Energy, Power, and Climate Change 8.9 The Greenhouse Effect

Energy, Power, and Climate Change8.9 The Greenhouse Effect

INTERACTION BETWEEN LIGHT AND ATOMSRecall that solar radiation strikes the earth at a rate of 1380 W m-2 or less, the farther from the equator you are. That energy is carried in the form of photons, which are quanta of light. The atmosphere is made up of gases, which are the first layer of matter that the sun's rays interact with. If a photon is at the precise energy for an electron to jump to a different energy level in an atom, it will be absorbed.

FYI: A photon of the "wrong" energy will pass right through the gas without interacting.

FYI: A photon of the "right" energy will be absorbed by the atom, and the electron will jump to a new energy level.

FYI: This process is called EXCITING THE ELECTRON. The electron is in an EXCITED STATE.

FYI: If the "right" photon is of sufficiently large energy, the electron can be completely freed from the atom. This is called IONIZATION.


INTERACTION BETWEEN LIGHT AND MOLECULESMolecules can also absorb light energy.

A relatively useful model of molecules (in this case diatomic) has two masses joined by a spring.First, we know that there is a potential energy associated with a spring given by U = (1/2)kx2, where k is the spring constant (representing how stiff the spring is) and x is the amount the spring is displaced from equilibrium.

NN

Second, we know that there is a kinetic energy associated with moving masses (two of them in this case). Note that a diatomic molecule has many modes of motion:

NN NN NN NN

Linear Vibration

NN

Rotation

N

N Translation

FYI: A photon of the "right" energy will cause the molecule to vibrate with a larger amplitude. This is called RESONANCE.

FYI: Resonant frequencies for molecules tend to be low, so the "right" photons are from the INFRARED region of the spectrum.


INTERACTION BETWEEN LIGHT AND SOLIDSSolids contain atoms which interact with each other in such a way that electrons do not need to exist in so few discrete energy levels.Solids therefore can absorb photons of a very wide range of energies (and therefore frequencies: E = hf).The allowable energies of a solid are called bands.

Solids will increase in temperature more readily than gases because of their increased absorption of photons.As the solid heatS up it emits low-energy radiation in the infrared region of the spectrum.Surfaces of solids, liquids, and clouds absorb and reflect light.

The ratio of reflected to absorbed light is called the albedo.

albedo =reflected lightabsorbed light

Albedo

FYI: The albedo for snow is one of the highest, and about 0.9 or 90%. Cloud cover has about the same albedo as snow.

FYI: The albedo for a dark forest is about 0.1 or 10%.

FYI: The average albedo for the earth is about 30%.


INTERACTION BETWEEN LIGHT AND BLACK BODIESYou may recall that black bodies are perfect absorbers of radiation (and also perfect emitters).Suppose we take our cavity blackbody and place a detector at the opening, and heat the cavity blackbody to successively higher temperatures, while measuring the frequencies of the emitted radiation. We get a family of graphs that looks like this:

Inte

nsity

Wavelength (nm)1000 2000 3000 4000 5000

UV

ra

dia

tio

n

IR r

ad

iati

on

Two trends emerge:(1) The higher the temperature the greater the intensity at all wavelengths.(2) The higher temperature the smaller the wavelength of the maximum intensity.

maxT = 2.9010-3 mK Wien's displacement law

vis

ible

ra

dia

tio

n


POWER OUTPUT OF RADIATING MATTERRadiation emitted by hot objects is called thermal radiation.Recall that the total radiation power emitted is proportional to T4, where T is the absolute Kelvin temperature.

P = AeT4 FYI: Stefan's Power law

where = 5.6710-8 W/m2K4 Stefan-Boltzmann constant

A is the surface area of the object, and e is the emissivity of the object and is a unitless number between 0 and 1 that depends on the material emitting the radiation.

FYI: Since all bodies are above absolute zero, all bodies emit thermal radiation.


POWER OUTPUT OF RADIATING MATTERWhat is the energy of one photon of 450 nm light?

E = hf c = fUsing and we can get E in terms of :

f = c E = hc

Energy of a photon of wavelength

E =(6.6310-34)(3108)

45010-9

E = 4.410-19 J 1 eV1.610-19 J

E = 2.8 eV


POWER OUTPUT OF RADIATING MATTERWhat is the wavelength of the most intense radiation coming from the sun if its surface temperature is 6000 K?

maxT = 2.9010-3 mK

max(6000) = 2.9010-3

max = 48310-9 m = 483 nm

What is the power per square meter emitted by the sun? Treat the sun like a perfect emitter.

P = AeT4

P / A = eT4

e = 1

P / A = (5.6710-8)(1)(6000)4

P / A = 7.35107 W m-2 FYI: This is 74 MW per square meter!


POWER OUTPUT OF RADIATING MATTERThe radius of the sun is 7108 m. What is the total power radiated by the sun?

A = r2 = (7108)2 = 1.51018 m2

P / A = 7.35107 W m-2

P = (7.35107 W m-2)A

P = (7.35107)(1.51018)

P = 1.11026 W

FYI: This is 1026 joules each second!


THE TEMPERATURE OF THE EARTHOur first model of the effect of the sun's energy on the temperature of the earth assumes no atmosphere.Ignoring the atmosphere we will estimate how warm the earth should be.The energy per second per unit area provided by the sun is 1360 W/m2. This energy is absorbed only by half of the earth. Why?

1360 W / m2This energy is contained in the disk in heavy yellow, which has a radius of the earth.

FYI: The radius of earth is R = 6.4106 m.

A = r2 = (6.4106)2 = 1.31014 m2

P = 1360A

P = 1360(1.31014)

P = 1.751017 W


Since the average albedo of the earth is 30%, this means that 70% of this power is absorbed:

FYI: The radius of earth is R = 6.4106 m.

P = (0.70)1.751017 W

P = 1.231017 W absorbed energy per secondTo calculate the energy absorbed by the earth, we assume that the whole surface area of the planet is absorbing this energy. Why?

FYI: The surface area of a sphere is A = 4r2.

A = 4r2 = 4(6.4106)2 = 5.21014 m2

P = AeT4

P = 5.6710-8(5.21014)(1)T4

P = 2.95107T4

Question: What assumption did we make about the emissivity of the earth?

From Stefan's power law we can get a handle of the expected temperature of earth (without atmosphere):

Question: If the temperature of the planet is to remain CONSTANT, at what rate must it be emitting energy if it is absorbing energy at a rate of 1.231017 W?

1.231017 = 2.95107T4

T4 = 4.17109

T = 254 K = -19°CFYI: The average temperature of the earth is 288 K = +15 C.

THE TEMPERATURE OF THE EARTH


This estimate is lower than the average temperature of the earth.We may conclude that other factors contribute to increasing the average temperature of the earth.The atmosphere absorbs some of the energy radiated by the earth before it escapes to space.

FYI: The absorption of this radiated energy (or TRAPPING of it) by the ATMOSPHERE is called THE GREENHOUSE EFFECT, since a greenhouse also traps radiated energy.

In our model we assumed that the emissivity was that of a block body. The earth does NOT have a perfect emissivity of 1. We can estimate the emissivity of earth using the Stefan power law and the actual temperature of 288 K:

P = AeT4

P = 5.6710-8(5.21014)e(288)4

P = 2.031017e

1.231017 = 2.031017e

e = 0.61

FYI: Another factor contributing to a higher surface temperature is the fact that the earth contains radioactive elements which produce heat as a byproduct of their disintegration. This is, after all, the energy behind volcanoes and plate tectonics!

FYI: The actual emissivity of earth depends on many things - not just the atmosphere. Clouds, ice packs, deserts, forests, lakes, etc. all contribute to a variable emissivity (and albedo).

THE EMISSIVITY OF THE EARTH


Obviously the atmosphere is the first layer of earth to be struck by incoming light, and so the atmosphere has first dibs on extracting energy from the sun.

ATMOSPHERIC HEAT ABSORPTION

Different gases in the atmosphere absorb different frequencies of light - this is because of the resonant properties of each gas.The atmosphere absorbs about 50% of the incoming solar radiation before it strikes the ground.

incoming solar

radiation

The Sankey diagram for incoming solar radiation looks like this:

20% UV and X-rays

(Ozone)

30% IR(H2O, CO2)

30% arrives at ground


ATMOSPHERIC HEAT ABSORPTION

incoming solar

radiation

20% UV and X-rays

(Ozone)


30% IR(H2O, CO2)


Not all of the remaining 50% of the solar radiation is absorbed by the ground.

SURFACE HEAT ABSORPTION

Depending on the albedo of the ground, some of the radiation is reflected back to the atmosphere.The atmosphere WILL NOT absorb and of the ground-reflected radiation. Why?

incoming solar

radiation

The remaining radiation is absorbed by the ground, increasing its temperature.

20% UV and X-rays

(Ozone)

30% IR(H2O, CO2)


The ground temperature will radiate its own heat according to Wien's displacement law:

Note that CO2 and H2O can both absorb IR radiation.

FYI: This is in the IR region of the spectrum.

maxT = 2.9010-3 mKmax(288) = 2.9010-3

max = 1.0110-6 m = 10100 nm

FYI: Do not confuse the radiation REFLECTED by the ground with the radiation PRODUCED by the ground.

FYI: You can think of the ground as a frequency converter. It absorbs radiation of a variety of frequencies and converts it to infrared radiation.


Absorption by the ground is very complex. Ice, snow, water, sand, forest, crops, cities, etc. all have different heat capacities.

SURFACE HEAT ABSORPTION

incoming solar

radiation

20% UV and X-rays

(Ozone)

30% IR(H2O, CO2)


We can lump all of the specific heat capacities into one overall surface heat capacity Cs, which is defined to be the amount of heat Q needed to raise the temperature of 1 m2 of the ground by 1K.For earth, Cs is estimated to be about 4108 J/K m2.


It is the absorption in the atmosphere of the IR radiation produced by the warm earth that we call the greenhouse effect.

THE GREENHOUSE EFFECT

Since H2O and CO2 are the gases in the atmosphere that are able to absorb IR, we call them greenhouse gases.

incoming solar

radiation

20% UV and X-rays

(Ozone)

30% IR(H2O, CO2)


FYI: Water vapor and carbon dioxide are the principal greenhouse gases.

FYI: If it wasn't for the greenhouse gases, all of the heat radiated by the earth would pass through the atmosphere and be lost to space.

FYI: The greenhouse gases account for much of the difference between the earth's calculated temperature of 254 K, and the earth's actual temperature of 288 K.


THE GREENHOUSE EFFECT

FYI: By the way, since the greenhouse gases absorb IR (both incoming and outgoing) they radiate IR just as the ground does, contributing to the overall temperature of the earth. This makes for a rather involved Sankey diagram!

WITHOUT GREENHOUSE GASES

the ground

the atmosphere

incoming solar

radiation342 W/m2

77

reflection by

atmosphere

reflection by

ground

30

235

235

radiated by earth

WITH GREENHOUSE GASES

greenhouse gases

34277

reflection by

atmosphere

reflection by

ground

30

168

67

absorbed by

greenhouse gases

radiated by earth

390

40

235

324

FYI: At each interface the energy in equals the energy out:

FYI: This means that the temperature of the earth is constant.

Energy, Power, and Climate Change 8.9 The Greenhouse Effect

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