Review for Final

1. Review for Final Physics 313 Professor Lee Carkner Lecture 25

2. Final Exam • Final is Tuesday, May 18, 9am • 75 minutes worth of chapters 9-12 • 45 minutes worth of chapters 1-8 • Same format as other tests (multiple choice and short answer) • Worth 20% of grade • Three formula sheets given on test (one for Ch 9-12 and previous two) • Bring pencil and calculator

3. Exercise #24 Maxwell • Set escape velocity equal to maximum Maxwell velocity • (2GM/R)½ = 10(3kT/m) ½ • m = (150KTR/GM) • Planetary atmospheres • Earth: m > 9.5X10-27 kg (NH3, O2) • Jupiter: m > 1.4X10-28 kg (He, NH3, O2) • Titan: m > 5.6X10-26 kg (None) • Moon: m > 2.2X10-25 kg (None)

4. Thermal Equilibrium Two identical metal blocks, one at 100 C and one at 120 C, are placed together. Which transfers the most heat? • Two objects at different temperatures will exchange heat until they are at the same temperature • Zeroth Law: Two systems in thermal equilibrium with a third are in thermal equilibrium with each other

5. Heat Transfer • Heat: Q = mcDT = mc(Tf-Ti) • Conduction: dQ/dt = -KA(dT/dx) Q/t = -KA(T1-T2)/x • Radiation dQ/dt = Aes(Tenv4-T4)

6. Temperature How would you make a tube of mercury into a Celsius thermometer? A Kelvin thermometer? • Thermometers defined by the triple point of water • A system at constant temperature can have a range of values for the other variables • Isotherm

7. Measuring Temperature • Thermometers T (X) = 273.16 (X/XTP) • Temperature scales T (R) = T (F) + 459.67 T (K) = T (C) + 273.15 T (R) = (9/5) T (K) T (F) = (9/5) T (C) +32

8. Equations of State If the temperature of an ideal gas is doubled while the volume stays the same, what happens to the pressure? • Equation of state detail how properties change with temperature • Increasing T will generally increase the force and displacement terms

9. Mathematical Relations • General Relations: dx = ( x/y)zdy + (x/z)ydz (x/y)z = 1/(y/x)z (x/y)z(y/z)x(z/x)y = -1 • Specific Relations: • Volume Expansivity: b = (1/V)(dV/dT)P • Isothermal Compressibility: k=-(1/V)(dV/dP)T • Linear Expansivity: a = (1/L)(dL/dT)t • Young’s modulus: Y = (L/A)(dt/dL)T

10. Work How much work is done in an isobaric compression of a gas at 1 Pa from 2 to 1 m3? • The work done a system is the product of a force term and a displacement term • No displacement, no work • Compression is positive, expansion is negative • Work is area under PV (or XY) curve • Work is path dependant

11. Calculating Work dW = -PdV W = - PdV • For ideal gas P = nRT/V • Examples: • Isothermal ideal gas: W = -nRT  (1/V) dV = -nRT ln (Vf/Vi) • Isobaric ideal gas: W = -P  dV = -P(Vf-Vi)

12. First Law Rank the following processes in order of increasing internal energy: Adiabatic compression Isothermal expansion Isochoric cooling • Energy is conserved • Internal energy is a state function, work and heat are not

13. First Law Equations DU = Uf-Ui = Q+W dU = dQ +dW dU = CdT - PdV

14. Ideal Gas • If the volume of an ideal gas is doubled and the pressure is tripled isothermally, how does the internal energy change? lim (PV) = nRT (dU/dP)T = (dU/dV)T = 0 (dU/dT)V = CV CP = CV + nR dQ = CVdT+PdV = CPdT-VdP

15. Adiabatic Processes • Can an adiabatic process keep constant P, V, or T? PVg = const TVg-1 = const T/P(g-1)/g = const W = (PfVf - PiVi)/g-1

16. Kinetic Theory • If the rms velocity of gas molecules doubles what happens to the temperature and internal energy (1/2)mv2 = (3/2)kT U = (3/2)NkT T = mv2/3k

17. Engines • If the heat entering an engine is doubled and the work stays the same what happens to the efficiency? • Engines are cycles • Change in internal energy is zero • Composed of 4 processes h = W/QH = (QH-QL)/QH = 1 - QL/QH QH = W + QL

18. Types of Engines • Otto • Adiabatic, Isochoric h = 1 - (T1/T2) • Diesel • Adiabatic, isochoric, isobaric h = 1 - (1/g)(T4-T1)/(T3-T2) • Rankine (steam) • Adiabatic, isobaric • Stirling • Isothermal, isochoric

19. Refrigerators • Transfer heat from low to high T with the addition of work • Operates in cycle • Transfers heat with evaporation and condensation at different pressures K = QL/W K = QL/(QH-QL)

20. Second Law • Is an ice cube melting at room temperature a reversible process? • Kelvin-Planck • Cannot convert heat completely into work • Clausius • Cannot move heat from low to high temperature without work

21. Carnot • What two processes make up a Carnot cycle? How many temperatures is heat transferred at? • Adiabatic and isothermal h = 1 - TL/TH • Most efficient cycle • Efficiency depends only on the temperature

22. Second Law • The second law of thermodynamics can be stated: • Engine cannot turn heat completely into work • Heat cannot move from low to high temperatures without work • Efficiency cannot exceed Carnot efficiency • Entropy always increases

23. Entropy • Entropy change is zero for all reversible processes • All real processes are irreversible • Can compute entropy for an irreversible process by replacing it with a reversible process that achieves the same result • Entropy change of system + entropy change of surroundings = entropy change of universe (which is > 0)

24. Determining Entropy • Can integrate dS to find DS dS = dQ/T DS =  dQ/T (integrated from Ti to Tf) • Examples: • Heat reservoir (or isothermal process) DS = Q/T • Isobaric DS = CP ln (Tf/Ti)

25. Pure Substances • Can plot phases and phase boundaries on a PV, PT and PTV diagram • Saturation • condition where substance can change phase • Critical point • above which substance can only be gas • where (dP/dV) =0 and (d2P/dV2) = 0 • Triple point • where fusion, sublimation and vaporization curves intersect

26. Properties of Pure Substances cP = (dQ/dT)P (per mole) cV = (dQ/dT)T (per mole) b = (1/V)(dV/dT)P k = -(1/V)(dV/dP)T • cP, cV and b are 0 at 0 K and rise sharply to the Debye temperature and then level off • cP and cV end up near the Dulong and Petit value of 3R • k is constant at a finite value at low T and then increases linearly

27. Legendre Transform: df = udx +vdy g= f-ux dg = -xdu+vdy Useful theorems: (dx/dy)z(dy/dz)x(dz/dx)y=-1 (dx/dy)f(dy/dz)f(dz/dx)f=1 dU = -PdV +T dS dH = VdP +TdS dA = - SdT - PdV dG = V dP - S dT (dT/dV)S = - (dP/dS)V (dT/dP)S = (dV/dS)P (dS/dV)T = (dP/dT)V (dS/dP)T = -(dV/dT)P Characteristic Functions and Maxwell’s Relations

28. Key Equations • Entropy T dS = CV dT + T (dP/dT)V dV T dS = CP dT - T(dV/dT)P dP • Internal Energy (dU/dV)T = T (dP/dT)V - P (dU/dP)T = -T (dV/dT)P - P(dV/dP)T • Heat Capacity CP - CV = -T(dV/dT)P2 (dP/dV)T cP - cV = Tvb2/k

29. Joule-Thomson Expansion • Can plot on PT diagram • Isenthalpic curves show possible final states for an initial state m = (1/cP)[T(dv/dT)P - v] = slope • Inversion curve separates heating and cooling region m = 0 • Total enthalpy before and after throttling is the same • For liquefaction: hi = yhL + (1-y)hf

30. Clausius-Clapeyron Equation • Any first order phase change obeys: (dP/dT) = (sf -si)/(vf - vi) = (hf - hi)/T (vf -vi) • dP/dT is slope of phase boundary in PT diagram • Can change dP/dT to DP/DT for small changes in P and T

31. Open Systems • For a steady flow open systems mass and energy are conserved: Smin = Smout Sin[Q + W + mq] = Sout [Q + W + mq] • Where q is energy per unit mass or: q = h + ke +pe (per unit mass) • Chemical potential = m = (dU/dn) mi = mf • For open systems in equilibrium: