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 (NH 3, O 2 )  Jupiter: m > 1.4X10 -28 kg (He, NH 3, O 2 )  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 = mc  T = mc(T f -T i )  Conduction: dQ/dt = -KA(dT/dx) Q/t = -KA(T 1 -T 2 )/x  Radiation dQ/dt = A  (T env 4 -T 4 )

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/X TP )  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) z dy + (  x/  z) y dz (  x/  y) z = 1/(  y/  x) z (  x/  y) z (  y/  z) x (  z/  x) y = -1  Specific Relations:  Volume Expansivity:  = (1/V)(dV/dT) P  Isothermal Compressibility:  =-(1/V)(dV/dP) T  Linear Expansivity:  = (1/L)(dL/dT)   Young’s modulus: Y = (L/A)(d  /dL) T

10. Work How much work is done in an isobaric compression of a gas at 1 Pa from 2 to 1 m 3 ?  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 (V f /V i )  Isobaric ideal gas: W = -P  dV = -P(V f -V i )

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  U = U f -U i = 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 = C V C P = C V + nR dQ = C V dT+PdV = C P dT-VdP

15. Adiabatic Processes  Can an adiabatic process keep constant P, V, or T? PV  = const TV  -1 = const T/P (  -1)/  = const W = (P f V f - P i V i )/  -1

16. Kinetic Theory  If the rms velocity of gas molecules doubles what happens to the temperature and internal energy (1/2)mv 2 = (3/2)kT U = (3/2)NkT T = mv 2 /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  = W/Q H = (Q H -Q L )/Q H = 1 - Q L /Q H Q H = W + Q L

18. Types of Engines  Otto  Adiabatic, Isochoric  = 1 - (T 1 /T 2 )  Diesel  Adiabatic, isochoric, isobaric  = 1 - (1/  )(T 4 -T 1 )/(T 3 -T 2 )  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 = Q L /W K = Q L /(Q H -Q L )

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  = 1 - T L /T H  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  S dS = dQ/T  S =  dQ/T (integrated from T i to T f )  Examples:  Heat reservoir (or isothermal process)  S = Q/T  Isobaric  S = C P ln (T f /T i )

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 (  P/  V) =0 and (  2 P/  V 2 ) = 0  Triple point  where fusion, sublimation and vaporization curves intersect

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

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

28. Key Equations  Entropy T dS = C V dT + T (  P/  T) V dV T dS = C P dT - T(  V/  T) P dP  Internal Energy (  U/  V) T = T (  P/  T) V - P (  U/  P) T = -T (  V/  T) P - P(  V/  P) T  Heat Capacity C P - C V = -T(  V/  T) P 2 (  P/  V) T c P - c V = Tv  2 / 

29. Joule-Thomson Expansion  Can plot on PT diagram  Isenthalpic curves show possible final states for an initial state  = (1/c P )[T(dv/dT) P - v] = slope  Inversion curve separates heating and cooling region  = 0  Total enthalpy before and after throttling is the same  For liquefaction: h i = yh L + (1-y)h f

30. Clausius-Clapeyron Equation  Any first order phase change obeys: (dP/dT) = (s f -s i )/(v f - v i ) = (h f - h i )/T (v f -v i )  dP/dT is slope of phase boundary in PT diagram  Can change dP/dT to  P/  T for small changes in P and T

31. Open Systems  For a steady flow open systems mass and energy are conserved:  m in =  m out  in [ Q + W + m  ] =  out [ Q + W + m  ]  Where  is energy per unit mass or:  = h + ke +pe (per unit mass)  Chemical potential =  = (  U/  n)  i =  f  For open systems in equilibrium: