The relationship between `P_(C), V_(C)` and `T_(C)` is

Options

A. `P_(C) V_(C) = RT`
B. `P_(C) V_(C) = 3 RT_(C)`
C. `P_(C) V_(C) = (3)/(5) RT_(C)`
D. `P_(C) V_(C) = (3)/(8) RT_(C)`

Explanation

To find the relationship between critical pressure (P_C), critical volume (V_C), and critical temperature (T_C), we can start by using the formulas provided in the video transcript.

### Step-by-Step Solution:

1. **Identify the formulas**:
- The critical temperature (T_C) is given by:
\[ T_C = \frac{8A}{27RB} \]
- The critical pressure (P_C) is given by:
\[ P_C = \frac{A}{27B^2} \]
- The critical volume (V_C) is given by:
\[ V_C = 3B \]

2. **Relate T_C and P_C**:
- From the formula for T_C, we can express A in terms of P_C and B:
\[ A = 27B^2 P_C \]
- Substitute this expression for A into the formula for T_C:
\[ T_C = \frac{8(27B^2 P_C)}{27RB} = \frac{8B P_C}{R} \]

3. **Relate T_C, P_C, and V_C**:
- Now, we know that V_C = 3B. We can express B in terms of V_C:
\[ B = \frac{V_C}{3} \]
- Substitute this expression for B into the equation for T_C:
\[ T_C = \frac{8\left(\frac{V_C}{3}\right) P_C}{R} = \frac{8V_C P_C}{3R} \]

4. **Rearranging the equation**:
- Rearranging the equation gives us:
\[ P_C V_C = \frac{3R}{8} T_C \]

5. **Final relationship**:
- Thus, we can express the relationship as:
\[ P_C V_C = \frac{3R}{8} T_C \]

### Conclusion:
The relationship between critical pressure (P_C), critical volume (V_C), and critical temperature (T_C) is given by:
\[ P_C V_C = \frac{3R}{8} T_C \]