Thermodynamic Systems
Physics 313
Professor Lee Carkner
Lecture 5
Exercise #3 Equations of State
Ideal gas pressure:
P = RT/v = (8.31)(150)/(1.1733) = 1062.39 kPa

Beattie-Bridgeman pressure:
P = (RT/v2)(1-(c/vT3))(v+B)-(A/v2)
P = [(8.31)(150)/(1.1733)2][1((4.2X104)/(1.1733)(150)3)](1.1733+0.05076)(133.193/1.17332) = 999.84 kPa

Savings
Design A requires 1062.39-999.84 = 62.55 kPa more

Temperature Dependence

Can use the equation of state to find
dependence

Can use differential theorems to relate
Generic Relations
Consider a system with interdependent
properties x, y and z:
dz = (z/x)y dx + (z/y)x dy
(x/y)z = 1/( y/ x)z
(x/y)z(y/z)x = -(x/z)y
Can use these along with:

Tabulated x,y,z dependencies (expansivity, bulk
modulus etc.)
Stretched Wire
A wire under tension is a
thermodynamic system that can be
described with three variables:



differential changes can be related by:
dL = (L/T)t dT + (L/t)T dt
Wire Relations
Linear Expansivity:

a = (1/L)(L/T)t
Isothermal Young’s Modulus:

Y = (L/A)(t/L)T

These are well known for most normal
conditions
Wires and Sound
Vibrating strings can produce notes of a
given frequency

Frequency depends on wave speed and
wavelength, which are properties of the
string:
l is usually fixed
based on string
m (linear density) is usually fixed

How does the tension change?
Surfaces
Surfaces (such as films) act like 2-D wires



The surface tension is a force that pulls in the
plane of the surface

Surface tension relations often depend on the
type of system

e.g. vapor above liquid, oil film on water
Boundaries as Surfaces
For surface defined as the boundary between
a liquid and its vapor:
g = g0[1 - (T/TC)]n
where:

•
• n is between 1 and 2
• Higher T means lower tension
•
Oil on Water
A film of oil on water increases the
surface tension:
(g - gw)A = aT


Sort of a 2-D equation of state
Electrochemical Cell
A battery produces emf through chemical
reactions

The emf depends on the amount of charge
transferred

Batteries can be recharged
Equation of State
We can relate the emf to 2 other variables



The equation of state is:
e = e20 + a(T-20) + b(T-20)2 + g(T-20)3

 Constants depend on materials and chemicals
Dielectric Slab
Material in an electric field will undergo
polarization (molecules become polar)
The total polarization depends on the electric
field and the temperature



Equation of state:
P/V = [a + (b/T)]E
Where P/V is the polarization per unit volume


Thermal “forces” compete with electrical
Paramagnetic Rod
Paramagnetic materials develop magnetization
in a magnetic field

Non-magnetic materials become magnetic
Properties:



Equation of state:
M = CH/T

M decreases at higher temperature
This assumes a long thin shape
The Eagle Nebula - Interstellar Dust
Paramagnetism and
Interstellar Dust
Intensive
Independent of mass

Tension

emf

Magnetic field
Extensive
Proportional to mass

Length

Charge

Total magnetization
Concepts
How do system properties vary with
temperature?

What are the differential relations?

How can the differential relations be
rewritten?
