The pressure P, Volume V and temperature T of a gas in the jar A and the other gas in the jar B at pressure `2P`, volume `V//4` and temperature `2 T`, then the ratio of the number of molecules in the jar A and B will be

Options

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

Answer

To find the ratio of the number of molecules in jar A and jar B, we can use the ideal gas equation, which is given by:

\[ PV = nRT \]

where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 1: Write the Ideal Gas Equation for Jar A
For jar A, we have:
- Pressure = \( P \)
- Volume = \( V \)
- Temperature = \( T \)

Using the ideal gas equation:

\[ PV = n_A RT \]

where \( n_A \) is the number of moles in jar A.

### Step 2: Write the Ideal Gas Equation for Jar B
For jar B, we have:
- Pressure = \( 2P \)
- Volume = \( \frac{V}{4} \)
- Temperature = \( 2T \)

Using the ideal gas equation:

\[ (2P) \left(\frac{V}{4}\right) = n_B R(2T) \]

where \( n_B \) is the number of moles in jar B.

### Step 3: Simplify the Equation for Jar B
Simplifying the equation for jar B:

\[ \frac{2PV}{4} = n_B (2RT) \]

This simplifies to:

\[ \frac{PV}{2} = n_B (2RT) \]

### Step 4: Rearranging the Equation for Jar B
Now, we can rearrange this to find \( n_B \):

\[ n_B = \frac{PV}{4RT} \]

### Step 5: Set Up the Ratio of Number of Moles
Now we have two equations:
1. For jar A:
\[ n_A = \frac{PV}{RT} \]
2. For jar B:
\[ n_B = \frac{PV}{4RT} \]

### Step 6: Calculate the Ratio of Number of Moles
Now, we can find the ratio of the number of moles \( n_A \) and \( n_B \):

\[ \frac{n_A}{n_B} = \frac{\frac{PV}{RT}}{\frac{PV}{4RT}} \]

### Step 7: Simplifying the Ratio
This simplifies to:

\[ \frac{n_A}{n_B} = \frac{PV}{RT} \cdot \frac{4RT}{PV} = 4 \]

### Conclusion
Thus, the ratio of the number of molecules in jar A to jar B is:

\[ \frac{n_A}{n_B} = 4:1 \]

### Final Answer
The ratio of the number of molecules in jar A and jar B is \( 4:1 \).

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