The van der wasl's equation of a gas is `(P+(aT^(2))/V) V^(c)=(RT+b)`. Where a, b, c and R are constant. If the isotherm is represented by `P=AV^(m)-BV^(n)`, where A and B depends on temperature:

Options

A. `m=-c, n=-1`
B. `m=+c, n=-1`
C. `m=-c, n=+1`
D. `m=c, n=-1`

Explanation

To solve the problem, we need to analyze the given Van der Waals equation and the isothermal representation of pressure. Let's go through the steps systematically.

### Step 1: Write down the Van der Waals equation
The Van der Waals equation is given as:
\[ \left(P + \frac{aT^2}{V}\right) V^c = RT + b \]
where \(a\), \(b\), \(c\), and \(R\) are constants.

### Step 2: Rearrange the equation to isolate \(P\)
We can rearrange the equation to express \(P\) in terms of \(V\):
\[ P + \frac{aT^2}{V} = \frac{RT + b}{V^c} \]
Now, subtract \(\frac{aT^2}{V}\) from both sides:
\[ P = \frac{RT + b}{V^c} - \frac{aT^2}{V} \]

### Step 3: Combine the terms on the right-hand side
To combine the terms, we need a common denominator, which is \(V^c\):
\[ P = \frac{RT + b - aT^2 V^{c-1}}{V^c} \]

### Step 4: Identify the form of the isothermal equation
The isothermal equation is given as:
\[ P = A V^m - B V^n \]
where \(A\) and \(B\) depend on temperature.

### Step 5: Compare the two expressions for \(P\)
From our rearranged Van der Waals equation, we can see that:
\[ P = \frac{RT + b}{V^c} - \frac{aT^2}{V} \]
This can be rewritten as:
\[ P = \frac{RT + b - aT^2 V^{c-1}}{V^c} \]

### Step 6: Determine the powers of \(V\)
From the expression \(\frac{RT + b}{V^c}\), we can see that:
- The term \(\frac{1}{V^c}\) suggests that \(m = -c\).
- The term \(-\frac{aT^2}{V}\) suggests that \(n = -1\).

### Step 7: Write down the values of \(m\) and \(n\)
Thus, we have:
\[ m = -c \quad \text{and} \quad n = -1 \]

### Conclusion
The values of \(m\) and \(n\) are:
- \(m = -c\)
- \(n = -1\)