Question

A 100.-L flask contains a mixture of methane \(\left(\mathrm{CH}_{4}\right)\) and argon gases at \(25^{\circ} \mathrm{C}\). The mass of argon present is \(228 \mathrm{~g}\) and the mole fraction of methane in the mixture is \(0.650 .\) Calculate the total kinetic energy of the gaseous mixture.

Short answer

The total kinetic energy of the gaseous mixture containing methane and argon is approximately 11,937.8 Joules.

Step-by-step solution

Calculate the moles of argon

Using the Ideal Gas Law \(PV = nRT\), we can find the moles of argon. We're given the mass of argon, the volume of the flask, and the temperature, but we don't have the pressure. However, since the mole fraction of methane is given, we can find the total moles and use it to back-calculate the moles of argon. We know: Mass of Argon = 228 g Molecular weight of Argon = 39.95 g/mol Volume = 100 L Temperature = 25°C = 298.15 K Mole fraction of methane = 0.650 R (Gas Constant) = 0.0821 L atm / (K mol) First, let's find the moles of argon: Moles of argon = Mass of argon / Molecular weight of argon Moles of argon = \( \frac{228}{39.95} \) Moles of argon = 5.707 mol

Calculate the moles of methane

Using the mole fraction equation and the moles of argon, we can find the moles of methane: Mole fraction of methane = 0.650 = \(\frac{n_{CH_4}}{n_{CH_4}+n_{Ar}}\) 0.650 = \( \frac{n_{CH_4}}{n_{CH_4}+5.707} \) Now, solve for the moles of methane: \(n_{CH_4} = \frac{0.650 \times (n_{CH_4}+5.707)}{1-0.650}\) \(n_{CH_4} = 10.595~mol\)

Calculate the total kinetic energy

To find the total kinetic energy of the gaseous mixture, we sum up the individual kinetic energies of methane and argon, using the formula: \(KE = \frac{3}{2}nRT\) Calculate the kinetic energy of methane: \(KE_{CH_4} = \frac{3}{2} \times 10.595 \times 0.0821 \times 298.15\) \(KE_{CH_4} = 7789.7~J\) Calculate the kinetic energy of argon: \(KE_{Ar} = \frac{3}{2} \times 5.707 \times 0.0821 \times 298.15\) \(KE_{Ar} = 4148.1~J\) Now add these two kinetic energies for the total kinetic energy of the gaseous mixture: Total kinetic energy = \(KE_{CH_4} + KE_{Ar}\) Total kinetic energy = \(7789.7+4148.1\) Total kinetic energy = \(11937.8~J\) The total kinetic energy of the gaseous mixture is approximately 11,937.8 Joules.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry that describes how gases behave under various conditions. It is expressed as \( PV = nRT \), where

• \( P \) is the pressure of the gas
• \( V \) is the volume
• \( n \) is the number of moles
• \( R \) is the gas constant
• \( T \) is the absolute temperature in Kelvin

In this problem, knowing the mass of argon, its molecular weight, and the conditions of the flask allows you to find the number of moles of argon. From here, you can use the mole fraction and total moles in the mixture to find the moles of methane. Remember that the Ideal Gas Law makes an important assumption that gas molecules do not interact and occupy no volume themselves.

Mole Fraction

Mole fraction is a way of expressing the concentration of a component in a mixture. It is the ratio of the moles of one component to the total moles in the mixture. Mathematically, the mole fraction \( X_A \) of a component \( A \) is given by:\[ X_A = \frac{n_A}{n_{total}} \]where \( n_A \) is the number of moles of the component \( A \), and \( n_{total} \) is the total number of moles of all components in the mixture.
In the exercise, you know the mole fraction of methane to be 0.650, which helps us determine the moles of methane once we have the moles of argon. Using the relationship \[ X_{CH_4} = \frac{n_{CH_4}}{n_{CH_4} + n_{Ar}} \]allows you to solve for the moles of methane after plugging in the known values.

Gas Constant

The gas constant \( R \) is a crucial factor in the Ideal Gas Law. It links the various units used for pressure, volume, and temperature to the quantity of gas in moles.
Its usual value is \( 0.0821 \, \text{L atm} / (\text{K mol}) \), but it can also appear in other units, depending on the context and system you're working in.

• Remember that \( R \) is fundamental for calculations involving gases at standard or described conditions.
• In kinetic energy calculations like those shown in the exercise, \( R \) works seamlessly with Kelvin temperatures to assist in energy computations.

Using \( R \) accurately is pivotal as it ensures all parts of your gas calculations, such as kinetic energy, align correctly with the physical conditions and laws being applied.

Molecular Weight

Molecular weight, also known as molar mass, is the weight of one mole of a substance and is typically measured in grams per mole (g/mol). Each element has its specific molecular weight, which helps determine how much one mole weighs.
The molecular weight of argon is 39.95 g/mol, which has been used in this exercise to find the number of moles of argon from its given mass.

• Knowing the molecular weight allows for conversion from mass to moles and vice versa, using the formula: \( \text{Moles} = \frac{\text{Mass (g)}}{\text{Molecular Weight (g/mol)}} \).
• This concept is especially useful in stoichiometry and other calculations where proper scaling of quantities is crucial.

Understanding the molecular weight helps bridge the gap between the macroscopic world (mass you can measure) and the microscopic world (moles of particles, molecules, or atoms).