The correct expression for the van der Waal's equation of state is

Options

A. `(p + a//n^(2) V^(2)) (V - nb) = nRT`
B. `(p + an^(2)//V^(2)) (V - nb) = Delta n RT`
C. `(P + an^(2)//V^(2)) (V - b) = n RT`
D. `(p + an^(2)//V^(2)) (V - nb) = nRT`

Explanation

To solve the question regarding the correct expression for the van der Waals equation of state, we will follow these steps:

### Step 1: Understand the van der Waals equation
The van der Waals equation is a modified version of the ideal gas law that accounts for the volume occupied by gas molecules and the attractive forces between them. The general form of the van der Waals equation is:

\[ \left( P + \frac{a n^2}{V^2} \right) (V - nb) = nRT \]

Where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = temperature in Kelvin
- \( a \) = van der Waals constant related to the attraction between particles
- \( b \) = van der Waals constant related to the volume occupied by the particles

### Step 2: Identify the correct expression
From the van der Waals equation, we can rewrite it in a more recognizable form:

\[ P + \frac{a n^2}{V^2} = \frac{nRT}{V - nb} \]

This equation shows how pressure \( P \) is adjusted for the interactions and volume occupied by the gas molecules.

### Step 3: Match with the options given
According to the video transcript, the correct expression for the van der Waals equation is:

\[ P + \frac{a n^2}{V^2} = \frac{nRT}{V - nb} \]

The correct option mentioned in the transcript is option C.

### Step 4: Analyze incorrect options
The transcript also discusses why other options are incorrect. Common mistakes include:
- Missing terms (like \( n \) or \( R \))
- Incorrect arrangement of terms
- Misinterpretation of constants \( a \) and \( b \)

### Final Answer
The correct expression for the van der Waals equation of state is:

\[ P + \frac{a n^2}{V^2} = \frac{nRT}{V - nb} \]

And the correct option is **C**.

---