In the kinetic theory of gases, which of these statements is/are true ?
(i) The pressure of a gas is proportional to the mean speed of the molecules. (ii) The root mean square speed of the molecules is proportional to the pressure. (iii) The rate of diffusion is proportional to the mean speed of the molecules. (iv) The mean translational kinetic energy of a gas is proportional to its kelvin temperature.

Options

A. (ii) and (iii) only
B. (i), (ii) and (iv) only
C. (i) and (iii) only
D. (iii) and (iv) only

Answer

To determine which statements are true in the context of the kinetic theory of gases, we will analyze each statement one by one.

### Step 1: Analyze Statement (i)
**Statement (i): The pressure of a gas is proportional to the mean speed of the molecules.**

In the kinetic theory of gases, the pressure \( P \) of an ideal gas is given by the equation:
\[ P = \frac{1}{3} n m \overline{v^2} \]
where \( n \) is the number density of molecules, \( m \) is the mass of a molecule, and \( \overline{v^2} \) is the mean of the square of the speed of the molecules.

The mean speed \( \overline{v} \) is related to \( \overline{v^2} \) but they are not directly proportional. Therefore, this statement is **false**.

### Step 2: Analyze Statement (ii)
**Statement (ii): The root mean square speed of the molecules is proportional to the pressure.**

The root mean square speed \( v_{rms} \) is defined as:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a molecule.

From the equation for pressure, we can see that as pressure increases (at constant volume), the temperature must also increase, leading to an increase in \( v_{rms} \). Thus, this statement is **true**.

### Step 3: Analyze Statement (iii)
**Statement (iii): The rate of diffusion is proportional to the mean speed of the molecules.**

According to Graham's law of effusion, the rate of diffusion of a gas is inversely proportional to the square root of its molar mass, and directly proportional to the average speed of the molecules. Therefore, this statement is **true**.

### Step 4: Analyze Statement (iv)
**Statement (iv): The mean translational kinetic energy of a gas is proportional to its Kelvin temperature.**

The mean translational kinetic energy \( KE \) of a gas is given by:
\[ KE = \frac{3}{2} k T \]
This shows that the mean kinetic energy is directly proportional to the Kelvin temperature \( T \). Therefore, this statement is **true**.

### Conclusion
Based on the analysis:
- Statement (i): False
- Statement (ii): True
- Statement (iii): True
- Statement (iv): True

### Final Answer
The true statements are (ii), (iii), and (iv).

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