Problem 68 - Chemistry

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 \(\mathrm{He},\) and 0.42 atm for \(\mathrm{Ne} .\) (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.

Step-by-Step Solution

Answer

The total pressure of the mixture is 0.89 atm. The volume occupied by He and Ne at STP (if the N2 is selectively removed) is approximately 1.1 liters.

Step 1: Calculate the total pressure

The total pressure of a mixture of gases can be calculated simply by adding up the partial pressures of each gas in the mixture. Therefore, the total pressure \( P_t \) can be found using the equation: \(P_t = P_{N2} + P_{He} + P_{Ne}\). Substituting the given values and calculating, \(P_t = 0.32 atm + 0.15 atm + 0.42 atm = 0.89 atm \)

Step 2: Calculate the Moles of He and Ne

Next, we calculate the number of moles of the gases He and Ne using the ideal gas law, rearranged to solve for n: \(n = P*V/R*T\). Here \(R = 0.0821 L.atm/mol.K\) which is a common value for the gas constant and the temperature T is converted to Kelvin (15 C = 288 K). So the moles of He and Ne can be calculated as: \(n_{He} = P_{He}*V/R*T \rightarrow n_{He} = 0.15*2.5/0.0821*288 = 0.0129 mol\), \(n_{Ne} = P_{Ne}*V/R*T \rightarrow n_{Ne} = 0.42*2.5/0.0821*288 = 0.0361 mol\)

Step 3: Calculate the volume

After calculating the moles of He and Ne, we next calculate the volume they would occupy at STP. We know that any number of moles of gas at STP occupies 22.4 L. Hence the volume of He and Ne at STP can be calculated as: \(V_{He} = n_{He}*22.4 = 0.0129*22.4 = 0.2896 L\), \(V_{Ne} = n_{Ne}*22.4 = 0.0361*22.4 = 0.8086 L\). Hence the total volume \(V_t = V_{He} + V_{Ne} = 0.2896 + 0.8086 = 1.0982 L \)

Key Concepts

Partial Pressure

In a gas mixture, each individual gas contributes to the total pressure of the system. This contribution is known as the **partial pressure**. It represents the pressure that a gas would exert if it occupied the entire volume alone.

For example, in the given exercise, we have three different gases: Nitrogen (\(\mathrm{N}_2\)), Helium (\(\mathrm{He}\)), and Neon (\(\mathrm{Ne}\)). Their partial pressures are given as 0.32 atm, 0.15 atm, and 0.42 atm, respectively.

Calculating the total pressure in the mixture involves simply adding up these partial pressures since according to Dalton's Law of Partial Pressures:

• The total pressure (\(P_t\)) of the gas mixture is the sum of the partial pressures of each individual gas.
• In this case: \(P_t = 0.32 \text{ atm} + 0.15 \text{ atm} + 0.42 \text{ atm} = 0.89 \text{ atm}\).

This concept underscores the importance of understanding how gases behave independently within a mixture, yet contribute collectively to the overall pressure.

Gas Mixture

A **gas mixture** involves combining different gas molecules within a single container where they share the same volume but retain their individual characteristics.

No chemical reactions between the gases are considered, meaning each gas behaves independently whilst occupying the entire volume of the container.

For example, in our exercise, the mixture involves \(\mathrm{N}_2\), \(\mathrm{He}\), and \(\mathrm{Ne}\) gases. Each of these gases has its partial pressure, contributing to the total pressure of the mixture.

Understanding the characteristics of gas mixtures is crucial as it allows us to predict and calculate properties such as total pressure, as seen in the exercise, where each gas within the flask contributes to the total pressure, depicting the independent yet simultaneous behavior of each gas in a mixture.

STP (Standard Temperature and Pressure)

**STP**, or Standard Temperature and Pressure, is a common reference point used in chemistry to provide consistent conditions for comparing different gas behaviors.

At STP, the defined conditions are a temperature of 0°C (273.15 K) and a pressure of 1 atm.

Under these conditions, one mole of an ideal gas occupies 22.4 liters of volume.

In the given exercise, after calculating the moles of Helium and Neon present in the flask, we determined the volume these gases would occupy at STP. This involved multiplying the number of moles by 22.4 liters to find their STP volumes.

• For Helium, the volume is \(0.0129 \text{ mol} \times 22.4 \text{ L/mol} = 0.2896 \text{ L} \).
• For Neon, the volume is \(0.0361 \text{ mol} \times 22.4 \text{ L/mol} = 0.8086 \text{ L} \).

The concept of STP aids in simplifying complex gas calculations by offering standard references, making it easier to relate volumes and moles of gases under these conditions.