Question

A cylindrical container of volume 4.0\( \times \)10\(^{-3}\) m\(^3\) contains one mole of hydrogen and two moles of carbon dioxide. Assume the temperature of the mixture is 400 K. The pressure of the mixture of gases is :
[Take gas constant as 8.3 J mol\(^{-1}\)K\(^{-1}\)]

• A. 24.9\( \times \)10\(^5\) Pa
• B. 24.9\( \times \)10\(^3\) Pa
• C. 24.9 Pa
• D. 249\( \times \)10\(^1\) Pa

Hint:

For a mixture of non-reacting ideal gases, the ideal gas law applies to the mixture as a whole. Simply use the total number of moles (\(n_{total}\)) in the equation \(PV=nRT\) to find the total pressure.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We have a mixture of two gases (hydrogen and carbon dioxide) in a container of known volume and temperature. We need to calculate the total pressure of the gas mixture.
Step 2: Key Formula or Approach:
We can treat the gas mixture as an ideal gas. The ideal gas law is given by \(PV = nRT\). For a mixture of gases, `n` represents the total number of moles of all gases in the mixture. This is based on Dalton's Law of partial pressures.
Step 3: Detailed Explanation:
Given values:
Volume, \(V = 4.0 \times 10^{-3}\) m\(^3\).
Temperature, \(T = 400\) K.
Gas constant, \(R = 8.3\) J mol\(^{-1}\)K\(^{-1}\).
Number of moles of hydrogen, \(n_{H_2} = 1\) mole.
Number of moles of carbon dioxide, \(n_{CO_2} = 2\) moles.
First, calculate the total number of moles in the mixture:
\[ n_{total} = n_{H_2} + n_{CO_2} = 1 + 2 = 3 \text{ moles} \] Now, apply the ideal gas law to the mixture:
\[ P_{total}V = n_{total}RT \] Rearrange to solve for the total pressure, \(P_{total}\):
\[ P_{total} = \frac{n_{total}RT}{V} \] Substitute the known values:
\[ P_{total} = \frac{(3 \text{ mol}) \times (8.3 \text{ J mol}^{-1}\text{K}^{-1}) \times (400 \text{ K})}{4.0 \times 10^{-3} \text{ m}^3} \] \[ P_{total} = \frac{3 \times 8.3 \times 400}{4.0 \times 10^{-3}} \] \[ P_{total} = \frac{3 \times 8.3 \times 100}{10^{-3}} = 24.9 \times 100 \times 10^3 \] \[ P_{total} = 24.9 \times 10^5 \text{ Pa} \] Step 4: Final Answer:
The pressure of the mixture of gases is 24.9\( \times \)10\(^5\) Pa. This corresponds to option (A).