Blackbody Radiation – Mathematica Analysis
Due:
Tuesday, Jan. 27 or Thursday, Jan 29
Format: Does not need to follow lab-report format -- can be Mathematica printed output.
However… make sure you label the problems as you go so that it is clear
which problem you are answering when.
Also, make sure name is on report
You do not need to turn in the introductory exercises that we did as a group.
Planck’s Equation for Blackbody Radiation
8π hc
ρ ( λ ,T ) = 5 h c / λ kT
λ (e
−1)
deBroglie Relationship (for matter)
p=h/λ
Values for Selected Constants (in SI units)
Planck’s Constant
Boltzmann Constant
Speed of Light
Electron Mass
Proton Mass
Energy Conversions
h
k
c
me
mp
1 eV
= 6.626 x 10-34
= 1.380 x 10-23
= 3.000 x 108
= 9.109 x 10-31
= 1.673 x 10-27
= 1.602 x 10-19
J·s
J/K
m/s
kg
kg
J;
1 cm-1 = 1.986 x 10-23 J
Exercise 1:
A) Use Plain Text to put your name at the top of the Mathematica document.
B) Using Plank’s equation for blackbody radiation, define an expression in Mathematica for
the power density as a function of the variables λ and T.
- You will want to define any universal constants before defining the equation.
C) Use the defined Blackbody equation to graph the power density at T = 5000K, 6000K, and
8000K. The x-axis range of the graph should be broad enough to include all the peaks.
[Hint: The peak for 5000K is around 650nm. What is a starting point guess that you could
use for the x-axis (λ) min. and max. range if you are plotting λ in SI units?]
E) Use Plain Text on your document to describe in words how the different graphs vary
from one another as a function of temperature.
Gentry, 2015
Exercise 2:
A) What is the temperature of a blue flame coming from a Bunsen burner?
(Will need to make iterative guesses for T to find correct λmax)
B) Estimate the temperature of a yellow flame in a candle.
Exercise 3:
A) Temperature not only affects the peak wavelength. It also affects the total energy density
coming from the blackbody. Calculate (integrate) the total emittance at 4000K and at
10,000K.
c
Total Emittance = ∫ ρ ( λ ,T )d λ where c = speed of light
4
Evaluate the integration from 10-9m to 10-3 m
Should get answers on order of 10+7 or 10+8. If not then use smaller λ range.
Exercise 4:
There has been a large push to replace traditional incandescent light bulbs with modern
compact fluorescent lights (CFL’s) or with solid-state LED lights. The reason for the
change is the poor efficiency that incandescent bulbs demonstrate for lighting rooms. The
color temperature of an incandescent bulb is approximately 3000K.
A) Integrate to find the radiant energy (emittance) emitted by a light bulb across the total
wavelength region of 180nm to 10-4m.
B) Find the partial amount and the percentage-split of emittance for each of the following
regions: UV (180-350nm), Visible (350-750nm), and IR ( 750nm - 10-4 m).
C) Which region has the largest emittance? Which has the smallest emittance?
D) What do your results imply about the energy efficiency of using traditional incandescent
lights to provide visible reading light in a room?
Exercise 5:
Use deBroglie’s Relationship as well as classical expressions for momentum and KE,
to find the wavelength of:
A) Electron with kinetic energy = 1.5eV,
B) H2 molecule with KE=kT at 300K (i.e. has thermal energy at room temperature)
C) Tennis ball (57g) at 80km/h,
Hint: when in doubt when doing calculations, convert everything to SI units since h is
given in SI units. This hint holds for most calculations in our P. Chem. class.
(J, W, kg, m, m3 (1000L), sec, Kelvin, Coulomb)
Classical expressions: p = mυ and KE = ½ mυ2 = p2/2m,
where KE = kinetic energy, υ = velocity, m = mass,
Show work. Answers can be found in lecture outline.