The correct order for `T_(i), T_(B)` and `T_(C)` is

Options

A. `T_(I) lt T_(C) lt T_(B)`
B. `T_(B) lt T_(C) lt T_(i)`
C. `T_(C) lt T_(B) lt T_(i)`
D. `T_(i) lt T_(B) lt T_(C)`

Explanation

To determine the correct order of \( T_i \), \( T_b \), and \( T_c \), we first need to understand what each of these temperatures represents and how they are mathematically defined.

### Step 1: Define the Temperatures
- **Inversion Temperature (\( T_i \))**: This is defined as:
\[ T_i = \frac{2a}{r_b} \]
- **Boiling Temperature (\( T_b \))**: This is defined as:
\[ T_b = \frac{a}{r_b} \]
- **Critical Temperature (\( T_c \))**: This is defined as:
\[ T_c = \frac{a}{27r_b} \]

### Step 2: Identify the Common Terms
In all three equations, the term \( \frac{a}{r_b} \) is common. We can compare the coefficients in front of this term to determine their relative sizes.

### Step 3: Compare the Coefficients
- For \( T_i \): The coefficient is \( 2 \).
- For \( T_b \): The coefficient is \( 1 \).
- For \( T_c \): The coefficient is \( \frac{1}{27} \).

### Step 4: Determine the Order
Now we can compare the coefficients:
- \( 2 > 1 > \frac{1}{27} \)

Thus, the order from greatest to least is:
\[ T_i > T_b > T_c \]

### Step 5: Conclusion
The correct order for \( T_i \), \( T_b \), and \( T_c \) is:
\[ T_i, T_b, T_c \]