Question

If gas in a cylinder is maintained at a constant temperature​ T, the pressure P is related to the volume V by a formula of the form $P = \frac{nRT}{V - nb} - \frac{an^2}{V^2}$.

Answer 1

The given question is incomplete. The complete question is as follows.

If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form

P = $\frac{nRT}{(V - nb)} - \frac{an^2}{V^2}$, in which a, b, n, and R are constants. Find $\frac{dP}{dV}$.

Explanation:

We will use the quotient rule for each of the two terms on the right side as follows.

        P = $\frac{nRT}{V - nb} - \frac{an^{2}}{V^{2}}$

$\frac{dP}{dV} = \frac{0(V - nb) - nRT(1)}{(V - nb)^{2}} - \frac{0(V)^{2} - an^{2}(2V)}{V^{4}}$

            = $\frac{-nRT}{(V - nb)^{2}} - \frac{-2an^{2}V}{V^{4}}$

            = $\frac{-nRT}{(V - nb)^{2}} + \frac{2an^{2}}{V^{3}}$

            = $\frac{2an^{2}}{V^{3}} - \frac{nRT}{(V - nb)^{2}}$

   $\frac{dP}{dV} = \frac{2an^{2}}{V^{3}} - \frac{nRT}{(V - nb)^{2}}$

Thus, we can conclude that the value of $\frac{dP}{dV} = \frac{2an^{2}}{V^{3}} - \frac{nRT}{(V - nb)^{2}}$.