The volume of vessel A is twice the volume of another vessel B and both of them are filled with the same gas. If the gas in A is at twice the temperature and twice the pressure in comparison to the gas in B, what is the ratio of number of gas molecule in A and B ?

Options

A. `1/2`
B. `2/1`
C. `3/2`
D. `2/3`

Explanation

To solve the problem, we will use the Ideal Gas Law, which states that:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles (or number of gas molecules)
- \( R \) = Universal gas constant
- \( T \) = Temperature

### Step-by-Step Solution:

1. **Define the Variables:**
- Let the volume of vessel B be \( V \).
- Therefore, the volume of vessel A is \( 2V \) (since it is twice the volume of B).
- Let the pressure of gas in vessel B be \( P \).
- Therefore, the pressure of gas in vessel A is \( 2P \) (since it is twice the pressure of B).
- Let the temperature of gas in vessel B be \( T \).
- Therefore, the temperature of gas in vessel A is \( 2T \) (since it is twice the temperature of B).

2. **Apply the Ideal Gas Law for Vessel A:**
- For vessel A, we can write:
\[ P_A V_A = n_A R T_A \]
Substituting the values:
\[ (2P)(2V) = n_A R (2T) \]
Simplifying this:
\[ 4PV = 2n_A RT \]
Rearranging gives:
\[ n_A = \frac{4PV}{2RT} = \frac{2PV}{RT} \]

3. **Apply the Ideal Gas Law for Vessel B:**
- For vessel B, we can write:
\[ P_B V_B = n_B R T_B \]
Substituting the values:
\[ P(V) = n_B R(T) \]
Rearranging gives:
\[ n_B = \frac{PV}{RT} \]

4. **Find the Ratio of Number of Gas Molecules:**
- Now, we can find the ratio of the number of gas molecules in A and B:
\[ \frac{n_A}{n_B} = \frac{\frac{2PV}{RT}}{\frac{PV}{RT}} = \frac{2PV}{PV} = 2 \]

5. **Conclusion:**
- Therefore, the ratio of the number of gas molecules in vessel A to vessel B is:
\[ \frac{n_A}{n_B} = 2:1 \]

### Final Answer:
The ratio of the number of gas molecules in A and B is \( 2:1 \).

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