PHYSICS 215 - Thermodynamics and Modern Physics
Practice Final Exam
Fall 2012
Name:
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PID:
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Problem #1
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Problem #2
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Problem #3
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Problem #4
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Problem #5
_____/6
Problem #6
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Problem #7
_____/6
Problem #8
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Extra Credit
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Total
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EC ____
EC ____
EC ____
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.
Useful Constants:
1 calorie = 4.186 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/m2K4
Avogadro’s number, NA = 6.02E23 mol-1
Coulomb’s constant, (1/4πε0) = 8.99E9 N.m2/C2
Speed of light, c = 3.00E8 m/s
Charge of an electron, -e = -1.6E-19 C
Mass of the electron, me = 9.1E-31 kg = 511 keV/c2 = 5.49E-4 u
Mass of the proton, mp = 1.67E-27 kg = 938.3 MeV/c2 = 1.00728 u
Mass of the neutron, mn = 1.675E-27 kg = 939.6 MeV/c2 = 1.00866 u
Mass of the α particle, mα = 3727.4 MeV/c2 = 4.00151 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, λc = h/mec = 2.4263E-12 m
The Bohr Magneton, µB = 5.79E-5 eV/T
Atomic mass unit, u = 1.66E-27 kg = 931.5 MeV/c2
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.
4
PRAD = σεAT where ε = emissivity and A = area.
1st Law of Thermodynamics: ΔQ = ΔW + ΔU
Ideal gas law: PV = nRT
Work done, ΔW =
Molar specific heats: CV = ΔU/nΔT
CP = ΔQ/nΔT
Adiabatic ==> ΔQ = 0, and PVγ = constant.
vrms = √(3RT/M)
∫PdV
γ = CP/CV
CP = CV + R
Entropy change: ΔS = ∫dQ/T
Carnot engine efficiency, εC = 1 – QC/QH = 1 – TC/TH
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 – β2)
Length Contraction: L′ = L/γ
Addition of Velocities: v′ = (v + u)/(1 + vu/c2)
Momentum – Energy relations:
Planck’s Relation:
E = hν
2
2 2
Time Dilation, T′ = γT
Relativistic Doppler Effect:
√(1 + β)
2 4
E =p c +m c
E = γmc2
p = γmv
ν′ = √(1 – β) ν
K = E – mc2
Einstein’s Photoelectric Law:
hν = K + φ
Compton Effect:
Δλ = λ’ – λ = (1 – cosθ)h/mec
Electrostatic potential at a distance R from a charge Q:
V = (1/4πε0)Q/R
Bohr Quantization Relation: L = mvr = nħ
Atomic Radii: rn = n2a0/Z
where a0 = 5.29E-11 m
Impact parameter:
Atomic Energies:
b = Z1 Z2 e2 cot(θ/2)
8πε0K
n = ρNA/A
f = πb2nt
Fraction of α’s scattered through θ or greater:
4
2
2
N(θ) = Ni n t e Z1 Z2 _____
16 (4πε0)2 r2 K2 sin4(θ/2)
Rutherford Scattering:
de Broglie wavelength: λ = h/p
Heisenberg Uncertainty Principle:
Probability = ψ2
En = -Z2E0/n2
where E0 = 13.6 eV
Bragg’s Law:
ΔpxΔx ≥ ħ/2
Normalization condition:
nλ = 2dsinθ
ΔEΔt ≥ ħ/2
2
∫ψ dx = 1
En = n2π2ħ2/2mL2
Infinite Square Well Potential in 1-dim:
ψ = √2/L sin(nπx/L)
Infinite Square Well Potential in 3-dims:
Simple Harmonic Oscillator: V = ½kx2
E = π2ħ2 (n12/L12 + n22/L22 + n32/L32)
2m
2
ω = k/m
En = (n+½)ħω
Quantum number relations:
L = √l(l+1) ħ
n>0 l<n
|ml| ≤ l
s = ±½ S = √s(s+1) ħ
L=
S
0
P
1
Zeeman Effect:
D
2
F
3
G
4
J=L+S
Lz = mlħ
j=l±s
Spectroscopic Notation:
n2s+1Lj
.
VB = -µ B = µBBml or 2µBBms
Anomalous Zeeman Effect: VB = µBBgmj where g = Landé g-factor = 1 + J(J+1)+S(S+1)-L(L+1)
2J(J+1)
Radioactive decay law:
N = Noe-λt
Activity:
R = λN
Q-value:
Q = (Mx + MX – My – MY)c2
with t1/2 = 0.693/λ
1 Becquerel (Bq) = 1 decay/s
1. [6 points] A Carnot cycle operates between the temperatures of 577 0C and 27 0C. The engine
performs 1200 J of work each cycle, which takes 0.25 s.
(a)
What is the average power of this engine?
(b)
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?
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.
(a)
What is the velocity of the small rocket relative to the Earth?
(b)
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?
3. [6 points] A photon with an energy of 120 keV scatters from a free electron at rest. The angle of
scattering is 600.
(a)
What is the initial wavelength of the photon?
(b)
What is the final energy of the photon?
(c)
What is the final kinetic energy of the scattered electron?
4. [6 points] A proton is trapped in a one-dimensional parabolic potential described by V = ½kx2 (with
k = 16500 N/m) such that the proton executes simple harmonic motion about the origin (x = 0.).
(a)
What is the frequency of its oscillation (in Hz)?
(b)
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?
5. [6 points] Consider the 4f state of atomic hydrogen.
(a)
Sketch all possible spatial orientations of the angular momentum vector L.
(b)
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?
6. [6 points] The radioactive isotope 18F9 has a half-life of 109.8 min. An amount having an activity
of 1E7 Bq is given to a patient.
(a)
What is the activity 48 hours later?
(b)
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.
7. [6 points]
(a)
Calculate the number of α and β particles produced from the decays of 230Th90 down to the stable
end product 206Pb82.
(b)
If the masses of the isotopes are mTh = 230.033127u and mPb = 205.974449u, what is the total
kinetic energy produced in the decay sequence (in MeV)?
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
(a)
Which of the above particles was predicted as the “mediating” particle of the strong interactions?
(b)
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!)