Containers A and B have same gases. Pressure, volume and temperature of A are all twice that of B, then the ratio of number of molecules of A and B are

Options

A. `1:2`
B. 2
C. `1:4`
D. 4

Answer

To solve the problem, we will use the Ideal Gas Law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles of the gas
- \( R \) = Ideal gas constant
- \( T \) = Temperature

### Step-by-Step Solution:

1. **Identify the Variables for Both Containers:**
- For Container A:
- Pressure \( P_A = 2P \)
- Volume \( V_A = 2V \)
- Temperature \( T_A = 2T \)
- For Container B:
- Pressure \( P_B = P \)
- Volume \( V_B = V \)
- Temperature \( T_B = T \)

2. **Apply the Ideal Gas Law for Both Containers:**
- For Container A:
\[ n_A = \frac{P_A V_A}{RT_A} = \frac{(2P)(2V)}{R(2T)} = \frac{4PV}{2RT} = \frac{2PV}{RT} \]
- For Container B:
\[ n_B = \frac{P_B V_B}{RT_B} = \frac{PV}{RT} \]

3. **Calculate the Ratio of Moles:**
- Now, we can find the ratio of the number of moles \( n_A \) to \( n_B \):
\[ \frac{n_A}{n_B} = \frac{\frac{2PV}{RT}}{\frac{PV}{RT}} = \frac{2PV}{RT} \cdot \frac{RT}{PV} = 2 \]

4. **Conclusion on the Ratio of Number of Molecules:**
- Since the number of molecules is directly proportional to the number of moles, we can conclude that:
\[ \text{Ratio of number of molecules of A to B} = \frac{n_A}{n_B} = 2:1 \]

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

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