University of California, Berkeley Physics 7B, Lecture 001, Spring 2009 (Xiaosheng Huang)
Monday, 2/23/2009 6:00-8:00 PM
Physical Constants: Avogadro's number, NA: 6.02×1023 Gas Constant, R: 8.315J/mol·K Boltmann's Constant, kB: 1.38×10-23 J/K Stefan-Boltzmann Constant, σ: 5.67×10-8 W/m2·K4 Specific heat for water: c=4.19×103 J/kg·oC Heat of vaporization for water: LV=22.6×105 J/kg Heat of fusion for water: LF= 3.33×105 J/kg Standard Temperature and Pressure (STP): T=273K, P=1atm=1.01×105 Pa Atomic mass unit (1u): 1.6605×10-27 kg Note: You are allowed one formula sheet (3½ by 5, double sided) and a calculator (without wireless capabilities). Do NOT just write down an answer in the answer box; show your steps. Formulaic answers may only involve the quantities given in a problem and constants. Good Luck!
#1 #2 #3 #4
1. (25 pts.) A spherical solid aluminum sphere with mass M, density ρ, specific heat c, and emissivity e, floats in the vacuum of intergalactic space. The sphere is initially (t=0) at temperature T1. a) How long does it take for the aluminum sphere to cool to the cosmic microwave background (CMB) temperature, T0, assuming T1>T0? (For this part, ignore the radiation energy that the aluminum sphere will absorb from the (CMB)). Formulaic Answer: If T1 = 300K and T0 = 2.7K at t=0 and the mass of the aluminum sphere is M= 100 kg, find the numerical value for the cooling time. For aluminum, e = 0.02, ρ = 2.7×103 kg/m3, and c= 0.90 ×103 J/kg•K. Numerical Answer:
For parts b) and c), take into consideration that the aluminum sphere will absorb radiation energy from the CMB. b) If the temperature of the aluminum sphere at time t is T, what is the net heat loss (|Qnet loss|) of the aluminum sphere at this point? Formulaic Answer: |Qnet loss| = c) Find a differential equation whose solution is T(t), the temperature of the aluminum sphere as a function of time, t. Formulaic Answer:
2. (25 pts.) The mint flavor comes from menthol, which has a molecular mass of M=156.27, in atomic units. A bottle of mint oil at thermal equilibrium with the air in a room at STP is opened at one end of the room at t=0. Assume the menthol molecule is spherical in shape with a radius of R=0.5nm. Also assume the air in the room consists only of N2 (ignore the O2 part) and treat the N2 as a spherical molecule with radius r=0.3nm. Ignore all intermolecular interactions except elastic collisions. a) Find the number density (number of molecules per unit volume, N/V) of the N2 in the room. Numerical Answer: b) Find the rms average speed, vrms, for menthol molecule. Formulaic Answer:
c) Derive an expression for the mean free path l for the menthol molecule in terms of the quantities given in the problem and N/V. (You don’t need to worry about factors of
2 .) Formulaic Answer: d) Given x2=sl2, where x is the displacement, l the mean free path, and s the number of collisions the molecule has suffered traveling through x, find the time, t, it takes for a menthol molecule to reach a displacement of x on average. Formulaic Answer:
e) Suppose the room is x=20m across, find the numerical value for the average time, t, it takes for a menthol molecule to go from one end of the room to the other. Numerical Answer: f) (1 bonus pt.) Does your answer agree with what you know from experience? If not, why not?
3. (25 pts.) At a steam power plant, steam engines work in pairs, the heat output of the first one being the approximate heat input of the second. The operating temperatures of the first are T1= 800 0C and T2 = 410 0C, and of the second T3 = 400 0C and T4 = 250 0C. a) If the heat of combustion of coal is b = 3.0×107 J/kg, at what rate must coal be burned if the plant is to put out p=1000 MW of power? Assume the efficiency of the engines is f = 50% of the ideal (Carnot) efficiency. Formulaic Answer Numerical Answer
b) Water is used to cool the power plant. How much heat per second (ΔQw/Δt) does the water have to absorb? Formulaic Answer Numerical Answer
c) If the water temperature is allowed to increase by no more than ΔT = 6 0C, estimate how much water must pass through the plant per second. You may use ΔQw/Δt in your formulaic answer. Formulaic Answer Numerical Answer:
4. (25 pts.) An ideal gas in a thermally insulated box is separated by a thermally conducting partition into two parts. There are n moles of gas in each part. Initially, the gas in part A has temperature T1 and volume V1 and in B temperature T2 and volume V2. The partition can slide without friction and the two parts have the same pressure, P. a) What is the final temperature when thermal equilibrium is reached? Formulaic Answer b) Calculate the entropy change for the gas in A and for the gas in B. (Please specify the reversible path that you choose for this calculation.) Formulaic Answer
c) Calculate the entropy change of the combined system of A and B, ΔStotal. Formulaic Answer d) (4 bonus pts.) Show that ΔStotal ≥ 0.