Two vessels A and B contain the same gas. If the pressure, volume and absolute temperature of the gas in A are two times as compared to that in B, and if the mass of the gas in B is x grams, the mass of the gas in A will be

Options

A. 4x g
B. `x/2` g
C. 2x g
D. x g

Answer

To solve the problem, we need to use the ideal gas law and the relationships between pressure, volume, temperature, and the number of moles of gas. Let's break it down step by step.

### Step 1: Understand the Ideal Gas Law
The ideal gas law is given by the equation:
\[ PV = nRT \]
where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature

### Step 2: Define Variables for Both Vessels
Let’s denote the parameters for vessel B as:
- Pressure \( P \)
- Volume \( V \)
- Temperature \( T \)
- Mass of gas in B = \( x \) grams

For vessel A, the parameters are:
- Pressure = \( 2P \)
- Volume = \( 2V \)
- Temperature = \( 2T \)

### Step 3: Calculate Moles of Gas in Vessel B
Using the ideal gas law for vessel B:
\[ n_B = \frac{PV}{RT} \]
Since the mass of gas in B is \( x \) grams, we can relate the number of moles to mass:
\[ n_B = \frac{x}{m} \]
where \( m \) is the molar mass of the gas.

### Step 4: Calculate Moles of Gas in Vessel A
Using the ideal gas law for vessel A:
\[ n_A = \frac{(2P)(2V)}{R(2T)} \]
This simplifies to:
\[ n_A = \frac{4PV}{2RT} = \frac{2PV}{RT} \]

### Step 5: Relate Moles of Gas in A to Mass
Now, substituting the expression for \( n_B \) into the equation for \( n_A \):
\[ n_A = \frac{2PV}{RT} = 2 \left(\frac{PV}{RT}\right) = 2n_B \]
Thus, we can express the mass of gas in A as:
\[ n_A = \frac{y}{m} \]
where \( y \) is the mass of gas in A.

### Step 6: Set Up the Equation
Since \( n_A = 2n_B \), we have:
\[ \frac{y}{m} = 2 \left(\frac{x}{m}\right) \]

### Step 7: Solve for Mass in Vessel A
Cancelling \( m \) from both sides:
\[ y = 2x \]

### Conclusion
The mass of the gas in vessel A is \( 2x \) grams.

### Final Answer
The mass of the gas in vessel A will be \( 2x \) grams.

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