The root mean square speed of an ideal gas is given by : `u_("rms") = sqrt((3RT)/M)` Thus we conclude that `u_("rms")` speed of the ideal gas molecules is proportional to square root of the temperature and inversely proportional to the square root of the molar mass. The translational kinetic energy per mole can also be given as `1/2Mu_(rms)^(2)` . The mean free path (`lambda`) is the average of distances travelled by molecules in between two successive collisions whereas collision frequency (C.F.) is expressed as number of collisions taking place in unit time. The two terms `lambda` and C.F. are related by : `C.F = (u_("rms")/lambda)`
If n represents number of moles, n0 is number of molecules per unit volume, k is Boltzmann constant, R is molar gas constant, T is absolute temperature and NA is Avogadro's number then which of the following relations is wrong ?

Options

A. `P = n_(0) kTN_(A)`
B. `P=n_(0)RT`
C. `P=(nKN_(A)T)/V`
D. `n_(0) = N_(A) xx n/V`

Answer

To solve the question regarding the relationships involving the root mean square speed, mean free path, collision frequency, and other parameters of an ideal gas, we will analyze each relationship step by step.

### Step-by-Step Solution:

1. **Understanding the Root Mean Square Speed**:
The root mean square speed of an ideal gas is given by the formula:
\[ u_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

**Hint**: Remember that \( u_{\text{rms}} \) is proportional to the square root of temperature and inversely proportional to the square root of molar mass.

2. **Translational Kinetic Energy**:
The translational kinetic energy per mole of an ideal gas can be expressed as:
\[ KE = \frac{1}{2} M u_{\text{rms}}^2 \]
This indicates that the kinetic energy depends on the square of the root mean square speed.

**Hint**: Kinetic energy is related to the speed of the molecules; higher speed means higher kinetic energy.

3. **Mean Free Path and Collision Frequency**:
The mean free path \( \lambda \) is the average distance traveled by gas molecules between collisions. The collision frequency \( C.F. \) is defined as the number of collisions per unit time. These two are related by:
\[ C.F. = \frac{u_{\text{rms}}}{\lambda} \]

**Hint**: Collision frequency increases with speed and decreases with mean free path.

4. **Number of Molecules per Unit Volume**:
The number of molecules per unit volume \( n_0 \) can be expressed in terms of the number of moles \( n \) and Avogadro's number \( N_A \):
\[ n_0 = \frac{n}{V} \times N_A \]

**Hint**: Always relate the number of moles to the number of molecules using Avogadro's number.

5. **Boltzmann Constant**:
The relationship between the Boltzmann constant \( k \) and the molar gas constant \( R \) is given by:
\[ k = \frac{R}{N_A} \]
This shows how the two constants are related.

**Hint**: Remember that \( k \) is a constant per molecule, while \( R \) is per mole.

6. **Ideal Gas Equation**:
The ideal gas equation can be expressed in different forms. One common form is:
\[ PV = nRT \]
From this, we can derive:
\[ P = \frac{n}{V} RT = n_0 kT \]

**Hint**: The ideal gas law connects pressure, volume, temperature, and number of moles.

7. **Identifying the Wrong Relation**:
Now, we need to check the given relationships to identify which one is incorrect. The relationships provided include:
- \( P = n_0 RT \) (this is incorrect)
- \( P = n_0 kT \) (this is correct)
- \( P = \frac{n}{V} RT \) (this is correct)
- \( P = n_0 kT \) (this is correct)

The incorrect relationship is:
\[ P = n_0 RT \]

**Hint**: Check the units and dimensions of each equation to verify their correctness.

### Conclusion:
The wrong relation among the given options is:
\[ P = n_0 RT \]
Thus, the answer is option B.