Blackbody Radiation Flashcards
The Ideal Gas Law
PV = μRT
P = pressure V = volume μ = no. of moles R = 8.31 (ideal gas constant) T = temperature
First Law of Thermodynamics
ΔQ = ΔU - ΔW
ΔQ = heat supplied to a system ΔU = increase in internal energy of the system ΔW = work done by the system on the environment, this is a negative number as the system is losing the energy
What is the formula for the total work done in changing the volume of a system from Va to Vb at a constant pressure?
W = (Vb,Va) ∫ P dv = P(Vb-Va)
What are degrees of freedom?
-the number of ways that it is possible for kinetic or potential energy to rise
How many degrees of freedom does a particle moving in free space have?
-three degrees of freedom for translational motion corresponding to the x, y, and z directions
How many degrees of freedom does a monatomic gas have?
-three translational degrees of freedom
How many degrees of freedom does a diatomic gas have?
- three translational degrees of freedom
- three rotational degrees of freedom
- one vibrational degree of freedom
- one electronic degree of freedom
- -so a diatomic molecule should have 8 degrees of freedom
Heat Capacity Formula
in terms of energy
C = dQ/dT
C = heat capcity dQ = a small change in energy supplied to the system dT = a small increase in temperature
Heat Capacity of a Gas at Constant Volume
Cv = (z/2)μR
Cv = heat capacity z = no. of degrees of freedom μ = no. of moles R = 8.31 (ideal gas constant)
Heat Capacity of a Gas at Constant Pressure
Cp = μR (z/2 +1)
Cp = heat capacity z = no. of degrees of freedom μ = no. of moles R = 8.31 (ideal gas constant)
Why is heat capacity greater when a gas is at a constant pressure than when it is at a constant volume?
- because when the pressure is constant, the volume can increase
- as the gas expands it does work
- so a proportion of the energy supplied is used to work instead of increasing the internal energy and temperature
- this means that the gas will increase in temperature by a smaller amount than the same gas at constant volume when supplied with the same energy
Predictions of Kinetic Theory
Monatomic Gas
-a monatomic gas has 3 degrees of freedom
-so kinetic theory predicts that:
Cv = 3/2μR
Cp = 5/2μR
-this is strongly supported by experimental data
Predictions of Kinetic Theory
Diatomic Gas
-a diatomic gas has 8 degrees of freedom
-so kinetic theory predicts that:
Cv = 4μR
Cp = 5μR
-this completely disagreed with experimental data
-data showed some agreement if 5 degrees of freedom were assumed instead of 8
-and 7 degrees of freedom seemed a closer approximation to the high temperature data
Boltzmann Factor
n / n0 = e^-(E/kT)
n = number of excited particles
n0 = total number of particles
-the equation tells us the proportion of particles in the state E at temperature T
-it relates the thermal energy of particles to other types of energy (e.g. potential or kinetic)
Boltzmann Factor
Examples
n / no = e^-(mgh/kT)
- isothermal atmosphere
- this Boltzmann factor gives the proportion of particles with potential energy mgh at a temperature T that account for the pressure difference at height h
Why do oscillating charges emit radiation?
- because they are thermally agitated
- ‘kinks’ in the electric field of a charge developed during acceleration are observed as radiation
- this radiation can be characterised by the temperature of the body, and the wavelength or frequency of emission
- a charge that is not accelerating does not emit radiation
Blackbody
Definition
a body that absorbs all of the radiation that falls on it
Whitebody
Definition
a body that reflects all of the radiation that falls on it
Intensity Against Frequency Graph
- constant rate of increase of intensity with frequency to a peak
- decrease in intensity with frequency after that
- the rate of decrease is greater than the rate of decrease
- at a higher temperature, the peak moves to a higher frequency, and the intensity is greater at every frequency
Wien’s Displacement Law
-as temperature increases, peak intensity moves to longer wavelengths/higher frequencies
λmax*T = 2.8978x10^-3
What conditions are required to calculate the cavity radiation?
- must establish thermal equilibrium between the object and the surroundings
- closed system - rates of emission and absorption of radiation are equal
Wave Vector
k = 2π/λ
-points in the direction of motion
Example of a Blackbody
- a box with a single small hole in it
- we also define the boundary condition that only waves with 0 displacement at the edges of the box can fit in
displacement: x=Asin(kx) - so for a wave to fit in the box, Asin(kx) = 0 at x=0 and x=L
How to define all of the wave vectors that will fit in the box?
-for a wave to fit in the box, Asin(kx) = 0 when x=0 and x=L sin(0)=0 gives no solutions sin(kL) = 0 kL = jπ
k = jπ/L where j is an integer
How is a cavity box a quantised system?
- the only waves that can fit in the box must have a wave vector of the form k=jπ/L
- this makes the box a quantised system as there is a set of variables with no values in between them since j has to be an integer
De Broglie Wavelength Formula
λ = h / mv
λ = de Broglie wavelength h = Planck's constant m = mass v = velocity
Density of States
g(k)dk³ = V/8π³ * d³k
Momentum of a Quantum Particle
Equation
ρ = ℏK
ℏ
-represents a reduced Plack’s constant
ℏ = h/2π
What is d³k?
a volume in k-space bounded by k and dk
Separation between adjacent k values that are allowed in the box
dk = π/L
Number of States Present in a Volume, V
g(f)df = 8Vπf² / c³ df
Rayleigh-Jeans Law
Energy Density Equation
ρ(f,T)df = 8πf²/c³ * kB*T
kB = boltzmanns constant T = temperature ρ = energy per unit volume inside the cavity
What does a successful theory of blackbody radiation need to explain?
- must fit the measured energy density data for blackbody radiation
- the energy radiated must be proportional to T^4 by the Stefan-Boltzmann Law
- Wein’s Displacement Law
Stefan-Boltzmann Law
radiance of a blackbody is directly proportional to its temperature
j = Tσ
j = blackbody radiant emittance σ = Stefan-Boltzmann constant, 5.67x10^-8
Number of Waves in the Box
Equation
g(k)d³k = V/(2π)³ d³k
-where d³k is shorthand for a small volume equal to dkxdkydkz, small changes in the three components of the vector k
What is the difference between g(k) and g(k)dk ?
- g(k)dk means the number of waves with k vectors between k and k+dk
- if you integrate g(k)dk over all values of k then you get the total number of waves in the box
Number of Waves in the Box in an Angular Frequency Range dω
g(ω)dω = Vω² / π²c³ dω
-this equation is found by substituting k=ω/c into g(k)dk
Number of Waves in the Box in a Frequency Range df
g(f)df = 8πVf²/c³ df
Number of Waves in the Box in a Wavelength Range d
g(λ)dλ = 8πV/λ^4 dλ
Rayleigh-Jeans Law
Intensity Equation
If(T) = 2f²/c² *kBT
I = intensity, i.e. energy radiated from a surface per unit frequency
Rayleigh-Jeans Law
Total Energy Density Across all Frequencies
E = ∫(2f²/c² *kBT)df
=[2f³/3c² *kBT] (0,∞)
The Ultra-Violet Catastrophe
-when compared with experimental results, Rayleigh-Jeans formula works at low frequencies but diverges at high frequencies
Assumptions Made by Rayleigh-Jeans Law
-assumed that each mode has average energy
= kBT
Planck’s Solution to the Blackbody Problem
-Planck developed the following formula for the energy of each mode
E = hf/(e^(hf/kBT)-1)
-this formula describes the probability that a photon with energy hf exists at temperature, T
Planck’s Radiation Law
frequency form
ρ(f,T) = 8πf²/c³ * hf/(e^(hf/kBT)-1)
Format of Planck’s Equation
density of states(the existing allowed states in the system) * the occupation function appropriate to the particles
How to find Wien’s Displacement Law From Planck’s Law
-differentiate the Planck formula and set it to zero to find the frequency or wavelength at which the peak occurs
Total Energy Density in a Cavity from Planck’s Law
ρ(T) = 8π^5kBT^4/15c³h³
For what objects are the blackbody formulae valid?
- for perfect absorbers/emiters
- in practise most objects are not perfect blackbodies
- to allow us to use the blackbody equations for ‘greybodies’ we can introduce emissivity (ɛ) which is 1 for an ideal body and less for a ‘greybody’
