Partial Derivatives and Gas Laws

Consider a surface representing a function of two variables, x and y. The partial derivatives give the local slopes of the surface when one variable changes, while the other remains constant.

The partial derivative of the function z in terms of x at constant y is represented using 'curly d' notation.

For functions of two variables, the total differential sums all partial derivatives, each multiplied by the infinitesimal change in its respective variable.

Now, imagine a cylinder filled with an ideal gas. Using the ideal gas law, the change of pressure with temperature at constant volume and number of moles can be evaluated.

One approach involves rearranging the ideal gas law and then differentiating both sides by temperature, keeping all other variables constant.

This partial derivative can also be calculated from the slope of the straight line formed by plotting the pressure of an ideal gas at various temperatures, but at constant volume.

More generally, for three variables linked by an equation of state, the three partial derivatives, each taken while holding one variable constant, multiply to minus one, which is Euler’s chain rule.