2gm of hydrogen is present in a closed vessel at S.T.P. If the same quantity of another gas 'X' when introduced into the vessel the pressure becomes 1.5 atm. The gas 'X' would be

Options

A. `CH_(4)`
B. `SO_(2)`
C. He
D. `O_(2)`

Answer

To solve the problem, we need to determine the identity of gas 'X' based on the information provided. Here’s a step-by-step solution:

### Step 1: Calculate the number of moles of hydrogen
Given:
- Mass of hydrogen (H₂) = 2 g
- Molar mass of hydrogen (H₂) = 2 g/mol

Using the formula for number of moles:
\[ \text{Number of moles} = \frac{\text{Mass}}{\text{Molar mass}} = \frac{2 \text{ g}}{2 \text{ g/mol}} = 1 \text{ mole} \]

### Step 2: Use the ideal gas law to find the volume of the vessel
At STP (Standard Temperature and Pressure):
- Pressure (P) = 1 atm
- Temperature (T) = 273 K
- Universal gas constant (R) = 0.0821 L·atm/(K·mol)

Using the ideal gas equation:
\[ PV = nRT \]
Substituting the known values:
\[ 1 \text{ atm} \cdot V = 1 \text{ mole} \cdot 0.0821 \text{ L·atm/(K·mol)} \cdot 273 \text{ K} \]
Calculating the volume (V):
\[ V = 1 \cdot 0.0821 \cdot 273 = 22.414 \text{ L} \]

### Step 3: Determine the pressure exerted by gas 'X'
When gas 'X' is introduced, the total pressure in the vessel becomes 1.5 atm. The pressure exerted by hydrogen is 1 atm, so the pressure exerted by gas 'X' is:
\[ P_X = 1.5 \text{ atm} - 1 \text{ atm} = 0.5 \text{ atm} \]

### Step 4: Calculate the number of moles of gas 'X'
Using the ideal gas law again for gas 'X':
\[ P_X V = n_X RT \]
Where \( n_X \) is the number of moles of gas 'X'. Rearranging gives:
\[ n_X = \frac{P_X V}{RT} \]
Substituting the values:
\[ n_X = \frac{0.5 \text{ atm} \cdot 22.414 \text{ L}}{0.0821 \text{ L·atm/(K·mol)} \cdot 273 \text{ K}} \]
Calculating \( n_X \):
\[ n_X = \frac{11.207}{22.414} = 0.5 \text{ moles} \]

### Step 5: Calculate the molar mass of gas 'X'
Given that the mass of gas 'X' is also 2 g, we can find its molar mass (M):
\[ M = \frac{\text{Mass}}{\text{Number of moles}} = \frac{2 \text{ g}}{0.5 \text{ moles}} = 4 \text{ g/mol} \]

### Step 6: Identify gas 'X'
The gas with a molar mass of 4 g/mol is helium (He).

### Final Answer:
Gas 'X' is **helium (He)**.

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