents how much pressure is created by a certain amount of gas at a given temperature and is key to understanding how gases behave under different conditions.

In the context of our exercise, \( R \) is used in calculations to determine the partial pressures of hydrogen and nitrogen gases at a set temperature and volume. Without \( R \) being a constant, it would be impossible to predict how a gas will respond to changes in temperature or volume.

Molar Volume

The concept of molar volume is particularly important when dealing with gases. Molar volume is the volume occupied by one mole of any gas at a specified temperature and pressure. For gases, this is often referred to under standard temperature and pressure (STP), which is at 0°C (273.15K) and 1 atm pressure. At STP, one mole of an ideal gas occupies 22.4 liters.

Molar volume helps us understand that all ideal gases, regardless of their chemical identity or properties, will occupy the same volume under identical temperature and pressure conditions. For our exercise, we are given that the vessel volume is 22.4 L, which corresponds to the molar volume of an ideal gas at STP for one mole of gas.

This proportionality allows simplifications in our calculations. In real-life applications, it's common to calculate how a gas expands or compresses with varying numbers of moles, which directly correlates with the molar volume.