A gas `(1 g)` at `4` bar pressure. If we add `2 gm` of gas `B` then the total pressure inside the container is `6` bar. Which of the following is true?

Options

A. `M_(A)=2M_B`
B. `M_(B)=2M_A`
C. `M_A=4M_B`
D. `M_B=4M_A`

Explanation

To solve the problem step by step, we will use the ideal gas law principles and the relationship between pressure, volume, and the number of moles of gas.

### Step 1: Understand the initial conditions
We have:
- Gas A: 1 g at 4 bar pressure.
- Gas B: 2 g is added, resulting in a total pressure of 6 bar.

### Step 2: Use the relationship between moles and pressure
Since the volume and temperature are constant, we can use the relationship:
\[ \frac{n_1}{p_1} = \frac{n_2}{p_2} \]
Where:
- \( n_1 \) = number of moles of gas A initially
- \( p_1 \) = initial pressure (4 bar)
- \( n_2 \) = total number of moles of gas A and gas B after adding gas B
- \( p_2 \) = total pressure (6 bar)

### Step 3: Calculate the number of moles
The number of moles can be calculated using the formula:
\[ n = \frac{\text{mass}}{\text{molar mass}} \]
Let \( M_A \) be the molar mass of gas A and \( M_B \) be the molar mass of gas B.

For gas A:
\[ n_1 = \frac{1 \text{ g}}{M_A} \]

For gas B (after adding 2 g):
\[ n_B = \frac{2 \text{ g}}{M_B} \]

Thus, the total number of moles after adding gas B is:
\[ n_2 = n_1 + n_B = \frac{1}{M_A} + \frac{2}{M_B} \]

### Step 4: Set up the equation using pressures
Substituting into the pressure relationship:
\[ \frac{\frac{1}{M_A}}{4} = \frac{\frac{1}{M_A} + \frac{2}{M_B}}{6} \]

### Step 5: Cross-multiply and simplify
Cross-multiplying gives:
\[ 6 \cdot \frac{1}{M_A} = 4 \left( \frac{1}{M_A} + \frac{2}{M_B} \right) \]
Expanding this:
\[ 6 \cdot \frac{1}{M_A} = \frac{4}{M_A} + \frac{8}{M_B} \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ 6 \cdot \frac{1}{M_A} - \frac{4}{M_A} = \frac{8}{M_B} \]
This simplifies to:
\[ \frac{2}{M_A} = \frac{8}{M_B} \]

### Step 7: Solve for the relationship between molar masses
Cross-multiplying gives:
\[ 2M_B = 8M_A \]
Thus:
\[ M_B = 4M_A \]

### Conclusion
The correct statement is that the molar mass of gas B is four times the molar mass of gas A.