Calculate the ratio of change in the mass of the molecules of a gas to the initial mass, if its speed is reduced to half and the ratio of initial and final pressure is 3:4

Explanation

To solve the problem, we need to calculate the ratio of the change in the mass of the molecules of a gas to the initial mass when the speed of the gas is reduced to half and the ratio of initial to final pressure is 3:4.

### Step-by-step Solution:

1. **Understanding the Given Information**:
- The initial pressure \( P_i \) and final pressure \( P_f \) have a ratio of \( \frac{P_i}{P_f} = \frac{3}{4} \).
- The speed of the gas is reduced to half, meaning if the initial speed is \( V \), the final speed \( V_f = \frac{V}{2} \).

2. **Using the Kinetic Theory of Gases**:
- The average translational kinetic energy of one molecule of gas is given by:
\[ KE = \frac{1}{2} m V^2 \]
- For \( n \) molecules, the total kinetic energy can also be expressed as:
\[ KE = \frac{3}{2} n k T \]
- Here, \( k \) is the Boltzmann constant and \( T \) is the temperature.

3. **Relating Pressure, Volume, and Mass**:
- From the ideal gas law, we know:
\[ PV = n k T \]
- The pressure can also be expressed in terms of mass and velocity:
\[ P = \frac{m V^2}{3V} \]
- This implies:
\[ P \propto m V^2 \]

4. **Setting Up the Mass Ratio**:
- For the initial state:
\[ P_i = \frac{m_i V^2}{3V} \]
- For the final state:
\[ P_f = \frac{m_f \left(\frac{V}{2}\right)^2}{3V} \]
- This simplifies to:
\[ P_f = \frac{m_f \frac{V^2}{4}}{3V} = \frac{m_f V^2}{12V} \]

5. **Finding the Mass Ratio**:
- Using the relationship between initial and final pressures:
\[ \frac{m_f V^2}{12V} = \frac{4}{3} \cdot \frac{m_i V^2}{3V} \]
- Simplifying gives:
\[ \frac{m_f}{m_i} = \frac{4}{3} \cdot \frac{12}{3} = \frac{16}{3} \]

6. **Calculating the Change in Mass**:
- The change in mass \( \Delta m \) is given by:
\[ \Delta m = m_f - m_i \]
- Therefore, the ratio of change in mass to initial mass is:
\[ \frac{\Delta m}{m_i} = \frac{m_f - m_i}{m_i} = \frac{\frac{16}{3} m_i - m_i}{m_i} = \frac{16}{3} - 1 = \frac{16}{3} - \frac{3}{3} = \frac{13}{3} \]

### Final Answer:
The ratio of the change in the mass of the molecules of a gas to the initial mass is \( \frac{13}{3} \).

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