Thermodynamics/Derivatives

Derivatives
Much of thermodynamics involves the handling derivatives. In thermodynamics we write the partial derivatives of f(x,y) as:

$$\left (\frac{\partial f}{\partial x}\right )_y and \left (\frac{\partial f}{\partial y}\right )_x$$

We read the left derivative above as "the partial of f with respect to x, keeping y constant. Note that in thermodynamics we always specify was is being kept constant.

Exact Differentials
If f = f(x,y), then the total derivative, df is:

$$df = \left (\frac{\partial f}{\partial x}\right )_y dx + \left (\frac{\partial f}{\partial y}\right )_x dy$$

If the function f(x,y) can be written as a total derivative, then f is known as an exact differential.

Theorem
The equation

$$df=M(x,y)dx+N(x,y)dy$$

is an exact differential if and only if

$$\left (\frac{\partial M}{\partial y}\right )_x = \left (\frac{\partial N}{\partial x}\right )_y dy$$

This theorem will be important later, especially when we discuss Maxwell's relation. However, for now there is a more important result.

Path Independency and State Functions
If f(x,y) is an exact differential, then a definite integral is equal to the function f evaluated at the limits of integration:

$$\int^{(x_2,y_2)}_{(x_1,y_2)} f'(x,y)\, dxdy = f(x_2,y_2) - f(x_1,y_1)$$

In other words, the value of the integral is dependent on only its initial and final points. It does not depend on what the function does between those two points. We say such an integral is path independent.

Properties which are path independent are called State Functions.

State functions are very important in thermodynamics because we usually do not know what goes on internally within a system. But we do know the beginning and ending conditions of a system.

Pressure, volume, temperature, and internal energy are state functions. But heat and work are not.

Derivative Rules
The following rules are useful for manipulating derivatives. In these f = f(x,y,z).


 * $$\left (\frac{\partial f}{\partial z} \right )_x =\left (\frac{\partial f}{\partial y} \right )_x\, \left (\frac{\partial y}{\partial z} \right )_x$$
 * $$\left (\frac{\partial f}{\partial x} \right )_z =\left (\frac{\partial f}{\partial x} \right )_y + \left (\frac{\partial f}{\partial y} \right )_x\,\left (\frac{\partial y}{\partial x} \right )_z$$
 * $$\left (\frac{\partial y}{\partial x} \right )_z = \left (\frac{\partial x}{\partial z} \right )_y\, \left (\frac{\partial z}{\partial y} \right )_x$$
 * $$\left (\frac{\partial x}{\partial y} \right )_z\, \left (\frac{\partial y}{\partial z} \right )_x\, \left (\frac{\partial z}{\partial x} \right )_y=-1$$

Note that the cycle in number 4 is not one but minus 1 (rule 4 is sometimes known as Euler's relation). Rule 3 is just a rearrangement of rule 4 (remember that taking the reciprocal of a partial derivative is the same as switching the numerator and denominator). Rule 2 is very useful if you need to change what variable is kept constant.

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