PHYSICS Thermodynamics and Modern Physics. Practice Final Exam. Fall 2011

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1 PHYSICS Thermodynamics and Modern Physics Practice Final Exam Fall 2011 Name: PID: Problem #1 Problem #2 Problem #3 /6 /6 /6 Problem #4 /6 EC Problem #5 /6 Problem #6 /6 EC Problem #7 /6 Problem #8 /8 EC Extra Credit Total /50 I will attempt to avoid multiple jeopardy. Please help me by giving an algebraic symbol for the answer to each part if you use it in subsequent parts.

2 Useful Constants: 1 calorie = J Latent heat of vaporization of water = 539 cal/g = 2256 kj/kg Latent heat of fusion of water = 79.5 cal/g = 333 kj/kg Specific heat of water = 1 cal/g = 4.19 kj/kg 1 atmosphere = 1.01E5 Pa Universal Gas Constant, R = 8.31 J/mol.K Boltzmann s constant, k = 1.38E-23 J/K Stefan-Boltzmann constant, σ = 5.67E-8 W/m 2 K 4 Avogadro s number, N A = 6.02E23 mol -1 Coulomb s constant, (1/4πε 0 ) = 8.99E9 N.m 2 /C 2 Speed of light, c = 3.00E8 m/s Charge of an electron, -e = -1.6E-19 C Mass of the electron, m e = 9.1E-31 kg = 511 kev/c 2 = 5.49E-4 u Mass of the proton, m p = 1.67E-27 kg = MeV/c 2 = u Mass of the neutron, m n = 1.675E-27 kg = MeV/c 2 = u Mass of the α particle, m α = MeV/c 2 = u Planck s constant, h = 6.63E-34 J.s = 4.14E-15 ev.s Planck s reduced constant, ħ = h/2π = 1.05E-34 J.s = 6.58E-16 ev.s Compton Wavelength of the electron, λ c = h/m e c = E-12 m The Bohr Magneton, µ B = 5.79E-5 ev/t Atomic mass unit, u = 1.66E-27 kg = MeV/c 2 1 Curie = 3.7E10 Bq Useful Formulae ΔQ = mcδt where m = mass, c = specific heat. Heat conduction, I = ΔT/R in Watts where R = thermal resistance = Δx/kA and Δx = thickness, A = area and k = thermal conductivity of the material. P RAD = σεat 4 where ε = emissivity and A = area. 1 st Law of Thermodynamics: ΔQ = ΔW + ΔU Ideal gas law: PV = nrt Work done, ΔW = PdV v rms = (3RT/M) Molar specific heats: C V = ΔU/nΔT C P = ΔQ/nΔT C P = C V + R γ = C P /C V Adiabatic ==> ΔQ = 0, and PV γ = constant. Entropy change: ΔS = dq/t Carnot engine efficiency, ε C = 1 Q C /Q H = 1 T C /T H Potential energy lost by a charge q in traversing a potential difference of V is U = qv Wave relation: v = νλ where v = velocity, ν = frequency, λ = wavelength. β = v/c γ = 1/ (1 β 2 ) Length Contraction: L = L/γ Time Dilation, T = γt Addition of Velocities: v = (v + u)/(1 + vu/c 2 ) Relativistic Doppler Effect: ν = (1 β) ν (1 + β) Momentum Energy relations: E 2 = p 2 c 2 + m 2 c 4 E = γmc 2 p = γmv K = E mc 2 Planck s Relation: E = hν Einstein s Photoelectric Law: hν = K + φ

3 Compton Effect: Δλ = λ λ = (1 cosθ)h/m e c Electrostatic potential at a distance R from a charge Q: V = (1/4πε 0 )Q/R Bohr Quantization Relation: L = mvr = nħ Atomic Radii: r n = n 2 a 0 /Z Atomic Energies: E n = -Z 2 E 0 /n 2 where a 0 = 5.29E-11 m where E 0 = 13.6 ev Impact parameter: b = Z 1 Z 2 e 2 cot(θ/2) n = ρn A /A 8πε 0 K Fraction of α s scattered through θ or greater: f = πb 2 nt Rutherford Scattering: N(θ) = N i n t e 4 Z 1 2 Z (4πε 0 ) 2 r 2 K 2 sin 4 (θ/2) de Broglie wavelength: λ = h/p Bragg s Law: nλ = 2dsinθ Heisenberg Uncertainty Principle: Δp x Δx ħ/2 ΔEΔt ħ/2 Probability = ψ 2 Normalization condition: ψ2 dx = 1 Infinite Square Well Potential in 1-dim: ψ = 2/L sin(nπx/l) E n = n 2 π 2 ħ 2 /2mL 2 Infinite Square Well Potential in 3-dims: E = π 2 ħ 2 (n /L 1 + n 2 2 /L n 2 3 /L 2 3 ) 2m Simple Harmonic Oscillator: V = ½kx 2 ω 2 = k/m E n = (n+½)ħω Quantum number relations: n > 0 l < n L = l(l+1) ħ m l l L z = m l ħ s = ±½ S = s(s+1) ħ J = L + S j = l ± s S P D F G Spectroscopic Notation: n 2s+1 L j L = Zeeman Effect: V B = -µ.b = µ B Bm l or 2µ B Bm s Anomalous Zeeman Effect: V B = µ B Bgm j where g = Landé g-factor = 1 + J(J+1)+S(S+1)-L(L+1) 2J(J+1) Radioactive decay law: N = N o e -λt with t 1/2 = 0.693/λ Activity: R = λn 1 Becquerel (Bq) = 1 decay/s Q-value: Q = (M x + M X M y M Y )c 2

4 1. [6 points] A Carnot cycle operates between the temperatures of C and 27 0 C. The engine performs 1200 J of work each cycle, which takes 0.25 s. What is the average power of this engine? What is its efficiency? (c) How much energy is extracted as heat from the high temperature reservoir each cycle? (d) How much energy is delivered as heat to the low temperature reservoir each cycle? (e) What is the entropy change of the working substance of the engine for the energy transfer to it from the high temperature reservoir? (f) What is the net entropy change of the working substance over one cycle?

5 2. [6 points] A large spaceship is moving away from Earth with a speed of 0.8c relative to the Earth. An astronaut realizes that he has forgotten to pack his ipod so he returns to Earth in a small rocket that has a velocity of 0.9c relative to the large spaceship. What is the velocity of the small rocket relative to the Earth? The radio receiver at the Space Flight Center on Earth is tuned to a frequency of 95.0 MHz. To what frequency must the large spaceship set its radio transmitter in order to communicate with Earth? (c) What should be the frequency of the transmitter on the small rocket in order to communicate with Earth?

6 3. [6 points] A photon with an energy of 120 kev scatters from a free electron at rest. The angle of scattering is What is the initial wavelength of the photon? What is the final energy of the photon? (c) What is the final kinetic energy of the scattered electron?

7 4. [6 points] A proton is trapped in a one-dimensional parabolic potential described by V = ½kx 2 (with k = N/m) such that the proton executes simple harmonic motion about the origin (x = 0.). What is the frequency of its oscillation (in Hz)? What is the minimum possible energy for the harmonic oscillator? (c) What is the longest wavelength of light emitted in a transition between energy levels of this system? [Extra Credit] What is the color of this light?

8 5. [6 points] Consider the 4f state of atomic hydrogen. Sketch all possible spatial orientations of the angular momentum vector L. What is the minimum angle between L and the z-axis? (c) If the atom is placed in a magnetic field of B = 2.5 T, what is the energy difference between adjacent levels?

9 6. [6 points] The radioactive isotope 18 F 9 has a half-life of min. An amount having an activity of 1E7 Bq is given to a patient. What is the activity 48 hours later? How many grams of Fluorine-18 were initially in the sample? (c) Fluorine-18 is a β+ emitter. What nucleus does it decay into? (i.e. give A and Z for the final element.) [Extra Credit] Name the final element.

10 7. [6 points] Calculate the number of α and β particles produced from the decays of 230 Th 90 down to the stable end product 206 Pb 82. If the masses of the isotopes are m Th = u and m Pb = u, what is the total kinetic energy produced in the decay sequence (in MeV)?

11 8. [8 points] Consider the following particles:- The photon The neutron The π-meson (pion) The µ-meson (muon) The k-meson (kaon) The neutrino Which of the above particles was predicted as the mediating particle of the strong interactions? Which of the above particles is a gauge particle? (c) Which two of the above particles are leptons? (d) Which of the above particles possesses the property of strangeness (i.e. contains a strange quark)? (e) Which of the above particles is a baryon? (f) The present theory of the strong interactions, known as Quantum Chromodynamics, predicts that quarks can be found in how many colors? (g) Experiments have led physicists to believe that there are how many flavors of quarks? [Extra Credit] Name the flavors. (Hint: not chocolate, vanilla and strawberry!)

Notes for Special Relativity, Quantum Mechanics, and Nuclear Physics

Notes for Special Relativity, Quantum Mechanics, and Nuclear Physics 1. More on special relativity Normally, when two objects are moving with velocity v and u with respect to the stationary observer, the