Day 3

Energy

The capacity to do work.

Power

The rate at which energy is flowing.

What does 1 W equal?

1 J/s

What does 1 Hp equal?

740 W

Internal Energy

How fast the atoms and molecules in the object are moving.

Temperature

A measure of the internal energy of an object.

Unit conversion from Celsius to Kelvin

K = C + 273.15

Photons

Small discrete packages of energy.

Wavelength

Characteristic size.

Visible

Seen with the human eye.

Infrared

Beyond the red end of the visible spectrum.

Ultraviolet

Wavelength is beyond the violet end of the visible spectrum.

Blackbody

Idealized physical body that absorbs all incident electromagnetic radiation.

Wien’s Displacement Law

Relationship between an object’s temperature and the peak of its emission spectrum.

Wien’s Displacement Law Formula

λ_max = 2897/T

T = Temperature (K)

λ_max = Wavelength of the peak of the emission spectrum (µ)

Stefan-Boltzmann Equation

Relationship between the total power radiated by a blackbody and temperature.

Stefan-Boltzmann Equation Formula

P/a = σT⁴

P/a = Power emitted by a blackbody unit of surface area (W/m²)

σ is the Stefan–Boltzmann constant, σ = 5.67 x 10 -⁸ W/m²/K⁴

T = Temperature (K)

What happens to internal energy and temperature when E_in > E_out?

Increases

What happens to internal energy and temperature when E_out > E_in?

Decreases

What happens to internal energy and temperature when E_out = E_in?

Not changing

Grey Body (still an approximation)

A real-world object that, like a blackbody, has a spectral distribution of radiation that is a constant fraction of a blackbody's radiation at the same temperature, but that fraction is always less than one, making it an imperfect emitter.

Where in the EM spectrum is the visible range?

C

Wavenumber

Number of wavelengths per centimetre

Wavenumber formula

n = 1/λ

An EM wave has a wavenumber of 200 cm ^-1. What is the distance between its wave peaks?

1/200 cm ^-1 = 0.005cm

Why do we use both variables λ and n?

• Wavelength is more intuitive to most people. It allows you to visualize the size of objects that will interact with the wave you are studying (e.g. a 50 meter boat doesn’t care about λ=1 cm waves or λ=100 km waves )

• But wavenumber can be more easily generalized to three dimensions, and can encode directionality

Energy Flux

The rate of energy transfer through a surface.

What are the units for energy flux

W/m²

A solid cube has a volume of 1 m³, and is emitting a flux of 1 W/m2. What is the total power radiated from the cube?

6W, all sides of the box.

Which of the following is FALSE?

(A) All objects absorb and emit electromagnetic radiation

(B) A blackbody emits the same flux at every wavelength

(C) The total emitted flux depends on the temperature of the object

(D) A blackbody is able to absorb all wavelengths of EM flux equally well

(E) The total flux emitted by a blackbody is the area (integral) under the Planck function

Blackbody radiation: Planck Curve

• All objects absorb and emit electromagnetic radiation!

• Radiation occurs over a spectrum of wavelengths

• Warmer objects emit more EM radiation

• Only blackbodies are able to absorb at all wavelengths

If you integrate this curve (e.g. for 220K, calculate the area of the grey shaded region), the units of the integral are:

W/m²

Estimate the total flux (W/m²) emitted by a 300 K blackbody between 500 and 1000 cm^-1

Area = 500 × 0.5 = 250 W/m²

The area is smaller than 5 boxes

If you want the flux in a particular wavelength range

Estimate the area under the Planck curve.

If you want the total flux over all wavelengths

Use Stefan-Boltzmann equation AS the integral of the Planck curve: I = σT⁴

Grey Body Formula

I = εσT⁴

Emissivity (ε)

The RATIO of the actual energy emitted to that which a blackbody would emit.

Is the black body (at the same temperature) a reasonable approximation of the real object?

No: the black body emits more total energy than the real object

Is the grey body a reasonable approximation of the real object?

Yes: total energy emitted is similar

Important note about grey bodies

For graybodies, emissivity is NOT a function of temperature. I changes with temperature, the wavelength of maximum emitted radiation changes with temperature, but ε does not.

What does the grey curve represent?

The grey curve is from a grey body, with a temperature of about 300K.

A blackbody at 280K emits the same total energy as a graybody with emissivity ε = 0.8. What is the temperature of the graybody?

296K

Electromagnetic (EM) Radiation

EM radiation hitting an object (e.g. a gas layer in the atmosphere) can either be reflected, absorbed, or transmitted.

Energy conservation

1 = 𝛼 + abs + tr

1st Law of Thermodynamics

Energy is conserved – it cannot be created or destroyed.

2nd Law of Thermodynamics

“Energy cannot be transferred from a colder body to a hotter body without work being done on the system”.

Two walls are an isolated system. The left wall is initially hotter than the right wall (T_L > T_R). Is a net heat transfer from the left to the right wall physically possible?

Yes

Two walls are an isolated system. The left wall is initially hotter than the right wall (T_L > T_R). Is a net heat transfer from the right to the left wall physically possible?

No

Is there a net transfer of energy from

Not at all

Which wall will emit the most radiative flux?

T_R

The right wall is emitting more radiative flux than the left wall But the temperatures of the walls aren’t changing So, what can we say about how much flux is absorbed by the right wall compared to the flux absorbed by the left wall thinking about the conservation of energy?

Right wall must absorb more radiative flux than the left.

Notes about emissivity

• If a body in equilibrium emits the same amount of flux as it absorbs does not mean that the outgoing radiation will be at the same wavelength (e.g. the Earth’s surface absorbs shortwave and longwave radiation and emits only longwave radiation)

• This is key for the greenhouse effect!

• Different surfaces/layers have different colors because their reflectivity changes with wavelength

• Their emissivities and absorptivities also change with wavelength