Question

A mixture of \(\mathrm{H}_{2}\) and \(\mathrm{NH}_{3}\) has a volume of \(139.0 \mathrm{cm}^{3}\) at \(0.00^{\circ} \mathrm{C}\) and 1 atm. The mixture is cooled to the temperature of liquid nitrogen at which ammonia freezes out and the remaining gas is removed from the vessel. Upon warming the vessel to \(0.00^{\circ} \mathrm{C}\) and 1 atm, the volume is \(77.4 \mathrm{cm}^{3}\). Calculate the mole fraction of \(\mathrm{NH}_{3}\) in the original mixture.

Short answer

The mole fraction of ammonia (NH3) in the original mixture can be calculated by first finding the moles of the remaining hydrogen gas (n_H2) and the moles of the original mixture (n_total) using the ideal gas law (PV = nRT). Then subtract n_H2 from n_total to find the moles of ammonia (n_NH3), and finally calculate the mole fraction of NH3 by dividing n_NH3 by n_total.

Step-by-step solution

Calculate moles of remaining hydrogen gas

In the final step, we are given the volume of the remaining hydrogen gas (V2) as 77.4 cm^3. To find the moles of hydrogen gas at this stage, we will use the ideal gas law (PV = nRT). The gas constant R is 0.0821 L atm/mol K, which means we need to convert the volume to liters. V2 = 0.0774 L (77.4 cm^3 = 0.0774 L) Since the hydrogen gas is at 0°C and 1 atm: T2 = 273.15 K (0.00°C + 273.15 = 273.15 K) P2 = 1 atm Now, we can find the moles of hydrogen gas: n_H2 = PV / RT n_H2 = (1 atm)(0.0774 L) / (0.0821 L atm/mol K)(273.15 K)

Calculate moles of the original mixture

Now, we will use the initial volume of the gas mixture (V1) and the ideal gas law to find the moles of the original mixture. V1 = 0.139 L (139.0 cm^3 = 0.139 L) The temperature and pressure for the original mixture are the same as previously (0.00°C and 1 atm), meaning: T1 = 273.15 K P1 = 1 atm Now, we can find the moles of the original mixture: n_total = PV / RT n_total = (1 atm)(0.139 L) / (0.0821 L atm/mol K)(273.15 K)

Calculate mole fraction of ammonia in the original mixture

To calculate the mole fraction of ammonia in the original mixture, first, we need to find the moles of ammonia. To do that, we will subtract the moles of hydrogen from the total moles: n_NH3 = n_total - n_H2 Now, we can find the mole fraction: Mole_fraction_NH3 = n_NH3 / n_total It is important to express the final mole fraction as a decimal or percentage.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry and physics that relates the pressure, volume, temperature, and number of moles of a gas. The law is an approximation that provides a reasonable explanation of the behavior of gases under many conditions, although it has its limitations, particularly at high pressures and low temperatures where real gas behavior diverges from ideal behavior.

At its core, the Ideal Gas Law is represented by the equation \( PV = nRT \), where \( P \) stands for pressure in atmospheres (atm), \( V \) for volume in liters (L), \( n \) for the number of moles of gas, \( R \) for the ideal gas constant (0.0821 L atm/mol K), and \( T \) for temperature in Kelvin (K). It assumes that the gas particles do not interact with each other and occupy no space.

Using this law, we can calculate any one of the variables if the other three are known, making it an incredibly useful tool in chemical calculations and studies involving gases.

PV=nRT

When we dive deeper into the Ideal Gas Law equation, \( PV = nRT \) becomes a simple but powerful formula that chemists and physicists use to predict the behavior of an ideal gas. It enables us to connect the macroscopic properties of gases that we can measure, such as pressure (\( P \)), volume (\( V \)), and temperature (\( T \)), with the microscopic property we cannot directly measure, which is the amount of substance present in moles (\( n \)).

For example, to find the amount of gas in moles, we rearrange the equation to solve for \( n \): \( n = \frac{PV}{RT} \). This shows us that by knowing the pressure, volume, and temperature of a gas, we can find out how many moles of it we have. This part of the ideal gas law is frequently used to calculate the amount of a particular gas present in a mixture by isolating it and measuring its volume, pressure, and temperature.

Molar Volume

Molar volume is the volume occupied by one mole of a substance, and in the case of gases, it's highly influenced by conditions such as temperature and pressure. Under standard conditions of temperature and pressure (STP), which are 273.15 K (0°C) and 1 atm, the molar volume of an ideal gas is approximately 22.4 liters. It's important to convert volumes to liters when using the ideal gas law, as the value of the gas constant (\( R \)), is expressed in liters.

However, molar volume can change when conditions change. Whenever temperature increases, the molar volume increases as well (assuming pressure is constant), because gas molecules move faster and tend to occupy more space. Conversely, an increase in pressure (assuming temperature is constant) causes the molar volume to decrease, as the molecules are forced closer together.

Gas Mixture Composition

Understanding the gas mixture composition involves knowing the proportions of different gases in a mixture. This is often represented in terms of mole fraction, which is the ratio of the number of moles of a particular component to the total number of moles of all components in the mixture.

For a two-component system like the one in our exercise with hydrogen (\(H_2\)) and ammonia (\(NH_3\)), we'd denote the mole fraction of ammonia as \( \chi_{NH_3} \). It's calculated by dividing the moles of ammonia by the total moles of the mixture, \( \chi_{NH_3} = \frac{n_{NH_3}}{n_{total}} \).

To put it simply, if we know the volume, temperature, and pressure of the gas mixture as well as the volume after one gas is removed (in this case, \(NH_3\)), the ideal gas law allows us to calculate the mole fractions by first determining the number of moles of each component. Mole fractions are dimensionless numbers and can be expressed as decimals or percentages, providing a clear picture of the composition of the mixture.