If masses of all molecules of a gas are halved and the speed doubled. Then the ratio of initial and final pressure is :

Options

A. `2:1`
B. `1:2`
C. `4:1`
D. `1:4`

Explanation

To solve the problem, we need to determine the ratio of the initial and final pressures of a gas when the masses of all its molecules are halved and their speeds are doubled.

### Step-by-Step Solution:

1. **Understanding the Pressure Relation**:
The pressure \( P \) of an ideal gas can be expressed using the formula:
\[ P = \frac{1}{2} \rho v^2 \]
where \( \rho \) is the density of the gas and \( v \) is the speed of the gas molecules.

2. **Expressing Density**:
The density \( \rho \) can be expressed in terms of mass \( m \) and volume \( V \):
\[ \rho = \frac{m}{V} \]
Therefore, we can rewrite the pressure as:
\[ P = \frac{1}{2} \left(\frac{m}{V}\right) v^2 \]

3. **Initial Pressure**:
Let the initial mass of the gas molecules be \( m \) and the initial speed be \( v \). The initial pressure \( P_i \) is:
\[ P_i = \frac{1}{2} \left(\frac{m}{V}\right) v^2 \]

4. **Final Conditions**:
According to the problem, the mass of the molecules is halved, so the final mass \( m_f \) is:
\[ m_f = \frac{m}{2} \]
The speed is doubled, so the final speed \( v_f \) is:
\[ v_f = 2v \]

5. **Final Pressure**:
Now, substituting the final mass and speed into the pressure formula, the final pressure \( P_f \) is:
\[ P_f = \frac{1}{2} \left(\frac{m_f}{V}\right) v_f^2 = \frac{1}{2} \left(\frac{\frac{m}{2}}{V}\right) (2v)^2 \]
Simplifying this expression:
\[ P_f = \frac{1}{2} \left(\frac{\frac{m}{2}}{V}\right) (4v^2) = \frac{1}{2} \cdot \frac{m}{2V} \cdot 4v^2 = \frac{2mv^2}{2V} = \frac{mv^2}{V} \]

6. **Finding the Ratio of Pressures**:
Now, we can find the ratio of the initial pressure \( P_i \) to the final pressure \( P_f \):
\[ \frac{P_i}{P_f} = \frac{\frac{1}{2} \left(\frac{m}{V}\right) v^2}{\frac{mv^2}{V}} = \frac{\frac{1}{2} m v^2}{m v^2} = \frac{1}{2} \]

### Final Answer:
The ratio of the initial pressure to the final pressure is:
\[ \frac{P_i}{P_f} = \frac{1}{2} \]