Thermodynamics
is the macroscopic handmaiden of statistical mechanics. Much of the
mathematics used in thermodynamics requires the use of partial derivatives.
The following mathematical interlude is a crude but practical introduction to
the subject.
Differential
Forms
The familiar notation dF(x)/dxis
the total derivative of F with
respect toxand
gives the change in F due to a
change in x,
However, when F is a function of two (or more) independent variables, F(x,
y), the changes dF come from
changes dxand
dy in each of the variables. The
generalization for two variables is written as
(1)
where the coefficient is called the partial
derivative of F with respect tox.It is evaluated by doing ordinary
differentiation with respect to x
while treating any other independent variables as if they are constants.
ã°¡n
style='font:7.0pt "Times New Roman"'> A fundamental expression in
thermodynamics is Identify the
independent variables.
A
Fundamental Expression
An expression for internal energyU
is given by the first law of thermodynamics
where
Q is heat put into the system and W is work done on the system.
(The d௮ the right hand sign are spurious because heat and work are
themselves changes. However, we find it convenient to use this notation.)
Classical thermodynamics defines entropyS
by the expressionwhere heat is added reversibly (the thermodynamic counterpart of a friction-free
process). The equation governing reversible equilibrium thermodynamics is
then
It
helps to have a physical concept of entropy. Entropy is proportional to the average number of microstate changes
accessible to the system and the
product TS is the average energy
committed to microstate changes.
The dW term is replaced by reversible work,
often é ³tyle='mso-bidi-font-style:normal'>PdV,
(1)
Although
Eq.(1) applies to ideal reversible processes, it can be used to analyze
irreversible processes. We can use any artificial ideal process that carries
the system through the same beginning and ending state values as the real
system. The second law of thermodynamics requires that the entropy of any
isolated system must increase to a maximum.
Statistical
mechanics can provide us with an entropy equation. Then Eq.(1) can be
rewritten to express dS. Once a
system८tropy is calculated from statistical mechanics, all of the related
thermodynamic properties can be generated. Here is the principal trick that
lets us wring thermodynamic state equations from entropy. Eq.(1) expresses
the physics requirement:
(2)
This can be
compared with the mathematical identity,
(3)
to obtain relations
between S and the more accessible
variables T and P:
ã°¡n
style='font:7.0pt "Times New Roman"'> Apply the thermodynamic equations
above to the entropy
to produce the following state
equations for radiation,
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is called the partial
derivative of F with respect to x.
where heat is added reversibly (the thermodynamic counterpart of a friction-free
process). The equation governing reversible equilibrium thermodynamics is
then
