1.5
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.
In functions with multiple variables, partial derivatives describe how a function changes with respect to one variable while keeping the others constant. A partial derivative is calculated from the ordinary derivative of the function with respect to the desired variable, while treating the other variables as constants.
Consider the function z = f(x, y). The partial derivative of the function z with respect to x at constant y is written as (∂z/∂x)y, using 'curly d'. It essentially tells us how z changes when x varies, while y remains constant.
The total derivative of a function of two variables, z(x, y), is given as the sum of all its partial derivatives, each multiplied by the infinitesimal change in the appropriate variable. This equation describes infinitesimal changes in a function due to changes in the independent variables.
The ideal gas law, pV = nRT, is an equation of state that relates the pressure (p), volume (V), temperature (T), and number of moles (n) of an ideal gas, where R is the gas constant.
In the context of the ideal gas law, partial derivatives are used to determine how each variable (p, V, and T) changes with respect to another. For example, (∂V/∂T)p,n represents the change of volume with respect to temperature at constant pressure and constant number of moles in the gaseous system. Similarly, (∂p/∂T)V,n represents the change of pressure with temperature at constant volume and number of moles.
This partial derivative can be analytically calculated using the ideal gas law. The first step is to rewrite the ideal gas law so that pressure is all by itself on one side of the equation. Then, take the derivative of both sides with respect to T, treating everything else as a constant. A plot of pressure versus temperature at constant volume for an ideal gas is therefore a straight line with slope nR/V, consistent with this derivative.
Because the variables p, V, and T are related through an equation of state, their partial derivatives are not independent. They satisfy Euler’s chain rule for three interdependent variables: (∂V/∂T)p (∂T/∂p)V (∂p/∂V)T = -1.
This relation is fundamental in thermodynamics and is widely used to analyze how systems respond to changes in their state variables.
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.
From Chapter 1:
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