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 increases by 40%. Calculate the ratio of initial and finial pressures.

Explanation

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

### Step-by-Step Solution:

1. **Understand the relationship between pressure and mass**:
According to the ideal gas law, pressure is directly proportional to the number of moles (or mass) of the gas when temperature and volume are constant. This can be expressed as:
\[ P \propto m \]
where \( P \) is pressure and \( m \) is mass.

2. **Define initial and final mass**:
Let the initial mass of the gas be \( m_i \). After increasing the mass by 40%, the final mass \( m_f \) can be calculated as:
\[ m_f = m_i + 0.4 m_i = 1.4 m_i \]

3. **Set up the ratio of pressures**:
Since pressure is proportional to mass, we can write the ratio of the initial pressure \( P_1 \) to the final pressure \( P_2 \) as:
\[ \frac{P_1}{P_2} = \frac{m_i}{m_f} \]

4. **Substitute the expression for final mass**:
Substitute \( m_f = 1.4 m_i \) into the ratio:
\[ \frac{P_1}{P_2} = \frac{m_i}{1.4 m_i} \]

5. **Simplify the ratio**:
The \( m_i \) terms cancel out:
\[ \frac{P_1}{P_2} = \frac{1}{1.4} = \frac{10}{14} = \frac{5}{7} \]

6. **Conclusion**:
Thus, the ratio of the initial pressure to the final pressure is:
\[ \frac{P_1}{P_2} = \frac{5}{7} \]

### Final Answer:
The ratio of initial pressure to final pressure is \( \frac{5}{7} \).