A gas is filled in a vessel at a certain temperature and pressure. At the same temperature more gas is filled in the vessel so that its mass increased by 40%. Calculate the ratio of initial and final pressures.

Explanation

To solve the problem, we need to determine the ratio of the initial pressure (P1) to the final pressure (P2) after the mass of gas in the vessel is increased by 40% while keeping the temperature constant.

### Step-by-Step Solution:

1. **Understand the Ideal Gas Law**:
The ideal gas law states that:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.

2. **Relate Pressure to Mass**:
The number of moles \( n \) can be expressed in terms of mass \( m \) and molar mass \( M \):
\[ n = \frac{m}{M} \]
Substituting this into the ideal gas equation gives:
\[ P = \frac{nRT}{V} = \frac{mRT}{MV} \]
This shows that pressure \( P \) is directly proportional to the mass \( m \) of the gas when temperature and volume are constant.

3. **Initial and Final Mass**:
Let the initial mass of the gas be \( m_1 \). After increasing the mass by 40%, the final mass \( m_2 \) is:
\[ m_2 = m_1 + 0.4m_1 = 1.4m_1 \]

4. **Relate Initial and Final Pressures**:
Since pressure is directly proportional to mass, we can write:
\[ \frac{P_2}{P_1} = \frac{m_2}{m_1} \]
Substituting \( m_2 \):
\[ \frac{P_2}{P_1} = \frac{1.4m_1}{m_1} = 1.4 \]

5. **Calculate the Ratio of Initial and Final Pressures**:
To find the ratio of initial pressure to final pressure:
\[ \frac{P_1}{P_2} = \frac{1}{1.4} = \frac{5}{7} \]

### Final Answer:
The ratio of initial pressure to final pressure is:
\[ \frac{P_1}{P_2} = \frac{5}{7} \]