The equation of state of a gas is given by (P + aT²/V)Vᶜ = (RT +b) , where a, b, c and R are constants. The isotherms can be represented by P = AVᵐ - BVⁿ, where A and B depend only on temperature then

The correct option is m = -c and n = -1.

Let us derive the given relation step-by-step from the equation of state.

The given equation of state for the gas is:
$$

( P + a T 2 V ) V c = R T + b

We want to express this equation in terms of pressure $P$ to represent the isotherms (where temperature $T$ is constant). Let us first divide both sides of the equation by $V^c$:
$$

P + a T 2 V = R T + b V c

We can rewrite the term $\frac{1}{V^c}$ as $V^{-c}$:
$$

P + a T 2 V = ( R T + b ) V - c

Now, let us isolate the pressure $P$ on the left-hand side:
$$

P = ( R T + b ) V - c - a T 2 V

Using the exponent notation $\frac{1}{V}=V^{-1}$, we obtain:
$$

P = ( R T + b ) V - c - ( a T 2 ) V - 1

For an isotherm, the temperature $T$ is constant. Therefore, the coefficients $(RT+b)$ and $\left(a{T}^2\right)$ depend only on temperature. Let us define:
$A=RT+b$
$B=a{T}^2$

Substituting $A$ and $B$ into the equation, we get the isotherm equation:
$$

P = A V - c - B V - 1

Comparing this with the given standard isotherm form $P=A{V}^m-B{V}^n$, we match the exponents of $V$:
$m=-c$
$n=-1$