Question

A 1.00 -L gas sample at \(100 .^{\circ} \mathrm{C}\) and 600 . torr contains 50.0\(\%\) helium and 50.0\(\%\) xenon by mass. What are the partial pressures of the individual gases?

Short answer

The partial pressures of the individual gases in the 1.00 L gas sample containing 50.0% helium and 50.0% xenon by mass at 100°C and 600 torr are approximately \( P_{He} \approx 0.383 atm \) for helium and \( P_{Xe} \approx 0.406 atm \) for xenon.

Step-by-step solution

Convert mass percentage to mass

First, we need to calculate the mass of each gas for the given mass percentages. We can assume a total mass of 100g for the mixture, which will make the calculation easier. Thus, 50% helium and 50% xenon would mean that there are 50g of helium and 50g of xenon in the mixture.

Calculate moles of each gas

Next, we'll calculate the moles of each gas using their molar masses. The molar mass of helium is approximately 4g/mol, and the molar mass of xenon is approximately 131g/mol. For helium: Moles = mass / molar mass Moles of helium = \( \frac{50 g}{4 g/mol} \) = 12.5 mol For xenon: Moles of xenon = \( \frac{50 g}{131 g/mol} \) = 0.381 mol

Use the ideal gas law to find individual pressures

Now, we'll use the ideal gas law to compute the partial pressures for each gas. The ideal gas law is given by: \( PV = nRT \) Where: P = pressure V = volume n = moles R = ideal gas constant (0.0821 L.atm/(mol.K) ) T = temperature in Kelvin First, we need to convert the given temperature to Kelvin: Temperature in Kelvin = 100°C + 273.15 = 373.15 K Also, we need to convert the total pressure from torr to atm: Pressure in atm = \( \frac{600 torr}{760 torr/atm} \) = 0.789 atm For helium: \( P_{He}V = n_{He}RT \) \( P_{He}(1.00 L)= (12.5 mol)(0.0821 L.atm/(mol.K))(373.15 K) \) \( P_{He} = \frac{(12.5 mol)(0.0821 L.atm/(mol.K))(373.15 K)}{1.00 L} \) \( P_{He} \approx 0.383 atm \) For xenon: \( P_{Xe}V = n_{Xe}RT \) \( P_{Xe}(1.00 L)= (0.381 mol)(0.0821 L.atm/(mol.K))(373.15 K) \) \( P_{Xe} = \frac{(0.381 mol)(0.0821 L.atm/(mol.K))(373.15 K)}{1.00 L} \) \( P_{Xe} \approx 0.406 atm \)

Find the final result

Now, we have the partial pressures for each gas. Partial Pressure of Helium: \( P_{He} \approx 0.383 atm \) Partial Pressure of Xenon: \( P_{Xe} \approx 0.406 atm \) These values are the partial pressures for the individual gases within the 1.00 L mixture at 100°C and 600 torr total pressure.

Key concepts

Partial Pressure

Partial pressure is the pressure exerted by an individual gas within a mixture of gases. It is essential to understand that in a mixture, gases behave independently, each contributing to the total pressure based on its proportion in the mixture.

When trying to find the partial pressure of a gas, you can use Dalton's Law of Partial Pressures, which states that the total pressure of a gas mixture is equal to the sum of the partial pressures of each individual gas. This can be written as:

• \(P_{total} = P_1 + P_2 + \ldots + P_n\)

In the exercise, you calculated the partial pressures of helium and xenon by using the Ideal Gas Law, \(PV = nRT\). Here, the 'P' on the left side represents the partial pressure of the gas you are solving for, given its molar quantity and other conditions like volume and temperature.

Moles Calculation

In any chemical calculation involving gases, finding the number of moles is fundamental. Moles quantify the amount of a substance. Here, it lets you link between mass measured in grams and the volume of space a gas occupies.

The calculation begins by using the molar mass of each gas. The formula to calculate moles is:

• \(\text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}}\)

In the problem, you find the moles of helium and xenon from their respective masses. With helium's molar mass of 4 g/mol and xenon's 131 g/mol, you calculated 12.5 moles of helium and 0.381 moles of xenon.

The number of moles will later help in finding the partial pressures using the Ideal Gas Law. It's a direct conversion of the mass information into a form useful for further calculations.

Conversion to Kelvin

Temperature conversion, particularly to Kelvin, is a critical step for gas law calculations. The Kelvin scale is an absolute temperature scale starting from absolute zero, where all molecular motion stops.

Converting Celsius to Kelvin is straightforward, as Kelvin = Celsius + 273.15. So, for the given temperature of 100 °C in the problem, adding 273.15 gives you 373.15 K.

This conversion is necessary because the Ideal Gas Law, \(PV = nRT\), requires temperature measured in Kelvin to correctly relate pressure, volume, and the number of moles. The Kelvin scale ensures that calculations based on gas molecules' kinetic energy and speed are accurate, as the scale avoids negative temperatures.