Question

At the same pressure, the volume occupied by $ 5.6 \, \text{g} $ of gas "A" at $ 610 \, \text{K} $ is the same as $ 1 \, \text{g} $ of $ \text{H}_2 $ at $ 243.9 \, \text{K} $. What is the molar mass (in $ \text{g mol}^{-1} $) of "A"? (Assume that gas "A" and $ \text{H}_2 $ are ideal gases) ($ M_{\text{H}_2} = 2 \, \text{u} $)

• A. $ 56 $
• B. $ 28 $
• C. $ 44 $
• D. $ 60 $

Hint:

When comparing volumes of ideal gases at the same pressure, use the relationship $ \frac{n_1}{T_1} = \frac{n_2}{T_2} $ to relate the masses and molar masses.

Answer 1

The Correct Option is B

\textbf{Step 1: Recall the Ideal Gas Law} The ideal gas law is given by: $$ PV = nRT $$ where:
$ P $ is the pressure,
$ V $ is the volume,
$ n $ is the number of moles,
$ R $ is the universal gas constant,
$ T $ is the temperature.
Since the volumes of the two gases are equal at the same pressure, we can use the relationship: $$ \frac{n_1}{T_1} = \frac{n_2}{T_2} $$ where:
$ n_1 $ and $ T_1 $ are the number of moles and temperature of gas "A",
$ n_2 $ and $ T_2 $ are the number of moles and temperature of $
\text{H}_2 $. Step 2: Express Moles in Terms of Mass and Molar Mass
For gas "A":
Mass of gas "A": $ m_A = 5.6 \, \text{g} $
Molar mass of gas "A": $ M_A $ (to be determined)
Number of moles of gas "A":
$$ n_A = \frac{m_A}{M_A} = \frac{5.6}{M_A} $$ For $ \text{H}_2 $:
Mass of $ \text{H}_2 $: $ m_{\text{H}_2} = 1 \, \text{g} $
Molar mass of $ \text{H}_2 $: $ M_{\text{H}_2} = 2 \, \text{g mol}^{-1} $ Number of moles of $ \text{H}_2 $:
$$ n_{\text{H}_2} = \frac{m_{\text{H}_2}}{M_{\text{H}_2}} = \frac{1}{2} $$ Step 3: Apply the Volume Relationship
Using the relationship $ \frac{n_1}{T_1} = \frac{n_2}{T_2} $: $$ \frac{n_A}{T_A} = \frac{n_{\text{H}_2}}{T_{\text{H}_2}} $$ Substitute the known values: $$ \frac{\frac{5.6}{M_A}}{610} = \frac{\frac{1}{2}}{243.9} $$ Simplify: $$ \frac{5.6}{M_A \cdot 610} = \frac{1}{2 \cdot 243.9} $$ Cross-multiply: $$ 5.6 \cdot 2 \cdot 243.9 = M_A \cdot 610 $$ Solve for $ M_A $: $$ M_A = \frac{5.6 \cdot 2 \cdot 243.9}{610} $$ $$ M_A = \frac{2753.76}{610} \approx 28 \, \text{g mol}^{-1} $$ Step 4: Analyze the Options
Option (1): $ 56 $
Incorrect — does not match the calculated value. Option (2): $ 28 $
Correct — matches the calculated value. Option (3): $ 44 $
Incorrect — does not match the calculated value. Option (4): $ 60 $
Incorrect — does not match the calculated value.