1. 1 On which side of the Coors Event Center arena will the PHYS2010 final exam take place? A)North B)South C)East D) West E) I don’t know! Clicker Question Room Frequency BA Flatirons are west of campus!

2. CAPA assignment #15 is due on tonight at 10 pm. By Sunday final Clicker scores will be posted on CULearn By Monday all Recitation/Lab scores will be posted. If you find a discrepancy, send me email Monday evening to Edward.Kinney@colorado.edu Announcements 2

3. Final Exam is Tuesday morning, Dec 13, 10:30am – 1pm Exam will be held in Coors Event Center, on the West (mountain) side of the arena. Calc-based physics is on the east side; don’t sit down in the wrong side! We are in sections 20, 21, 22, with individual seat assignments. Your seat assignment are posted on CU Learn. Bring student ID. We will be checking that people are in their assigned seat. Practice Exams and Formula sheet are posted on CULearn; you can ignore Prob. 37 on the practice exam Final Exam News 3

4. 4 Solving Calorimetry Problems Problem solution is based on energy conservation: heat lost by hot object must equal heat gained by cold object Heat (transfer without phase change) Q = mcΔT Latent Heat (Phase change) Q = mL In problems with phase changes, you have to figure out the final phase or phases Final state can be all one phase or a mixture of several phases

5. 5 An ice cube is placed in a cup containing some liquid water. The water and ice exchange energy with each other but not with the outside world. After the water and ice come to the same temperature, is it possible the ice could freeze the water rather than the water melt some of the ice? A)Yes, the ice could freeze the water. B)No, the water will always melt some of the ice. Clicker Question Room Frequency BA If the ice is cold enough initially (well below zero), then the heat lost by the liquid to cool it and freeze it could be more than the heat gained by the initial cold ice!

6. 6 Calorimetry Example with Ice - 1 1 kg of ice at temperature -200°C is placed is a container of 100 mL of water at 20°C. What is the final temperature and phase of the system? Preliminary analysis: Will the final state be all ice, all water or a mixture of ice and water? Energy gain to heat ice to 0°C: Q ice =m ice c ice (200-0) with c ice = 2100 J/(kg°C) and m ice = 1 kg, Q ice = 420,000 J Ice will heat up, water will cool to 0°C and start to freeze; how much of the water will freeze? Look at total water freeze. Q water to ice = m water L ice-water = 0.1 kg*333,000 = 33,000 J Energy loss to cool water to 0°C: Q water =m water c water (0-20) with c water = 4186 J/(kg°C) and m water = 0.1 kg, Q water = 8372 J Original ice would gain 8372+33,000 J = 41372 J < 420,000 J Not enough to bring ice up to 0°C => All ice in the final state

7. 7 Calorimetry Example with Ice - 2 1 kg of ice at temperature -200°C is placed is a container of 100 mL of water at 20°C. Q ice =m ice c ice (T f - -200) c ice = 2100 J/(kg°C) Using c water = 4186 J/(kg°C), L water-ice = 333,000 J/kg |Q ice | =m ice c ice (T f + 200 ) = |Q water | =m water c water (20) + m water L water-ice + m water c ice (0 – T f ) Solve for T f and You’ll get Q water =m water c water (0-20) - m water L water-ice + m water c ice (T f - 0) T f = -164°CIs the final T between -200 and 0? Yes! Final analysis: Final state will be all ice. What is T f ?

8. 8 Calorimetry Example with Less Ice - 1 0.05 kg of ice at temperature -200°C is placed is a container of 100 mL of water at 20°C. What is the final temperature/phase of the system? Preliminary analysis: Will the final state be all ice, all water or a mixture of ice and water? Energy gain to heat ice to 0°C: Q ice =m ice c ice (0 - -200) with c ice = 2100 J/(kg°C) and m ice = 0.05 kg, Q ice = 21,000 J Ice will heat up, water will cool to 0°C and start to freeze; how much of the water will freeze? Look at total water freeze. Q water to ice = m water L ice-water = 0.1 kg*333,000 = 33,000 J Energy loss to cool water to 0°C: Q water =m water c water (0-20) with c water = 4186 J/(kg°C) and m water = 0.1 kg, Q water = 8372 J Heat Loss to freeze all water is 8372+33,000 J = 41372 J > 21,000 J Not enough to freeze all water => Ice water mixture in the final state

9. 9 Calorimetry Example with Less Ice - 2 Q ice =m ice c ice (0 - -200) c ice = 2100 J/(kg°C) Using c water = 4186 J/(kg°C), L water-ice = 333,000 J/kg |Q ice | =m ice c ice (200 ) = |Q water | =m water c water (20) + m freeze L water-ice Solve for m freeze : Q water =m water c water (0-20) - m freeze L water-ice + m water c ice (0 - 0) m freeze = 0.038 kgMust be less than original 0.1 kg of water Final analysis: Final state will be ice-water mixture (at 0 °C). How much ice and water will there be? m total-ice = 0.088 kg, m liquid water = 0.062 kg (62 mL)

10. 10 Gases and Absolute Temperature In 1600’s Robert Boyle find gas Pressure P times its Volume V is a constant at constant temperature: PV = constant In late 1700’s Jacques Charles find that at constant pressure, the Volume of the gas changes linearly with temperature T(°C) giving V = constant*(T+267) Inventing a new temperature scale T abs = T(°C) +267, we get Charles’ law V = constant * T abs In early 1800’s, Joseph Gay-Lussac finds Pressure P is proportional to T(K): P = constant*T abs In 1834 Clapeyron put all this together to get PV = nRT abs R is the same for many gases! n is “amount” of gas Avogadro says n is the number of molecules 1 mole = 6.02 x 10 23 molecules More refined T abs is Kelvin scale T(K) = T(°C) + 273.15

11. 11 Gas Constants In SI units, R = 8.314 J/(mole*K) If you want to see this law in terms of individual gas molecules, you divide R by Avogadro’s number to get k, the Boltzmann constant In SI units, k = 1.38 x 10 -23 J/K Now Ideal Gas Law is PV = NkT, where N is the number of gas molecules In many problems, N is a constant, so we have PV/T = constant

12. 12 A balloon full of gas of volume 1 m 3 at pressure 1 atm at temperature 20°C. If I heat the balloon to 40°C and double the pressure to 2 atm, what is the approximate volume of the balloon? A)0.5 m 3 B)1.0 m 3 C)2.0 m 3 D) 4.0 m 3 E) None of the above Clicker Question Room Frequency BA PV/T = constant = (1 atm)(1 m 3 )/293 K = (2 atm)(V)/313 K V =(313/293)/2 m 3 = 0.534 m 3

13. 13 Mechanics and Ideal Gas Law Consider a hollow cube of side length L filled with N atoms Each atom has mass m and average velocity v If the atoms bounces off a wall of the cube it exerts a force on the wall, and the wall exerts a force on the atom L v F

14. 14 Atom Force on Wall The average force is found from the change in the atom momentum F ave Δt = mΔv ≈2mv, so F ave = 2mv /Δt (single atom) For N atoms in cube, about 1/6 are heading towards a given wall Then on average for Δt = L/v,N/6 hit the wall each giving an average force F ave. Average total force F T on a wall is then (N/6)*(2mv)/(L/v) = Nmv 2 /3L Pressure P = F T /L 2 = Nmv 2 /3L 3 P = Nmv 2 /3V or PV = N(2/3)(mv 2 /2) = N(2/3)(KE atom ) v F Δv ≈ 2v

15. 15 Kinetic Theory! PV = NkT from experiment PV = N*number*KE from theory of molecules kT is average kinetic energy of molecules!!! Using proper exact averages you find In a real gas, there is a distribution of velocities square root of average of v 2 is called v rms and we now get formula At same T, molecule speed is faster for lighter molecules

16. 16 Two boxes, one filled with helium (m = 4u) and one filled with nitrogen (m =30 u) are brought to the same temperature T. The box holding nitrogen is twice as large as that holding helium. How are the average kinetic energies of the individual molecules of the two gases related? A)KE He < KE N B)KE He = KE N C) KE He > KE N D)not enough information Clicker Question Room Frequency BA 3kT/2 = average KE so same T, same average KE

17. 17 Two boxes, one filled with helium (m = 4u) and one filled with nitrogen (m =30 u) are brought to the same temperature T. The box holding nitrogen is twice as large as that holding helium. How are the average molecular speeds of the two gases related? A)v He < v N B)v He = v N C) v He > v N D)not enough information Clicker Question Room Frequency BA Nitrogen is heavier, hence v N is smaller

18. On behalf of Profs. Nagle and Uzdensky 18 Thank you for being a great class! Stay calm and rested and do well on the final exam!!!