Choose the correct options.

Options

A. In `p = (m)/(M) R T`, (m) is mass of gas per unit volume.
B. In `p V = (m)/(M) R T`, (m) is mass of one molecule of gas.
C. In `p = (1)/(3) (m N)/(V) v_(r m s)^2`, (m) is total mass of gas.
D. In `v_(r m s) = sqrt (3 k T)/(m)` , (m) is mass of one molecule of gas.

Explanation

To solve the question of choosing the correct options related to the ideal gas equation and properties of gases, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding the Ideal Gas Equation**:
The ideal gas equation is given by:
\[ PV = nRT \]
where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature.

2. **Analyzing Option A**:
The first option states:
\[ P = \frac{m}{M} RT \]
Here, \( m \) is the mass of gas per unit volume. We can express \( n \) (number of moles) as:
\[ n = \frac{m}{M} \]
where \( M \) is the molar mass. Substituting this into the ideal gas equation gives:
\[ PV = \left(\frac{m}{M}\right)RT \]
Dividing both sides by \( V \):
\[ P = \frac{m}{MV} RT \]
For unit volume, \( V = 1 \):
\[ P = \frac{m}{M} RT \]
Thus, this option is **correct**.

3. **Analyzing Option B**:
The second option states:
\[ m \text{ is the mass of one molecule of gas.} \]
In the previous derivation, \( m \) represents the total mass of the gas, not the mass of one molecule. Therefore, this option is **incorrect**.

4. **Analyzing Option C**:
The third option states:
\[ P = \frac{1}{3} \frac{mn}{V} v_{rms}^2 \]
Here, \( v_{rms} \) is the root mean square velocity. We know:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
Substituting this into the equation:
\[ P = \frac{1}{3} \frac{mn}{V} \left(\frac{3RT}{M}\right) \]
Simplifying gives:
\[ P = \frac{mnRT}{MV} \]
Here, \( m \) is the mass of one molecule, and \( n \) is Avogadro's number. Therefore, this option incorrectly states \( m \) as the total mass of gas. Thus, this option is **incorrect**.

5. **Analyzing Option D**:
The fourth option states:
\[ v_{rms} = \sqrt{\frac{3RT}{m}} \]
Here, \( m \) is claimed to be the mass of one molecule. We know:
\[ v_{rms} = \sqrt{\frac{3kT}{m}} \]
where \( k \) is Boltzmann's constant. Since \( m \) is indeed the mass of one molecule, this option is **correct**.

### Final Conclusion:
- **Correct Options**: A and D
- **Incorrect Options**: B and C