Question

A gas mixture contains 1.25 \(\mathrm{gN}_{2}\) and 0.85 \(\mathrm{g} \mathrm{O}_{2}\) in a \(1.55-\mathrm{L}\) container at \(18^{\circ} \mathrm{C} .\) Calculate the mole fraction and partial pressure of each component in the gas mixture.

Short answer

First, convert temperature to Kelvin: \(291.15\) K. Calculate moles for \(N_2\): \(0.0446 \text{mol}\) and for \(O_2\): \(0.0266 \text{mol}\). Total moles: \(0.0712 \text{mol}\). Mole fractions for \(N_2\) and \(O_2\) are \(0.626\) and \(0.374\) respectively. Total pressure using ideal gas law: \(1.494 \text{atm}\). Partial pressures are \(0.935 \text{atm} \) for \(N_2\) and \(0.559 \text{atm}\) for \(O_2\).

Step-by-step solution

Convert temperature to Kelvin

Convert the given temperature from Celsius to Kelvin by adding 273.15 to the Celsius temperature. For this problem, the temperature in Kelvin is: \( T = 18^{\text{\(^\circ\)C}} + 273.15 \) K.

Calculate moles of each gas

Use the molar mass of nitrogen (\(N_2\)) and oxygen (\(O_2\)) to find the moles of each gas. For \(N_2\), the molar mass is \(28.02 \text{g/mol}\). For \(O_2\), it is \(32.00 \text{g/mol}\). Use the formula \( n = \frac{m}{M} \), where \(n\) is the number of moles, \(m\) is the mass of the gas, and \(M\) is the molar mass.

Calculate the total moles of the gas mixture

Sum the moles of each component to get the total moles in the mixture: \( n_{\mathrm{total}} = n_{\mathrm{N}_2} + n_{\mathrm{O}_2} \).

Calculate mole fractions

Calculate the mole fractions of \(N_2\) and \(O_2\) by dividing the moles of each gas by the total moles: \( X_{\mathrm{N}_2} = \frac{n_{\mathrm{N}_2}}{n_{\mathrm{total}}} \) and \( X_{\mathrm{O}_2} = \frac{n_{\mathrm{O}_2}}{n_{\mathrm{total}}} \).

Calculate the total pressure using the ideal gas law

Utilize the ideal gas law \( PV = nRT \) to find the total pressure. Where \( P \) is pressure, \( n \) is the total number of moles of gas, \( R \) is the gas constant (\( 0.0821 \text{L} \cdot \text{mol}^{-1} \cdot \text{K}^{-1} \)), \( V \) is the volume of the container, and \( T \) is the temperature in Kelvin.

Calculate partial pressures

Find the partial pressure of each component by multiplying the mole fraction by the total pressure. The partial pressure of a gas component is given by \( P_{\mathrm{component}} = X_{\mathrm{component}} \times P_{\mathrm{total}} \).

Key concepts

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of that component to the total number of moles of all components in the mixture. To calculate it, we use the formula:
\[ X_i = \frac{n_i}{n_{total}} \]
where \( X_i \) is the mole fraction, \( n_i \) is the number of moles of the component, and \( n_{total} \) is the total number of moles in the mixture. This expression is dimensionless, meaning it does not have a unit, making it universal for comparing the proportions of components irrespective of the amount of mixture. It plays a crucial role in determining the partial pressure of each gas in a mixture, which is essential in many chemical calculations.

Ideal Gas Law

The ideal gas law is a fundamental equation in chemistry that relates the pressure, volume, temperature, and number of moles of an ideal gas. The equation is:
\[ PV = nRT \]
where \( P \) represents the pressure, \( V \) the volume, \( n \) the number of moles, \( R \) the ideal gas constant (0.0821 \( L\text{mol}^{-1}K^{-1} \)), and \( T \) the temperature in Kelvin. This equation assumes that the gas behaves ideally, meaning the particles are point masses without volume and no intermolecular forces between them. Real gases exhibit ideal behavior under certain conditions, particularly at high temperatures and low pressures, and thus the ideal gas law can be used as an approximation for many practical purposes.

Converting Temperature to Kelvin

Temperature measurements in chemistry are often required in Kelvin because this scale is used in various equations, including the ideal gas law. The Kelvin scale is an absolute temperature scale, meaning it starts at absolute zero, where theoretically no more thermal energy can be removed from a substance. To convert Celsius to Kelvin, which is a common conversion in chemistry problems, use the formula:
\[ T(\text{K}) = T(\text{\(^\text{\circ}C\)}) + 273.15 \]
By adding 273.15 to the Celsius temperature, you ensure that the calculations for gas laws and many other thermodynamic equations are accurate.

Calculating Moles from Mass

To find the amount of substance in moles from its mass, we use the molar mass of the substance—the mass of one mole. The molar mass is typically given in grams per mole (g/mol) and can be found on the periodic table for each element. For compounds, the molar mass is the sum of the molar masses of the individual elements that make up the compound, each multiplied by the number of atoms of that element in the compound. The formula to calculate the number of moles (\( n \)) from mass (\( m \)) is:
\[ n = \frac{m}{M} \]
where \( M \) represents the molar mass of the substance. Accurate determination of the number of moles is essential in stoichiometry and in predicting the behavior of gases using the ideal gas law.

Gas Mixture

A gas mixture contains more than one type of gas, each contributing to the total pressure exerted by the mixture. According to Dalton's Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each individual gas. The partial pressure of a gas is proportional to its mole fraction in the mixture. The relationship between mole fraction and partial pressure is crucial in applications like calculating the concentration of gases in the atmosphere, designing chemical reactors, and optimizing industrial gas processes. Understanding the properties and behavior of each component in a gas mixture allows chemists to predict how the mixture will react under varying conditions, which is essential for many practical and industrial applications.