A scientist proposed the following equation of state `P = (RT)/(V_(m)) - (B)/(V_(m)^(2)) +(C )/(V_(m)^(3))`. If this equation leads to the critical behavior then critical temperature is:

Options

A. `(8B)/(27RC)`
B. `(B)/(8RC)`
C. `(B^(2))/(3RC)`
D. None of these

Explanation

To find the critical temperature from the given equation of state \( P = \frac{RT}{V_m} - \frac{B}{V_m^2} + \frac{C}{V_m^3} \), we can follow these steps:

### Step 1: Understand the given equation of state
The equation provided is a modified form of the ideal gas law, where \( P \) is the pressure, \( R \) is the universal gas constant, \( T \) is the temperature, and \( V_m \) is the molar volume. The terms involving \( B \) and \( C \) account for deviations from ideal gas behavior.

### Step 2: Identify the critical constants
In the context of real gases, the critical temperature can be derived from the general form of the van der Waals equation or similar equations of state. The critical temperature \( T_c \) is related to the coefficients in the equation.

### Step 3: Compare with known equations
From the van der Waals equation, we know that the critical temperature can be expressed as:
\[ T_c = \frac{8a}{27Rb} \]
where \( a \) and \( b \) are constants related to the specific gas.

### Step 4: Relate the constants in the given equation to \( a \) and \( b \)
In the given equation:
- The term \( B \) corresponds to \( a \) (the attraction parameter).
- The term \( C \) can be related to \( b \) (the volume excluded due to finite size of molecules).

### Step 5: Establish relationships
From the comparison:
- We can assume \( B \) is negligible in the context of critical behavior.
- The relationship between \( C \) and \( B \) can be established as \( C = aB \).

### Step 6: Substitute into the critical temperature formula
Substituting \( B \) and \( C \) into the critical temperature formula:
\[ T_c = \frac{8B}{27R} \quad \text{and} \quad B = \frac{C}{b} \]
Thus,
\[ T_c = \frac{8 \left( \frac{C}{b} \right)}{27R} \]

### Step 7: Final expression for critical temperature
Rearranging gives:
\[ T_c = \frac{8C}{27Rb} \]
This shows how the critical temperature is derived from the coefficients in the equation of state.

### Final Answer
The critical temperature \( T_c \) can be expressed as:
\[ T_c = \frac{8B^2}{27RC} \]