Question

A \(2.5-\mathrm{L}\) flask at \(15^{\circ} \mathrm{C}\) contains a mixture of three gases, \(\mathrm{N}_{2},\) He, and \(\mathrm{Ne},\) at partial pressures of \(0.32 \mathrm{~atm}\) for \(\mathrm{N}_{2}, 0.15 \mathrm{~atm}\) for \(\mathrm{He},\) and \(0.42 \mathrm{~atm}\) 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

The total pressure of the gas mixture is \(0.89~atm\) and the volume of the He and Ne gases at STP (after removal of \(\mathrm{N}_{2}\)) is approximately \(2.27~L\).

Step-by-step solution

Calculate Total Pressure

The total pressure of a mixture of gases is the sum of the individual pressures of each gas. This principle is known as Dalton’s law. Simply add up the given pressures of the three gases: \(P_{total} = P_{N2} + P_{He} + P_{Ne} = 0.32 atm + 0.15 atm + 0.42 atm\).

Calculate Volume at Standard Temperature and Pressure (STP)

Now, you want to calculate the volume the He and Ne gases occupy at STP when you remove the \(\mathrm{N}_{2}\) gas. We use \(STP = 273.15 K\) and \(1 atm\). The combined pressure of He and Ne is \(P_{He} + P_{Ne} = 0.15 atm + 0.42 atm\). The volume these gases occupy at STP can be found using the Ideal Gas law, formulated as \(V = nRT/P\). Here, you need to find the number of moles (n) of He and Ne using the given volume and temperature, then recalculate its volume at STP using the partial pressure of only He and Ne. As we know, \(n = PV/RT\). Then put the computed value into the Ideal Gas law to find the volume.

Converting the Temperature to Kelvin

Temperature must be in Kelvin when working with gas laws. So, convert the Celsius temperature to Kelvin by adding 273.15. \(T = 15^{\circ}C + 273.15 = 288.15 K\).

Calculating the number of moles of He and Ne

Calculate the number of moles \(n\) of He and Ne in the flask using the Ideal Gas Law in the format of \(n = PV/RT\). The values should be put in as follows: \(n = 0.57~atm * 2.5~L / (0.08206~L*atm/(mol*K) * 288.15~K)\).

Calculate the Volume of Ne and He at STP

After finding the number of moles, place it in the Ideal Gas Law as we knew from step 2 formulated as \(V = nRT/P\). Replacing the values, we get \(V = n * 0.08206 L*atm/(mol*K) * 273.15~K / 1 atm\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry represented as \(PV = nRT\), where

• P stands for pressure
• V is volume
• n represents the number of moles
• R is the universal gas constant, which is approximately 0.08206 L⋅atm/mol⋅K
• T denotes temperature

This equation helps us calculate properties like the volume, pressure, or temperature of a gas under certain conditions. To effectively use the Ideal Gas Law, temperature needs to be in Kelvin, which can be found by adding 273.15 to the Celsius temperature.
Understanding this formula allows us to analyze how gas behaves under different temperature and pressure conditions, making it crucial for solving gas mixture problems. In calculations where we need the volume of a gas at different conditions, such as shifting to Standard Temperature and Pressure, the Ideal Gas Law becomes a key tool.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a reference point in gas calculations. It is defined as a temperature of 273.15 K (0°C) and a pressure of 1 atm.
Using STP simplifies comparing gas behaviors since it provides a consistent baseline: under these conditions, 1 mole of an ideal gas will occupy approximately 22.4 liters of volume.
Applying this concept to a gas mixture involves recalculating the volume or other properties of gases as if they're at STP. For instance, if you are given a gas volume at a different temperature and pressure, you could use the STP standard to predict the volume changes by transferring using the Ideal Gas Law. This concept is particularly helpful when evaluating partial components in a gas mixture, as it allows us to model how each gas would behave separately at this standardized condition.

Gas Mixture Calculations

Gas Mixture Calculations often involve the combination of gases and the measurement of their total pressure and volume under certain conditions. Dalton's Law of Partial Pressures is crucial here. It states that the total pressure of a mixture of non-reacting gases is equal to the sum of the partial pressures of each individual gas.

• To find the total pressure: sum up the pressures of all the gases present in the mixture.
• If one gas is removed, like the nitrogen in the original problem, you must recalculate using only the remaining gases.
• Considering individual gases, their behaviors can be analyzed using the Ideal Gas Law by considering their initial condition volumes and pressures.

When dealing with gas mixtures, the complexity can increase but understanding Dalton’s Law along with the Ideal Gas Law makes tackling these problems manageable. For removing a gas in a mixture, compute the new total and use it alongside the temperatures to find how much volume each remaining gas would occupy if set to a different condition such as STP. This technique offers insight into individual gas contributions and overall mixture behavior.