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Please solve for V. Please help me by giving step by step 

The equation provided is:

\[
\left( p + \frac{a}{V^2} \right) (V - b) = nRT
\]

You are asked to solve for \(V\). This equation is a form of the **Van der Waals equation** for real gases. Let's solve it step by step:

### Step 1: Expand the Equation
First, expand the left-hand side by distributing the terms:

\[
p(V - b) + \frac{a}{V^2}(V - b) = nRT
\]

### Step 2: Distribute the Terms
Distribute \(p\) and \(\frac{a}{V^2}\) across \(V - b\):

\[
pV - pb + \frac{aV}{V^2} - \frac{ab}{V^2} = nRT
\]

Simplify the fractions:

\[
pV - pb + \frac{a}{V} - \frac{ab}{V^2} = nRT
\]

### Step 3: Rearrange the Equation
Rearrange all terms to get a standard cubic equation in terms of \(V\):

\[
pV - pb + \frac{a}{V} - \frac{ab}{V^2} - nRT = 0
\]

To clear the fractions, multiply the entire equation by \(V^2\):

\[
pV^3 - pbV^2 + aV - ab = nRTV^2
\]

### Step 4: Combine Like Terms
Move all terms to one side to set the equation to zero:

\[
pV^3 - (pb + nRT)V^2 + aV - ab = 0
\]

### Step 5: Analyze the Equation
This is a cubic equation in \(V\):

\[
pV^3 - (pb + nRT)V^2 + aV - ab = 0
\]

### Step 6: Solve the Cubic Equation
Cubic equations can have one real root and two complex roots, or three real roots. The specific method for solving depends on the coefficients, but you can use:

1. **Numerical Methods**: Use methods like Newton-Raphson or software tools to find the real root.
2. **Cardano's Method**: An algebraic method for solving cubic equations, though it can be complex.

### Example Approach
Assuming you're looking for a numerical or approximate solution, the cubic equation can be solved using numerical software, or by graphing methods to find where the function crosses the \(V\)-axis.

If you're dealing with typical values for a gas, numerical methods are usually preferred. 

Would you like more details on numerical solving methods or more examples?

Here are five questions related to this:

1. How does the Van der Waals equation correct the Ideal Gas Law?
2. What are typical values for the constants \(a\) and \(b\) for common gases?
3. How does the compressibility factor relate to the Van der Waals equation?
4. Can this cubic equation be simplified under certain conditions, such as low pressure?
5. What is the physical significance of each term in the Van der Waals equation?

**Tip:** When dealing with cubic or higher-order polynomial equations, consider whether numerical methods or software might give you a more practical solution than algebraic manipulation.

Learn how to solve the Van der Waals equation step-by-step to find the volume (V) of a gas. This equation involves real gases and includes a cubic equation form. Explore numerical methods for solving cubic equations and understand the physical significance of the terms involved.

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