A gas in container `A` is in thermal equilibrium with another gas in container `B`. Both contain equal masses of the two gases. Which of the following can be true ?

Options

A. `P_(A) V_(A) = P_(B) V_(B)`
B. `P_(A) = P_(B), V_(A) != V_(B)`
C. `P_(A) != P_(B), V_(A) = V_(B)`
D. `(P_(A))/(V_(A)) = (P_(B))/(V_(B))`

Explanation

To solve the problem, we will analyze the conditions given and apply the ideal gas law to understand the relationships between the gases in containers A and B.

### Step-by-Step Solution:

1. **Understanding Thermal Equilibrium**:
- When two gases are in thermal equilibrium, they are at the same temperature. This means that the temperature \( T_A = T_B \).

2. **Equal Masses of Gases**:
- We have equal masses of gas in both containers, denoted as \( m_A = m_B \).

3. **Number of Moles**:
- The number of moles of a gas is given by the formula:
\[ N = \frac{m}{M} \]
where \( m \) is the mass of the gas and \( M \) is the molar mass of the gas.
- Since the gases in containers A and B are different, their molar masses \( M_A \) and \( M_B \) will be different. Thus:
\[ N_A = \frac{m_A}{M_A} \quad \text{and} \quad N_B = \frac{m_B}{M_B} \]
- Since \( m_A = m_B \), we can conclude that:
\[ N_A \neq N_B \quad \text{(because \( M_A \neq M_B \))} \]

4. **Applying the Ideal Gas Law**:
- The ideal gas law states:
\[ PV = nRT \]
- For container A:
\[ P_A V_A = N_A R T \]
- For container B:
\[ P_B V_B = N_B R T \]

5. **Taking the Ratio of Pressures and Volumes**:
- Dividing the two equations, we get:
\[ \frac{P_A V_A}{P_B V_B} = \frac{N_A}{N_B} \]
- Since \( N_A \neq N_B \) (from step 3), it follows that:
\[ P_A V_A \neq P_B V_B \]

6. **Analyzing the Given Options**:
- **Option 1**: \( P_A V_A = P_B V_B \) → Incorrect (as shown above).
- **Option 2**: If \( P_A = P_B \), then \( V_A \neq V_B \) → Correct (since \( N_A \neq N_B \)).
- **Option 3**: If \( V_A = V_B \), then \( P_A \neq P_B \) → Correct (since \( N_A \neq N_B \)).
- **Option 4**: \( \frac{P_A}{V_A} = \frac{P_B}{V_B} \) → Incorrect (not supported by our analysis).

### Conclusion:
- The correct options are **Option 2** and **Option 3**.