The correct order of temperature of a real gas is :
(I) Boyle's temperature
(II) Critical temperature
(III) Inversion temperature

Options

A. `III gt I gt II`
B. `I gt II gt III`
C. `II gt I gt III`
D. `I gt III gt II`

Explanation

To determine the correct order of temperatures for a real gas, we need to analyze the three temperatures given: Boyle's temperature (TB), critical temperature (Tc), and inversion temperature (Ti).

### Step-by-Step Solution:

1. **Understanding Boyle's Temperature (TB)**:
- Boyle's temperature is the temperature at which a real gas behaves ideally over a wide range of pressures.
- It is denoted as \( TB \) and can be expressed in terms of Van der Waals constants as:
\[ TB = \frac{A}{RB} \]
- Here, \( A \) and \( B \) are the Van der Waals constants.

2. **Understanding Critical Temperature (Tc)**:
- The critical temperature is the highest temperature at which a substance can exist as a liquid. Above this temperature, the substance cannot be liquefied regardless of the pressure applied.
- It is denoted as \( Tc \) and can be expressed as:
\[ Tc = \frac{8A}{27RB} \]

3. **Understanding Inversion Temperature (Ti)**:
- Inversion temperature is the temperature below which a non-ideal gas expands at constant enthalpy and experiences a temperature decrease. Above this temperature, the gas experiences a temperature increase.
- It is denoted as \( Ti \) and can be expressed as:
\[ Ti = \frac{2A}{RB} \]

4. **Comparing the Temperatures**:
- Now we need to compare the values of \( TB \), \( Tc \), and \( Ti \):
- From the expressions derived:
- \( TB = \frac{A}{RB} \)
- \( Ti = \frac{2A}{RB} \)
- \( Tc = \frac{8A}{27RB} \)
- To compare these, we can see that:
- \( Ti \) is greater than \( TB \) because \( \frac{2A}{RB} > \frac{A}{RB} \).
- To compare \( Ti \) and \( Tc \), we can convert \( Tc \) into a similar form:
- \( Tc = \frac{8A}{27RB} \approx 0.296A/RB \) (since \( \frac{8}{27} \) is less than 1)
- Therefore, \( Ti > Tc \).

5. **Final Order**:
- From the comparisons:
- \( Ti > TB \)
- \( TB > Tc \)
- Thus, the correct order of temperatures is:
\[ Ti > TB > Tc \]

### Conclusion:
The correct order of temperatures for a real gas is:
1. Inversion Temperature (Ti)
2. Boyle's Temperature (TB)
3. Critical Temperature (Tc)