Question

A \(6.0-\mathrm{L}\) flask contains a mixture of methane \(\left(\mathrm{CH}_{4}\right),\) argon, and helium at \(45^{\circ} \mathrm{C}\) and \(1.75 \mathrm{~atm} .\) If the mole fractions of helium and argon are 0.25 and \(0.35,\) respectively, how many molecules of methane are present?

Short answer

The number of methane molecules is approximately \(9.655 \times 10^{22}\).

Step-by-step solution

Convert the Temperature to Kelvin

First, the temperature needs to be converted from Celsius to Kelvin. Use the formula \(T(K) = T(°C) + 273.15\). So, \(T = 45 + 273.15 = 318.15K\)

Calculate the Total Number of Moles in the Flask

Use the Ideal Gas Law, \(PV = nRT\) to calculate the total number of moles \(n\). Here, \(P = 1.75 \, atm\), \(V = 6.0 \, L\), \(R = 0.0821 \, atm \, L/(mol \, K)\), and \(T = 318.15 \). Substituting these: \[n = \frac{PV}{RT} = \frac{(1.75)(6.0)}{(0.0821)(318.15)} \approx 0.401 \, mol\]

Calculate Moles of Helium and Argon

Using mole fractions: Number of moles of Helium \(n_{He}\) = mole fraction of He \(\times\) total moles \[ n_{He} = 0.25 \times 0.401 \approx 0.10025 \, mol\] Similarly, number of moles of Argon \(n_{Ar}\) = mole fraction of Ar \(\times\) total moles \[ n_{Ar} = 0.35 \times 0.401 \approx 0.14035 \, mol\]

Calculate Moles of Methane

The mole fraction of methane \(x_{CH_4}\) can be found by subtracting the mole fractions of Helium and Argon from 1. \[ x_{CH_4} = 1 - (0.25 + 0.35) = 0.40 \] Number of moles of methane \(n_{CH_4}\) = mole fraction of CH\textsubscript{4} \(\times\) total moles \[ n_{CH_4} = 0.40 \times 0.401 \approx 0.1604 \, mol \]

Calculate Number of Methane Molecules

Convert moles of methane to number of molecules using Avogadro's number \(6.022 \times 10^{23} \). \[\text{Number of methane molecules} = n_{CH_4} \times 6.022 \times 10^{23} \approx 0.1604 \times 6.022 \times 10^{23} \approx 9.655 \times 10^{22} \]

Key concepts

Ideal Gas Law

The Ideal Gas Law is a key concept in chemistry that relates the pressure, volume, temperature, and number of moles of a gas. The equation is expressed as:
\[ PV = nRT \]
Here,

• P represents the pressure of the gas, measured in atmospheres (atm).
• V represents the volume of the gas, measured in liters (L).
• n is the number of moles of gas.
• R is the ideal gas constant, which has a value of 0.0821 atm·L/(mol·K).
• T represents the temperature of the gas, measured in Kelvin (K).

To find the number of moles of a gas in a flask, you need to know the pressure, volume, and temperature of the gas. By rearranging the equation to solve for n, you can determine the number of moles:
\[ n = \frac{PV}{RT} \]
In our exercise, this formula helped calculate the total moles of gas in a mixture.

Mole Fraction

Mole fraction is a way to express the concentration of a component in a mixture. It is calculated by dividing the number of moles of one component by the total number of moles of all components in the mixture. For any component i in a mixture, the mole fraction (x_i) is:
\[ x_i = \frac{n_i}{n_{total}} \]
Here,

• \(n_i\) is the number of moles of the component i.
• \(n_{total}\) is the total number of moles in the mixture.

Mole fractions are important because they describe the proportion of each gas in the mixture. In the given exercise, we used mole fractions to find the number of moles of helium and argon. By subtracting the mole fractions of helium and argon from 1, we calculated the mole fraction of methane. This step was crucial to determine the number of moles and subsequently the number of methane molecules.

Temperature Conversion

Temperature conversions are often needed in chemistry when using equations that require the temperature in Kelvin. The temperature in Celsius (°C) can be converted to Kelvin (K) using the following formula:
\[ T(K) = T(°C) + 273.15 \]
This formula is straightforward and helps ensure accurate calculations in gas law problems. For instance, in the exercise provided, the initial temperature was given in Celsius. By converting it to Kelvin using the formula, we obtained:
\[ T = 45°C + 273.15 = 318.15 K \]
Having the temperature in Kelvin allowed us to properly apply the Ideal Gas Law, ensuring the accuracy of our calculations. Remember, always make sure your temperature is in Kelvin when using the Ideal Gas Law.