Problem 112 - Chemistry: The Central Science

Question

A glass vessel fitted with a stopcock valve has a mass of \(337.428 \mathrm{~g}\) when evacuated. When filled with Ar, it has a mass of \(339.854 \mathrm{~g}\). When evacuated and refilled with a mixture of Ne and Ar, under the same conditions of temperature and pressure, it has a mass of \(339.076 \mathrm{~g}\). What is the mole percent of Ne in the gas mixture?

Step-by-Step Solution

Answer

The mole percent of Ne in the gas mixture is approximately 25.6%.

Step 1: Calculate the mass of Ar and the mixture of Ne and Ar

First, we need to find the mass of Ar and the mixture of Ne and Ar in the vessel. Subtract the mass of the evacuated vessel from the mass of the vessel filled with Ar and the mixture. Mass of Ar = Mass when filled with Ar - Mass of evacuated vessel Mass of Ar = \(339.854 \mathrm{~g} - 337.428 \mathrm{~g} = 2.426 \mathrm{~g}\) Mass of the mixture (Ne and Ar) = Mass when filled with the mixture - Mass of evacuated vessel Mass of the mixture (Ne and Ar) = \(339.076 \mathrm{~g} - 337.428 \mathrm{~g} = 1.648 \mathrm{~g}\)

Step 2: Calculate the number of moles of Ar and the mixture of Ne and Ar

Now, we need to find the number of moles of Ar and the mixture of Ne and Ar using their respective molar masses: For Ar, Molar mass of Ar = \(39.948 \mathrm{~g/mol}\) Number of moles of Ar = \(\frac{mass}{molar~mass}\) Number of moles of Ar = \(\frac{2.426 \mathrm{~g}}{39.948 \mathrm{~g/mol}} = 0.0607 \mathrm{~mol}\) For the mixture (Ne and Ar), Assuming x moles of Ne with a molar mass of \(20.18 \mathrm{~g/mol}\) and (0.0607 - x) moles of Ar, Mass of the mixture (Ne and Ar) = \(x \times 20.18 + (0.0607 - x) \times 39.948\)

Step 3: Solve for the number of moles of Ne (x)

We know the mass of the mixture, so let's solve the equation to find the number of moles of Ne (x). \(1.648 \mathrm{~g} = x \times 20.18 + (0.0607 - x) \times 39.948\) Rearrange the equation and solve for x. \(x (39.948 - 20.18) = 1.648 - (0.0607 \times 39.948)\) \(x = \frac{1.648 - (0.0607 \times 39.948)}{19.768} \approx 0.0209 \mathrm{~mol}\)

Step 4: Calculate the mole fraction and mole percent of Ne in the mixture

To find the mole fraction of Ne, divide the moles of Ne by the total moles in the mixture (Ne and Ar): Mole fraction of Ne = \(\frac{moles \thinspace of \thinspace Ne}{moles \thinspace of \thinspace Ne + moles \thinspace of \thinspace Ar}\) Mole fraction of Ne = \(\frac{0.0209 \mathrm{~mol}}{0.0209 \mathrm{~mol} + 0.0607 \mathrm{~mol}} \approx 0.256\) Now, to convert the mole fraction to mole percent: Mole percent of Ne = Mole fraction of Ne \(\times\) 100% Mole percent of Ne = \(0.256 \times 100\% \approx 25.6\% \) Therefore, the mole percent of Ne in the gas mixture is approximately 25.6%.

Key Concepts

Molar Mass

Molar mass is a vital concept in chemistry that helps determine the mass of a given number of moles of a substance. It is expressed in grams per mole (g/mol). Each element has its unique molar mass, which can be found on the periodic table. For instance, the molar mass of argon (Ar) is 39.948 g/mol, while neon (Ne) has a molar mass of 20.18 g/mol.
Understanding molar mass is key when converting from mass to moles, using the formula:

• \[ \text{Number of Moles} = \frac{\text{mass}}{\text{molar mass}} \]

This conversion is crucial in solving stoichiometry problems where molar ratios are applied. By knowing the mass and molar mass, students can easily calculate the number of moles present in a sample, which is essential when dealing with chemical reactions and mixtures.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of one component to the total number of moles in the mixture. This concept is particularly useful in dealing with gas mixtures because it provides a way to quantify the composition of each gas present without the effects of temperature and pressure.
The formula for mole fraction (\(x\)) is:

• \[\text{Mole Fraction} = \frac{\text{moles of component}}{\text{total moles in mixture}}\]

In the example exercise, the mole fraction of Ne is calculated to be 0.256. This means that 25.6% of the gas mixture is composed of Ne, as demonstrated by the conversion from mole fraction to mole percent.

Ideal Gas Law

The Ideal Gas Law is an equation of state for a hypothetical gas called an "ideal gas." It is a good approximation for the behavior of many gases under a variety of conditions, although it fails at high pressures and low temperatures. The law combines several empirical laws, including Boyle's, Charles's, and Avogadro's laws into a single equation:

• \[PV = nRT\]

Here, \(P\) is the pressure of the gas, \(V\) is its volume, \(n\) is the number of moles, \(R\) is the universal gas constant, and \(T\) is the temperature in Kelvin.
This equation allows you to relate the physical properties of a gas, and it's instrumental when dealing with situations involving gas mixtures, as it provides the means to calculate changes in states and compositions.

Gas Mixtures

A gas mixture is a combination of two or more gases that do not react chemically with one another. Common examples include air, hydrogen and oxygen mixtures, and, as presented in the exercise, a mixture of neon and argon. In chemistry, gas mixtures are treated as single entities that behave like a single gas, as long as the gases do not interact chemically.
Because gases in a mixture contribute equally to the total pressure, the partial pressure of each gas can be helpful for determining its mole fraction through Dalton's Law of Partial Pressures.
This situation reflects our example exercise, where mixing Ne with Ar doesn’t alter their chemical properties. The mole percent of each gas can be found as long as one knows the total mass and the masses of the individual components. Calculating these concentrations helps when you need to know the precise composition needed for processes like gas reactions and phase changes.