Two cylinder having `m_(1)g` and `m_(2)g` of a gas at pressure `P_(1)` and `P_(2)` respectively are put in cummunication with each other, temperature remaining constant. The common pressure reached will be

Options

A. `(P_(1)P_(2)(m_(1)+m_(2)))/(P_(2)m_(1)+P_(1)m_(2))`
B. `(P_(1)P_(2)m_(1))/(P_(2)m_(1)+P_(1)m_(2))`
C. `(m_(1)m_(2)(P_(1)+P_(2)))/(P_(2)m_(1)(P_(1)+P_(2)))`
D. `(m_(1)m_(2)P_(2))/(P_(2)m_(1)+P_(1)m_(2))`

Explanation

To find the common pressure reached when two cylinders containing gases at different pressures are put in communication with each other, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two cylinders with masses of gas \( m_1g \) and \( m_2g \) at pressures \( P_1 \) and \( P_2 \) respectively. When they are connected, they will reach a common pressure \( P \) while maintaining a constant temperature.

2. **Use the Ideal Gas Law**: The ideal gas equation is given by:
\[ PV = nRT \]
This can also be expressed in terms of mass:
\[ PV = \frac{m}{M}RT \]
where \( m \) is the mass of the gas, \( M \) is the molar mass, \( R \) is the universal gas constant, and \( T \) is the temperature.

3. **Write the Equations for Each Cylinder**:
- For the first cylinder:
\[ P_1 V_1 = \frac{m_1}{M}RT \quad \text{(Equation 1)} \]
- For the second cylinder:
\[ P_2 V_2 = \frac{m_2}{M}RT \quad \text{(Equation 2)} \]

4. **Combine the Two Cylinders**: When the two cylinders are put in communication, they will have a total volume \( V_1 + V_2 \) and a total mass \( m_1 + m_2 \). The common pressure \( P \) can be expressed as:
\[ P(V_1 + V_2) = \frac{m_1 + m_2}{M}RT \quad \text{(Equation 3)} \]

5. **Substitute Equations 1 and 2 into Equation 3**:
- From Equation 1, we can express \( V_1 \) as:
\[ V_1 = \frac{m_1RT}{P_1M} \]
- From Equation 2, we can express \( V_2 \) as:
\[ V_2 = \frac{m_2RT}{P_2M} \]

6. **Substituting into Equation 3**:
\[ P\left(\frac{m_1RT}{P_1M} + \frac{m_2RT}{P_2M}\right) = \frac{m_1 + m_2}{M}RT \]

7. **Simplify the Equation**: Cancel \( RT \) and \( M \) from both sides:
\[ P\left(\frac{m_1}{P_1} + \frac{m_2}{P_2}\right) = m_1 + m_2 \]

8. **Rearranging for Common Pressure \( P \)**:
\[ P = \frac{(m_1 + m_2)}{\left(\frac{m_1}{P_1} + \frac{m_2}{P_2}\right)} \]

9. **Final Expression**: To express this in a more usable form, we can multiply the numerator and denominator by \( P_1P_2 \):
\[ P = \frac{(m_1 + m_2)P_1P_2}{m_1P_2 + m_2P_1} \]

### Final Answer:
The common pressure reached when the two cylinders are put in communication is:
\[ P = \frac{(m_1 + m_2)P_1P_2}{m_1P_2 + m_2P_1} \]