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Equations of State

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 / l max Star A l max = 6900 A, T = Star B: l max = 2200 A, T = Find area from Stefan-Boltzmann law: P = se AT 4

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Equations of State

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  1. Equations of State Physics 313 Professor Lee Carkner Lecture 4

  2. Exercise #2 Radiation • Size of Alberio stars • Find T for each from Wien’s law: T = 2.9X107/lmax • Star A lmax = 6900 A, T = • Star B: lmax = 2200 A, T = • Find area from Stefan-Boltzmann law: P = seAT4 • A = P / (seT4) • AA = (3X1029) / [(5.6703X10-8)(1)(4202)4] = • AB = (4.7X1028) / (5.6703X10-8)(1)(13182)4 = • Convert area to radius: A = 4pr2 • r = (A/4p)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 = • lmax = 2.9X107/T = • P = (5.6703X10-8)(1)(2)(310)4 ~

  3. Equilibrium • Mechanical • Chemical • Thermal • Thermodynamic

  4. 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

  5. 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

  6. Ideal Gas PV = nRT • or, since v = V/n (molar volume): • Remember ideal gas law is more accurate as the pressure gets lower

  7. 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

  8. Hydrostatic Systems • X,Y,Z are P,V,T • Many applications • Well determined equations of state

  9. Types of Hydrostatic Systems • Pure substances • Homogeneous mixture • Heterogeneous mixture

  10. 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

  11. 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

  12. 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

  13. State Relations in Hydrostatic Systems • How does the volume of a hydrostatic system change when P and T change? • Volume Expansivity: b = (1/V) (V/ T)P • Isothermal Compressibility: k = -(1/V) (V/ P)T • Both are empirically determined tabulated quantities

  14. 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

  15. 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 b and k with T

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