Question

A mixture of carbon dioxide and oxygen has volume 8310 cm³, temperature 300 K, pressure 100 kPa and mass 13.2 g. The number of moles of carbon dioxide and oxygen gases in the mixture respectively are _______.

• A. 0.15 and 0.18
• B. 0.25 and 0.08
• C. 0.21 and 0.12
• D. 0.13 and 0.20

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We can treat the gas mixture as a single ideal gas to find the total number of moles in the container.
Once the total moles are found, we can use the total mass and the molar masses of the constituent gases to set up a system of linear equations and solve for the individual moles.
Step 2: Key Formula or Approach:
Ideal Gas Law: \(PV = n_{total} RT\).
Total mass equation: \(m_{total} = n_1 M_1 + n_2 M_2\).
Total moles equation: \(n_{total} = n_1 + n_2\).
Molar mass of CO\(_2\) (\(M_1\)) = \(44 \text{ g/mol}\).
Molar mass of O\(_2\) (\(M_2\)) = \(32 \text{ g/mol}\).
Step 3: Detailed Explanation:
Convert given values to standard SI units:
\(P = 100 \text{ kPa} = 10^5 \text{ Pa}\).
\(V = 8310 \text{ cm}^3 = 8310 \times 10^{-6} \text{ m}^3 = 8.31 \times 10^{-3} \text{ m}^3\).
\(T = 300 \text{ K}\).
Calculate the total moles using the Ideal Gas Law:
\[ n_{total} = \frac{PV}{RT} = \frac{10^5 \times 8.31 \times 10^{-3}}{8.31 \times 300} \] \[ n_{total} = \frac{100 \times 8.31}{8.31 \times 300} = \frac{100}{300} = \frac{1}{3} \text{ moles} \] Let \(n_1\) be the moles of CO\(_2\) and \(n_2\) be the moles of O\(_2\).
We have two equations:
1) \(n_1 + n_2 = \frac{1}{3} \implies n_2 = \frac{1}{3} - n_1\)
2) \(44n_1 + 32n_2 = 13.2\)
Substitute the first equation into the second:
\[ 44n_1 + 32\left(\frac{1}{3} - n_1\right) = 13.2 \] \[ 44n_1 + \frac{32}{3} - 32n_1 = 13.2 \] \[ 12n_1 = 13.2 - \frac{32}{3} \] Multiply by 3 to clear the fraction:
\[ 36n_1 = 39.6 - 32 = 7.6 \] \[ n_1 = \frac{7.6}{36} = \frac{76}{360} = \frac{19}{90} \approx 0.211 \text{ moles} \] Now find \(n_2\):
\[ n_2 = \frac{1}{3} - \frac{19}{90} = \frac{30}{90} - \frac{19}{90} = \frac{11}{90} \approx 0.122 \text{ moles} \] The moles are approximately \(0.21\) and \(0.12\).
Step 4: Final Answer:
The number of moles of carbon dioxide and oxygen are \(0.21 \text{ and } 0.12\).