PHYS-307 Midterm Questions With 100% Correct Answers 2024/2025
Energy as a function of quantum number - ANS-= -13.6 eV (Z²/n²)
lifetime of an orbit/time to decay to lower energy level - ANS-t = E/(dE/dt) = E_n/P
= 4.7x10^-11 s (h⁶/Z⁴)
energy difference between two states, n1 and n2: - AN...
relation between kinetic energy and thermal energy of a particle - ANS-✔✔1/2 mv² = kT
,What is the maximum of the Maxwell-Boltzmann distribution, and how do we find it? What does it
represent? - ANS-✔✔v_p = (2kT/m)^1/2
It represents the most probable speed for a particle
average speed of a particle - ANS-✔✔<v> = 1.6 (kT/m)^1/2
mean kinetic energy per particle: - ANS-✔✔<E> = 3/2 kT
Rotational Broadening - ANS-✔✔seen in rapidly rotating objects
Which three types of broadening are all very general Doppler-based motion/line-of-sight effects? - ANS-
✔✔thermal, turbulent, and rotational broadening
optical depth - ANS-✔✔a measure of how much the intensity of light is attenuated by traveling through
the gas
T(x) = σN(x)
if N is constant:
T(x) = σnx
optically thin - ANS-✔✔T(x) << 1
σnx << 1
optically thick - ANS-✔✔T(x) >> 1
σnx >> 1
,mean free path (mfp) - ANS-✔✔average distance a photon will travel through a gas before being
absorbed
-for a gas with constant n and σ:
<x> = 1/nσ
mfp is the distance over which the optical depth grows from T=0 to T=1
Curve of growth:
Describe line and profile anatomy at low optical depth T₀<1 - ANS-✔✔the shape and width of the
absorption line is determined by thermal Doppler broadening
-Gaussian shape, width depends on temperature and molecular mass of absorbing particles. the wings
are weak
Curve of growth:
Describe line and profile anatomy at larger optical depth, 1<T₀<10⁴ - ANS-✔✔the line core saturates,
the line is black at the center because none of the photons with a wavelength enar the line center make
it through the absorbing gas. the width of the line only grows slowly with optical depth
Curve of growth:
Describe line and profile anatomy at very large optical depth, T₀>10⁴ - ANS-✔✔the cumulative effects
of the (v-v₀)² wings of the Lorentz profile are important. the column density of atoms is now so large
that photons far from the center (|v-v₀|>>γ) have a fair probability at being absorbed
-even at low resolution, absorption lines are easily detected and measured
equivalent width - ANS-✔✔width of a rectangle from continuum to zero with the same area as a line;
use FWHM or triangle approximation
W = ∫(I_cont-I_line)/I_cont dλ = ∫[1-e^(-Tλ)]dλ
, curve of growth - ANS-✔✔a plot of an absorption line's equivalent width W as a function of T₀ (or a
function of column density N of the absorbing atoms) with 3 parts
Local Thermodynamic Equilibrium - ANS-✔✔all particles characterized by the same temperature T,
where:
½mv² = (3/2)kT;
all molecules have frequent collisions, so they all have the same average kinetic energy and temperature
What do we need to achieve LTE with photons? - ANS-✔✔1. a high number density of photons and
massive particles (nuclei)
2. optically thick at all wavelengths- otherwise particles, photons are decoupled
What does the Boltzmann equation tell us? - ANS-✔✔gives the relationship between the number of
atoms in one excitation state to another, e.g., n₂/n₁, and allows us to relate what we see in spectral lines
to the physical conditions of the gas
Trends for a fixed E_B and E_A, and for a fixed T in the Boltzmann equation - ANS-✔✔1. As T → ∞,
N_B/N_A → g_B/g_A
2. As T → 0 , N_B/N_A → 0
When T is fixed:
1. As E_B increases, or E_A-E_B→∞, N_B/N_A→0
2. as E_A-E_B → -kT, N_B/N_A → e-¹
3. As E_A-E_B → 0 , N_B/N_A → g_B/g_A
What is different between the Saha and Boltzmann equations? - ANS-✔✔electron density dependence
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