Equations of State

Physics 313Professor Lee

CarknerLecture 4

Exercise #2 Radiation Size of Alberio stars Find T for each from Wien’s law: T = 2.9X107/max

Star A max = 6900 A, T = Star B: max = 2200 A, T =

Find area from Stefan-Boltzmann law: P = AT4

A = P / (T4) AA = (3X1029) / [(5.6703X10-8)(1)(4202)4] = AB = (4.7X1028) / (5.6703X10-8)(1)(13182)4 =

Convert area to radius: A = 4r2

r = (A/4)½

rA = [(1.7X1022) / (4)(p)] ½ = rB = [(2.8X1019) / (4)(p)] ½ = 3.68X1010 / 1.48X109 = 25 times larger (red star compared

to blue) Your blackbody radiation (T = 37 C)

Convert to Kelvin, T = 37 +273 = max = 2.9X107/T = P = (5.6703X10-8)(1)(2)(310)4 ~

Equilibrium

Mechanical

Chemical

Thermal

Thermodynamic

Non-Equilibrium

System cannot be described in macroscopic coordinates

If process happens quasi-statically, system is approximately in equilibrium for any point during the process

Equation of State

System with properties X, Y and Z

Equation relating them is equation of state:

Determined empirically

These constants can be looked up in tables Equations only useful over certain conditions

Ideal Gas

PV = nRT

or, since v = V/n (molar volume):

Remember ideal gas law is more accurate as the pressure gets lower

Constants In the previous formulation

R = universal gas constant (8.31 J/mol

K) We can rewrite in terms of:

Rs = specific gas constant (R/M)

The ideal gas law is then:Pvs = RsT

Hydrostatic Systems

X,Y,Z are P,V,T

Many applications Well determined equations of state

Types of Hydrostatic Systems

Pure substances

Homogeneous mixture

Heterogeneous mixture

Homogeneous Pure Gas : Equations of State

Pv = RT

(P + a/v2)(v - b) = RT

P = (RT/v2)(1 - c/vT3)(v+B)-(A/v2)

A = A0(1 - a/v) and B = B0(1 - b/v) Note: a, b and c are constants specific to a particular

gas and are determined experimentally (empirical relations) Ideal gas ignores interactions between particles, the other

two approximate interaction effects

Differentials

For small changes we use the differential notation, e.g. dV, dT, dP

P, V and T have no meaning for small numbers of molecules

Differential Relations

For a system of three dependant variables:

dz = (z/x)y dx + (z/y)x dy

The total change in z is equal to the change in z due to changes in x plus the change in z due to changes in y

State Relations in Hydrostatic Systems

How does the volume of a hydrostatic system change when P and T change?

Volume Expansivity:

= (1/V) (V/ T)P

Isothermal Compressibility:

= -(1/V) (V/ P)T

Both are empirically determined tabulated quantities

Two Differential Theorems

(x/y)z = 1/(y/x)z

(x/y)z(y/z)x = -(x/ z)y

If we know something about how a system changes, we can tabulate it

We can use the above theorems to relate these known quantities to other changes

Constant Volume Relations

For hydrostatic systems:dP = (P/ T)V dT + (P/ V)T dV

For constant volume:

But, -(P/ T)V = (P/ V)T (V/ T)P, so:

For constant and with T