Understanding Isotherms in P-V Diagrams and Calculating dP/dV

In a pressure-volume (P-V) diagram, an isotherm for an ideal gas is represented as a hyperbola, not a parabola. This article will explain why constant temperature lines are hyperbolic and demonstrate how to calculate the derivative dP/dV for an isotherm.

The Ideal Gas Law and Isotherms

The relationship between pressure (P), volume (V), and temperature (T) for an ideal gas is described by the ideal gas law:

[ PV nRT ]

Here, ( n ) is the number of moles of gas, ( R ) is the ideal gas constant, and ( T ) is the absolute temperature.

Why Isotherms are Hyperbolic

When the temperature ( T ) is constant, we can rearrange the ideal gas law to express pressure as a function of volume:

[ P frac{nRT}{V} ]

This equation describes a hyperbola in the P-V plane. As the volume ( V ) increases, the pressure ( P ) decreases, and vice versa, while the product ( PV nRT ) remains constant. A hyperbola is a graphic representation of an inverse relationship, which is why we see this shape in the P-V isotherms.

Calculating (frac{dP}{dV}) for an Isotherm

Let's now derive the derivative ( frac{dP}{dV} ) for an isotherm. Using the equation:

[ P frac{nRT}{V} ]

we can differentiate both sides with respect to ( V ):

[ frac{dP}{dV} -frac{nRT}{V^2} ]

This result shows that the rate of change of pressure with respect to volume along an isotherm is negative, indicating that as volume increases, pressure decreases. The negative sign reflects the inverse relationship between pressure and volume for an ideal gas at constant temperature.

Summary

The constant temperature line on a P-V diagram for an ideal gas is a hyperbola, not a parabola.

The derivative frac{dP}{dV} for an isotherm can be calculated as frac{dP}{dV} -frac{nRT}{V^2}.

Other Key Concepts

When dealing with an isothermal process, the product of pressure and volume is constant, ( PV constant ). This is because ( n ) and ( R ) are constants for a given gas and ( T ) is constant:

[ PV c ]

When ( P ) increases, ( V ) must decrease, and vice versa. This inverse relationship is represented by a rectangular hyperbola in the P-V plot. The mathematical representation of an isotherm in the P-V diagram is given by:

[ xy constant ]

This is the equation of a rectangular hyperbola.

Further Derivation

To find ( frac{dP}{dV} ) for an isotherm, we can also use the ideal gas law:

[ PV nRT ]

When differentiating both sides with respect to volume ( V ), we get:

[ PdV -VdP ]

Rearranging, we find:

[ frac{dP}{dV} -frac{P}{V} ]

Thus, the rate of change of pressure with respect to volume for an isothermal process is inversely proportional to the volume of the gas.