The mass of molecule `A` is twice the mass of molecule B. The rms speed of A is twice the rms speed of B. If two samples of A and B contain same number of molecules. If the pressure of gas B is 2atm then what will be the pressure of gas A ( atm) . If two samplest are taken in separate containers of equal volume ?

Options

A. 16
B. 32
C. 48
D. 64

Explanation

To solve the problem step by step, we will use the information provided about the gases A and B, their molecular masses, RMS speeds, and the relationship between pressure, volume, and temperature.

### Step-by-Step Solution:

1. **Identify Given Data:**
- Mass of molecule A (mA) = 2 * Mass of molecule B (mB)
- RMS speed of A (u_rms_A) = 2 * RMS speed of B (u_rms_B)
- Pressure of gas B (PB) = 2 atm
- Number of molecules in both samples is the same.
- Volume of containers for A and B is equal (VA = VB).

2. **Express Molecular Masses:**
- Let the mass of molecule B be mB. Then, the mass of molecule A is:
\[ mA = 2mB \]

3. **Express RMS Speeds:**
- Let the RMS speed of molecule B be u_rms_B. Then, the RMS speed of molecule A is:
\[ u_{rms_A} = 2u_{rms_B} \]

4. **Use the Ideal Gas Law:**
- The pressure of a gas can be expressed using the formula:
\[ P = \frac{1}{3} \frac{N}{V} m u_{rms}^2 \]
- Where N is the number of molecules, V is the volume, m is the mass of the molecule, and u_{rms} is the root mean square speed.

5. **Set Up Equations for Pressures:**
- For gas A:
\[ P_A = \frac{1}{3} \frac{N}{V} m_A u_{rms_A}^2 \]
- For gas B:
\[ P_B = \frac{1}{3} \frac{N}{V} m_B u_{rms_B}^2 \]

6. **Divide the Equations:**
- Since VA = VB, we can divide the two equations:
\[ \frac{P_A}{P_B} = \frac{m_A u_{rms_A}^2}{m_B u_{rms_B}^2} \]

7. **Substitute Known Values:**
- Substitute mA and u_{rms_A}:
\[ \frac{P_A}{P_B} = \frac{(2m_B)(2u_{rms_B})^2}{m_B (u_{rms_B})^2} \]
- Simplifying gives:
\[ \frac{P_A}{P_B} = \frac{2m_B \cdot 4u_{rms_B}^2}{m_B \cdot u_{rms_B}^2} = 8 \]

8. **Calculate Pressure of Gas A:**
- Given that PB = 2 atm:
\[ P_A = 8 \cdot P_B = 8 \cdot 2 \text{ atm} = 16 \text{ atm} \]

### Final Answer:
The pressure of gas A (PA) is **16 atm**.

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