Equations of State Physics 313 Professor Lee Carkner Lecture 4

Exercise #2 Radiation  Size of Alberio stars  Find T for each from Wien’s law: T = 2.9X10 7 / max  Star A max = 6900 A, T =  Star B: max = 2200 A, T =  Find area from Stefan-Boltzmann law: P =  AT 4  A = P / (  T 4 )  A A = (3X10 29 ) / [(5.6703X10 -8 )(1)(4202) 4 ] =  A B = (4.7X10 28 ) / (5.6703X10 -8 )(1)(13182) 4 =  Convert area to radius: A = 4  r 2  r = (A/4  ) ½  r A = [(1.7X10 22 ) / (4)(p)] ½ =  r B = [(2.8X10 19 ) / (4)(p)] ½ =  3.68X10 10 / 1.48X10 9 = 25 times larger (red star compared to blue)  Your blackbody radiation (T = 37 C)  Convert to Kelvin, T = =  max = 2.9X10 7 /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:   R s = specific gas constant (R/M)   The ideal gas law is then: Pv s = R s T

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/v 2 )(v - b) = RT  P = (RT/v 2 )(1 - c/vT 3 )(v+B)-(A/v 2 ) A = A 0 (1 - a/v) and B = B 0 (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 