Cooling takes place when real gas expands adiabatically at

Options

A. Inversion temperature `= (2a)/(Rb)`
B. Boyle's temperature = `(a)/(Rb)`
C. Critical temperature`= (8a)/(27Rb)`
D. None of the above

Answer

To solve the question "Cooling takes place when real gas expands adiabatically at," we need to analyze the conditions under which a real gas cools during adiabatic expansion. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand Adiabatic Expansion
Adiabatic expansion refers to a process in which a gas expands without exchanging heat with its surroundings. During this process, the internal energy of the gas changes, which can lead to a change in temperature.

**Hint:** Remember that in an adiabatic process, the system is insulated, meaning no heat is transferred in or out.

### Step 2: Identify the Role of Inversion Temperature
The inversion temperature is a specific temperature for real gases. It is the temperature above which a gas will heat up upon expansion and below which it will cool down.

**Hint:** Inversion temperature is a critical point that determines the behavior of real gases during adiabatic processes.

### Step 3: Analyze the Options
The options provided are:
1. Inversion temperature
2. Boyle's temperature
3. Critical temperature
4. None of the above

- **Inversion Temperature (Ti)**: Cooling occurs when the gas expands adiabatically below this temperature.
- **Boyle's Temperature (Tb)**: This is the temperature at which a real gas behaves ideally, but it does not relate to cooling during adiabatic expansion.
- **Critical Temperature (Tc)**: This is the temperature above which a gas cannot be liquefied, and it does not relate to adiabatic cooling.
- **None of the Above**: This option is incorrect since we have identified a valid answer.

**Hint:** Compare each option to the concept of adiabatic expansion and cooling to determine which is relevant.

### Step 4: Conclusion
Since cooling occurs when a real gas expands adiabatically below the inversion temperature, the correct answer is **Inversion Temperature**.

**Final Answer:** The correct answer is **Inversion Temperature**.