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

Answer

To solve the problem of finding the ratio of the number of molecules in containers A and B, we can follow these steps:

### Step 1: Identify the Variables
Let:
- Pressure of container A = \( P_A = 2P \)
- Volume of container A = \( V_A = 2V \)
- Temperature of container A = \( T_A = 2T \)

For container B:
- Pressure of container B = \( P_B = P \)
- Volume of container B = \( V_B = V \)
- Temperature of container B = \( T_B = T \)

### Step 2: Use the Ideal Gas Law
The Ideal Gas Law is given by:
\[ PV = nRT \]
Where:
- \( P \) = Pressure
- \( V \) = Volume
- \( n \) = Number of moles
- \( R \) = Universal gas constant
- \( T \) = Temperature

### Step 3: Express the Number of Moles for Both Containers
Using the Ideal Gas Law, we can express the number of moles 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:
\[ n_B = \frac{P_B V_B}{RT_B} = \frac{P V}{RT} \]

### Step 4: Find the Ratio of Number of Moles
Now, we can find the ratio of the number of moles \( n_A \) to \( n_B \):
\[ \frac{n_A}{n_B} = \frac{\frac{2PV}{RT}}{\frac{PV}{RT}} = \frac{2PV}{PV} = 2 \]

### Step 5: Relate Moles to Number of Molecules
The number of molecules \( N \) is related to the number of moles \( n \) by Avogadro's number \( N_A \):
\[ N = n \times N_A \]
Thus, the ratio of the number of molecules in containers A and B is:
\[ \frac{N_A}{N_B} = \frac{n_A \times N_A}{n_B \times N_A} = \frac{n_A}{n_B} = 2 \]

### Step 6: Final Ratio
Therefore, the ratio of the number of molecules in containers A and B is:
\[ N_A : N_B = 2 : 1 \]

### Conclusion
The final answer is that the ratio of the number of molecules in container A to container B is \( 2 : 1 \).

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