Lecture-Notes (Thermodynamics) - niser

4.7. USEFUL COEFFICIENTS 35
we cancel identical terms:
and get
∂U
∂V T
= T
∂
∂V
∂P
∂T V
∂U
=
∂T
∂U
∂T∂V
− P ⇒ energy equation. (4.5)
In the energy equation, the derivative of the internal energy is written in terms of measurable
quantities.
Examples: for an ideal gas
∂U
∂V T
= T nR
V
− P = 0,
which means that the internal energy of the ideal gas is not dependent on its volume:
Gay-Lussac experiment:
4.7 Useful coefficients
U = U(T) = CV T + const.
U = 3
2 NkBT → gas of monoatomic molecules,
U = 5
2 NkBT → gas of diatomic molecules.
As
we
saw in the last section, the energy equation relates the experimentally inaccessible
∂U
∂V T to the experimentally accessible
∂P
∂T V . We will show now that
∂P
can be
∂T V
related to other thermodynamic coefficients as well. For that purpose, we will make use
of the chain rule
∂x ∂y ∂z
= −1 ,
∂y z ∂z x ∂x y
(4.6)
which can be derived by considering the differential of function f(x, y, z):
df = ∂f ∂f ∂f
dx + dy +
∂x ∂y ∂z dz,
∂x
∂y
∂y
∂z
z
x
= − ∂f/∂y
∂f/∂x ,
= − ∂f/∂z
∂f/∂y ,