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PHAS0018: Astrophysical Processes – Nebulae to Stars
[Part marks]
The theoretical curve of growth for the CII 1334.5 A line is shown below, for three
values of the broadening parameter (b = 1, 5 and 20 kms 1
):
In this figure, the column density (N) is in cm 2 and the wavelength is in A(10 10m).
The table below gives the properties of different absorption lines of CII, in its ground
state, observed towards a particular source:
Wavelength (λ) Oscillator strength (f) Equivalent width (Wλ)
1100 A 1.12×10 2 20m A
1425 A 0.11 71m A
2350 A 2.39×10 4 2.35m A
2970 A 0.67 236m A
225μm 1.12 800 A
If we make the assumption that the same curve of growth applies to all CII transitions:
1. Identify on which part of the curve of growth each of these lines occur. [7]
2. Determine the column density of ground state CII (NCII) and the value of the
broadening parameter (b). [7]
3. If the curves of growth for other elements along the same line of sight were found
to have the same value of b, what value, or limit, can be placed on the turbulent
velocity [6]
UCL PHYSICS AND ASTRONOMY 2024 MIDTERM
PHAS 0023 ATOMIC AND MOLECULAR
PHYSICS
The following data may be used if required:
Speed of light in vacuum, c = 2.999 792 × 108 m · s
1
Electron charge, e = 1.602 177 × 10 19 C
Planck constant, h = 6.626 070 × 10 34 J s
Vacuum permittivity, 0 = 8.854 188 × 10 12 F m 1
Bohr Magneton, μB = 9.274 010 × 10 24 J/T
Electron mass, me = 9.109 384 × 10 31 kg
Proton mass, mp = 1.672 622 × 10 27 kg
Atomic mass unit, mu = 1.660 539 × 10 27 kg
Fine structure constant, α = e
2mec/2h = (137.036) 1
Bohr radius, a0 = 0.529 177 × 10 10 m
Rydberg constant, R∞ = 109 737.316 cm 1
1. Helium (He) is a two-electron system with Rydberg constant RHe =
109 722.267cm 1
.
(a) The He ground state electronic configuration is 1s2
.
i. Write down two possible terms describing the He ground
state, indicating which is allowed according to the Pauli
principle. [3]
ii. Write down general expressions for the spatial part of a
two-electron wavefunction. [2]
iii. By considering the exchange of electrons, explain whether
the ground state He spatial wavefunction is symmetric or
anti-symmetric. [3]
1
(b) The first excited state of He has the electronic configuration
1s2s.
i. What are the allowed terms describing this configuration
[2]
ii. Explain which of these terms you would expect to be higher
in energy with respect to the ground state, and why. [2]
iii. Given that energies of the n = 2 triplet and singlet S lev_xfffe_els can be described in terms of quantum defects 2
3S =
0.3108 and 2
1S = 0.1493, determine the energy differ ence between them in cm 1
. [2]
iv. Would it be possible to use a laser to drive transitions be tween the He n = 2 triplet and singlet S levels Explain
your answer in terms of electric dipole selection rules. [2]
v. Determine the possible J and MJ values for the 1s2s ex cited states and write down the description of the corre sponding states using spectroscopic notation. [2]
(c) With reference to indistinguishable fermions, briefly explain the
difference between the Pauli principle and the Pauli exclusion
principle. [2]
2
PHAS0024 Statistical Physics of Matter 2023-23
Answer ALL questions
The numbers in square brackets show the provisional allocation of
maximum marks per question or part of question.
[Part marks]
1. The partition function of a single particle in a cubic box of volume
V is given by:
z(V, T) = 2πmkBT
h
2

3/2
V.
(a) Determine the average energy of the particle in the box. [3]
(b) Calculate the Gibbs entropy. [3]
(c) Give an expression for the Helmholtz free energy in terms of
the average energy and the Gibbs entropy and calculate its
value for the single particle. [2]
(d) Show that dF = pdV SdT, and derive an expression for
the pressure in terms of the Helmholtz free energy and the
volume. [3]
(e) From your answer to question (d) derive an expression for the
average pressure exerted on the walls of the container by the
particle and show this is consistent with the equation of state
for an ideal gas. [3]
PHAS0024/2023-24 CONTINUED
1
2. An ideal gas of N particles undergoes an adiabatic free expansion
where no work is done, no heat is transferred and the temperature
does not change. The volume changes from V1 to V2 in this
process.
(a) Use an equivalent reversible thermodynamic process at
constant temperature to calculate the entropy. [4]
(b) Calculate the change in Boltzmann entropy when the
multiplicity is given by:
= C(N)V
N

2
3
β

3N
2
,
where C(N) is a function of particle number N, and β is the
inverse temperature. [2]
PHAS0024/2023-24 END OF EXAMINATION PAPER
2
PHAS0027 Environmental Physics ICA 2023-24
[R. Saakyan]
PHAS0027/2023-24 1
Answer ALL THREE questions
The numbers in square brackets show the provisional allocation of maximum
marks per question or part of question.
The following data can be used if required:
Standard Temperature and Pressure (STP)
To, Po
273.15 K, 1.01x105Pa
Density of air (at STP) ρ 1.29 kg m-3
Density of water ρw 1000 kg m-3
Saturation vapour pressure of water at STP,
es
0
610.78 Pa
Universal gas constant, R 8.31 J mol-1 K-1
Specific gas constant of air, Rs 287.1 J kg-1 K-1
Specific gas constant of water, Rs(H2O) 461.5 J kg-1K-1
Specific latent heat of evaporation of water,
Ls
2.26xl06 J kg-1
Gravitational acceleration, g 9.81 m s-2
Mean molecular weight of air, 28.96 g mol-1
Specific heat capacity of air, 1003.5 J kg-1 K-1
Stefan-Boltzmann constant, 5.67 10-8 W m-2 K-4
Radius of the Sun, 6.98 108 m
Radius of the Earth, 6400 km
Atomic mass unit, u 1.66 10-27 kg
Speed of light, c 3 108 m s-1
Planck constant, h 6.63 10-34 J s
The Avogadro constant, NA 6.02 1023 mol-1
M
PHAS0027 Environmental Physics ICA 2023-24
[R. Saakyan]
PHAS0027/2023-24 2
1. Using the ideal gas equation in the form = derive the value of
1 DU (Dobson Unit) in terms of column density expressed in molecules/m2
.
[6]
2. a) The radius of the orbit of Venus is 1.08 × 1011 m. Assuming the Sun and
Venus can be both modelled as black bodies and the “glass roof” model for the
Venus atmosphere, what is the expected temperature of its surface You can
assume the Sun’s temperature to be 6000 and the planet’s albedo is 0.75.
[5]
b) The actual surface temperature on Venus is 735 K. Compare the two values
and comment on reasons behind the difference.
[2]
3. The length, width and height of a room are 6m, 5m and 3m respectively. What
is the mass of water vapour in the air of the room if the relative humidity is 50%,
the temperature is 20
oC and the pressure is 1 atmosphere (1.01×105 Pa)
[7]