Question

A 2.5-L flask at \(15^{\circ} \mathrm{C}\) contains a mixture of \(\mathrm{N}_{2}\), \(\mathrm{He},\) and Ne at partial pressures of 0.32 atm for \(\mathrm{N}_{2}, 0.15\) atm for He, and 0.42 atm for Ne. (a) Calculate the total pressure of the mixture. (b) Calculate the volume in liters at STP occupied by He and Ne if the \(\mathrm{N}_{2}\) is removed selectively.

Short answer

Total pressure is 0.89 atm; volume of He and Ne at STP is approximately 1.39 L.

Step-by-step solution

Identify the Partial Pressures

First, note the partial pressures of the individual gases present in the flask. - The partial pressure of \( \text{N}_2 \) is 0.32 atm.- The partial pressure of \( \text{He} \) is 0.15 atm.- The partial pressure of \( \text{Ne} \) is 0.42 atm.

Calculate Total Pressure

The total pressure of a gas mixture is the sum of the partial pressures of each gas present. Using Dalton's Law of Partial Pressures: \[P_{\text{total}} = P_{\text{N}_2} + P_{\text{He}} + P_{\text{Ne}} \] Substitute in the known values: \[P_{\text{total}} = 0.32 \text{ atm} + 0.15 \text{ atm} + 0.42 \text{ atm} = 0.89 \text{ atm}\] Therefore, the total pressure is 0.89 atm.

Use Ideal Gas Law to Find the Volume of He and Ne at STP

To find the volume of \( \text{He} \) and \( \text{Ne} \) at STP (Standard Temperature and Pressure: 0°C and 1 atm), we need the moles of these gases. First, compute the moles of He and Ne using the Ideal Gas Law. The formula is \( PV = nRT \), where \( R = 0.0821 \; \text{L atm K}^{-1} \cdot \text{mol}^{-1} \).Correct the temperature to Kelvin: \( T = 15 + 273 = 288 \; \text{K} \).

Calculate Moles of Helium (He)

Using the Ideal Gas Law: \\[ n_{\text{He}} = \frac{P_{\text{He}} \cdot V}{R \cdot T} \] \Substituting in the values, \( P_{\text{He}} = 0.15 \; \text{atm} \), \( V = 2.5 \; \text{L} \), \( R = 0.0821 \; \text{L atm K}^{-1} \cdot \text{mol}^{-1} \), and \( T = 288 \; \text{K} \): \\[ n_{\text{He}} = \frac{0.15 \times 2.5}{0.0821 \times 288} \approx 0.016 \; \text{mol} \]

Calculate Moles of Neon (Ne)

Using the Ideal Gas Law: \\[ n_{\text{Ne}} = \frac{P_{\text{Ne}} \cdot V}{R \cdot T} \] \Substituting in the values, \( P_{\text{Ne}} = 0.42 \; \text{atm} \): \\[ n_{\text{Ne}} = \frac{0.42 \times 2.5}{0.0821 \times 288} \approx 0.046 \; \text{mol} \]

Combine Moles of He and Ne

Sum the moles of \( \text{He} \) and \( \text{Ne} \): \\[ n_{\text{total}} = n_{\text{He}} + n_{\text{Ne}} = 0.016 + 0.046 = 0.062 \; \text{mol} \]

Calculate New Volume at STP

Using the Ideal Gas Law rearranged for \( V \) and with \( P = 1 \; \text{atm} \), \( T = 273 \; \text{K} \): \\[ V = \frac{n \cdot R \cdot T}{P} \] \Substitute in the moles \( n = 0.062 \; \text{mol} \) and solve: \\[ V = \frac{0.062 \cdot 0.0821 \cdot 273}{1} \approx 1.39 \; \text{L} \] \ Therefore, the volume at STP for \( \text{He} \) and \( \text{Ne} \) is approximately 1.39 L.

Key concepts

Ideal Gas Law

The Ideal Gas Law is one of the fundamental equations in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. The formula is given by \( PV = nRT \), where:

• \( P \) is the pressure of the gas in atmospheres (atm),
• \( V \) is the volume of the gas in liters (L),
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant (0.0821 L atm K^-1 mol^-1), and
• \( T \) is the temperature in Kelvin (K).

To convert Celsius to Kelvin, simply add 273. This law assumes that the gas behaves ideally, meaning the gas particles do not interact and the volume of the particles themselves is negligible compared to the container. Ideal Gas Law helps to determine the unknown quantity among pressure, volume, temperature, or moles when the other three are known.

Partial Pressure

Partial pressure refers to the pressure exerted by a single type of gas in a mixture of gases. According to Dalton's Law of Partial Pressures, the total pressure of a mixture of gases is the sum of the partial pressures of all individual gases in the mixture. Mathematically:\[P_{\text{total}} = P_1 + P_2 + P_3 + ...\]For example, in the given problem, the partial pressures are given for nitrogen (\(\text{N}_2\)), helium (\(\text{He}\)), and neon (\(\text{Ne}\)). By adding these partial pressures, you determine the total pressure in the flask. This concept is vital for understanding how gases interact in a mixture and is used in various applications such as respiratory science and chemical engineering.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a reference point used in thermodynamics for the purpose of consistency in calculations involving gases. STP is defined as a temperature of 0°C (273 K) and a pressure of 1 atm. At STP, one mole of an ideal gas occupies 22.4 liters. This uniform benchmark allows scientists and engineers to compare different gases under the same conditions, making calculations more straightforward. In gas-related calculations, values of temperature and pressure are often converted to align with STP for easier analysis and comparison.

Gas Mixture Calculations

Solving problems involving gas mixtures often requires a clear understanding of both Dalton's Law and the Ideal Gas Law. When a gas mixture is described, you might need to find the total pressure from individual pressures or determine other properties like volume or number of moles under various conditions. For the given exercise:

• First, calculate the total pressure using the partial pressures of the gases involved using Dalton's Law.
• Then, use the Ideal Gas Law to find the volume at STP by first calculating the moles of the gases in the mixture.
• It involves understanding the effect of selectively removing components, and converting conditions to STP allows for standardization in calculations.

Gas mixture calculations help in industries such as atmospheric science, engineering, and pharmaceuticals by ensuring the right conditions are applied for reactions or in processes, like gas purifications or blends.