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Quantum Physics > Blackbody Radiation > Flashcards

Blackbody Radiation Flashcards

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1
Q

The Ideal Gas Law

A

PV = μRT

P = pressure
V = volume
μ = no. of moles
R = 8.31 (ideal gas constant)
T = temperature
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2
Q

First Law of Thermodynamics

A

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
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3
Q

What is the formula for the total work done in changing the volume of a system from Va to Vb at a constant pressure?

A

W = (Vb,Va) ∫ P dv = P(Vb-Va)

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4
Q

What are degrees of freedom?

A

-the number of ways that it is possible for kinetic or potential energy to rise

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5
Q

How many degrees of freedom does a particle moving in free space have?

A

-three degrees of freedom for translational motion corresponding to the x, y, and z directions

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6
Q

How many degrees of freedom does a monatomic gas have?

A

-three translational degrees of freedom

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7
Q

How many degrees of freedom does a diatomic gas have?

A
  • 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
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8
Q

Heat Capacity Formula

in terms of energy

A

C = dQ/dT

C = heat capcity
dQ = a small change in energy supplied to the system
dT = a small increase in temperature
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9
Q

Heat Capacity of a Gas at Constant Volume

A

Cv = (z/2)μR

Cv = heat capacity
z = no. of degrees of freedom
μ = no. of moles
R = 8.31 (ideal gas constant)
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10
Q

Heat Capacity of a Gas at Constant Pressure

A

Cp = μR (z/2 +1)

Cp = heat capacity
z = no. of degrees of freedom
μ = no. of moles
R = 8.31 (ideal gas constant)
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11
Q

Why is heat capacity greater when a gas is at a constant pressure than when it is at a constant volume?

A
  • 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
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12
Q

Predictions of Kinetic Theory

Monatomic Gas

A

-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

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13
Q

Predictions of Kinetic Theory

Diatomic Gas

A

-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

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14
Q

Boltzmann Factor

A

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)

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15
Q

Boltzmann Factor

Examples

A

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
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16
Q

Why do oscillating charges emit radiation?

A
  • 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
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17
Q

Blackbody

Definition

A

a body that absorbs all of the radiation that falls on it

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18
Q

Whitebody

Definition

A

a body that reflects all of the radiation that falls on it

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19
Q

Intensity Against Frequency Graph

A
  • 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
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20
Q

Wien’s Displacement Law

A

-as temperature increases, peak intensity moves to longer wavelengths/higher frequencies
λmax*T = 2.8978x10^-3

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21
Q

What conditions are required to calculate the cavity radiation?

A
  • must establish thermal equilibrium between the object and the surroundings
  • closed system - rates of emission and absorption of radiation are equal
22
Q

Wave Vector

A

k = 2π/λ

-points in the direction of motion

23
Q

Example of a Blackbody

A
  • 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
24
Q

How to define all of the wave vectors that will fit in the box?

A 
-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

25
Q

How is a cavity box a quantised system?

A
  • 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
26
Q

De Broglie Wavelength Formula

A

λ = h / mv

λ = de Broglie wavelength
h = Planck's constant
m = mass
v = velocity
27
Q

Density of States

A

g(k)dk³ = V/8π³ * d³k

28
Q

Momentum of a Quantum Particle

Equation

A

ρ = ℏK

29
Q

ℏ

A

-represents a reduced Plack’s constant

ℏ = h/2π

30
Q

What is d³k?

A

a volume in k-space bounded by k and dk

31
Q

Separation between adjacent k values that are allowed in the box

A

dk = π/L

32
Q

Number of States Present in a Volume, V

A

g(f)df = 8Vπf² / c³ df

33
Q

Rayleigh-Jeans Law

Energy Density Equation

A

ρ(f,T)df = 8πf²/c³ * kB*T

kB = boltzmanns constant
T = temperature
ρ = energy per unit volume inside the cavity
34
Q

What does a successful theory of blackbody radiation need to explain?

A
  • 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
35
Q

Stefan-Boltzmann Law

A

radiance of a blackbody is directly proportional to its temperature
j = Tσ

j = blackbody radiant emittance
σ = Stefan-Boltzmann constant, 5.67x10^-8
36
Q

Number of Waves in the Box

Equation

A

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

37
Q

What is the difference between g(k) and g(k)dk ?

A
  • 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
38
Q

Number of Waves in the Box in an Angular Frequency Range dω

A

g(ω)dω = Vω² / π²c³ dω

-this equation is found by substituting k=ω/c into g(k)dk

39
Q

Number of Waves in the Box in a Frequency Range df

A

g(f)df = 8πVf²/c³ df

40
Q

Number of Waves in the Box in a Wavelength Range d

A

g(λ)dλ = 8πV/λ^4 dλ

41
Q

Rayleigh-Jeans Law

Intensity Equation

A

If(T) = 2f²/c² *kBT

I = intensity, i.e. energy radiated from a surface per unit frequency

42
Q

Rayleigh-Jeans Law

Total Energy Density Across all Frequencies

A

E = ∫(2f²/c² *kBT)df

=[2f³/3c² *kBT] (0,∞)

43
Q

The Ultra-Violet Catastrophe

A

-when compared with experimental results, Rayleigh-Jeans formula works at low frequencies but diverges at high frequencies

44
Q

Assumptions Made by Rayleigh-Jeans Law

A

-assumed that each mode has average energy

= kBT

45
Q

Planck’s Solution to the Blackbody Problem

A

-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

46
Q

Planck’s Radiation Law

frequency form

A

ρ(f,T) = 8πf²/c³ * hf/(e^(hf/kBT)-1)

47
Q

Format of Planck’s Equation

A

density of states(the existing allowed states in the system) * the occupation function appropriate to the particles

48
Q

How to find Wien’s Displacement Law From Planck’s Law

A

-differentiate the Planck formula and set it to zero to find the frequency or wavelength at which the peak occurs

49
Q

Total Energy Density in a Cavity from Planck’s Law

A

ρ(T) = 8π^5kBT^4/15c³h³

50
Q

For what objects are the blackbody formulae valid?

A
  • 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’

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