Which has least value ?

Options

A. `(T_(1))/(T_(B))`
B. `(T_(i))/(T_(C))`
C. `(T_(C))/(T_(B))`
D. `(T_(B))/(T_(C))`

Explanation

To determine which temperature has the least value among the given options (Ti, Tb, and Tc), we will analyze the formulas for each temperature:

1. **Identify the formulas for the temperatures:**
- Inversion Temperature (Ti) = \( \frac{2a}{Rb} \)
- Boyle's Temperature (Tb) = \( \frac{a}{Rb} \)
- Critical Temperature (Tc) = \( \frac{8a}{27Rb} \)

2. **Compare the values:**
- We can express all three temperatures in terms of \( \frac{a}{Rb} \):
- Ti = \( 2 \times \frac{a}{Rb} \)
- Tb = \( 1 \times \frac{a}{Rb} \)
- Tc = \( \frac{8}{27} \times \frac{a}{Rb} \)

3. **Evaluate the coefficients:**
- Ti has a coefficient of 2.
- Tb has a coefficient of 1.
- Tc has a coefficient of \( \frac{8}{27} \) which is approximately 0.296.

4. **Determine the order of the temperatures:**
- Since 2 (Ti) > 1 (Tb) > \( \frac{8}{27} \) (Tc), we can conclude that:
- Ti is the greatest.
- Tb is in the middle.
- Tc is the least.

5. **Final conclusion:**
- Therefore, the temperature with the least value is **Tc**.