Physics 6 Tutorial 11 - Global Temperatures

This tutorial is for students of the Welsh Board and Eduqas.

 

It is quite a long tutorial, so you might want to work through it in stages.

 

Contents

Thermal Equilibrium

The Greenhouse Effect

Solar Energy Transfer

Black Body Radiation

Absorption of Solar Radiation

Wien's Law

Luminosity of the Sun

The Power of the Sun
Rising Sea Levels

Archimedes' Principle

 

Thermal Equilibrium

The Earth has the right environment for carbon-based life as we know it.  The characteristics of our planet are that:

The lowest temperature recorded on the Earth has been -88 oC, while the highest has been +58 oC.  However, the average temperature is a benign and comfortable 14 oC, even though there are climatic variation.  As I write these notes (July 2018), there have been several weeks of hot and dry weather in the United Kingdom.  However the weather in March and April was decidedly colder than average with a strong easterly wind (the "Beast from the East").

 

The Zeroth Law of Thermodynamics states that when two objects are in thermal contact, heat energy flows from the hotter object to the cooler object until the two reach the same temperature.  When both are at the same temperature, no energy is transferred, a state that is called thermal equilibrium.

 

The Earth is in Thermal Equilibrium with the Sun.  The Sun transfers heat by radiation to the Earth by day.  The Earth reradiates the radiation, mostly at night.

 

 

There is some re-radiation during the day, but most is at night.  This diagram shows a simple version of events, assuming that the radiation from the Sun is evenly distributed during the day, and that the radiation from the Earth is even at night.

 

Question 1

Give two reasons why the model shown in the diagram is too simplistic.

Answer

 

The heating of the Earth's surface is not even, leading to areas around the tropics being much warmer than areas near the pole.  Additionally the Earth rotates on an axis that is about 23 o to the vertical, as a result of which there are the seasons, which are more marked the closer to the poles you are.  These is two of the many reasons that the Earth has climatic and weather variations.  The atmosphere is very energetic as well as chaotic, which is why accurate weather forecasting is so very difficult.  Some of the world's most powerful computers are employed to process the fiendishly complex equations and models that describe the global weather.

 

The Greenhouse Effect

There is much concern about the greenhouse effect.  Let's explain this using a traditional model of a greenhouse;

 

Radiation of all wavelengths (white light) lands on the greenhouse.  Some of it is reflected, but most is transmitted by through the glass, to be absorbed by the soil.  The soil gets warm, and emits a mixture of infra-red radiations, shown by the red arrow with the dark red border.  The glass can transmit the short wavelength infra-red, but reflects the long wavelength infra red.  The glass reduces radiative transfer.  Therefore the greenhouse retains much of the internal energy of its interior, so its temperature is higher than the external temperature.

 

Question 2

Why is internal energy, not heat, referred to in the paragraph above?

Answer

 

The model referred to above is rather simplistic.  The convection of air heated by radiation from the ground has not been considered in the explanation.  Hot air rises (because it's less dense) and collects in the roof.  Opening a roof vent reduces the temperature considerably, by letting out the hot air.

 

 

In the Earth's atmosphere, radiative transfer is reduced by greenhouse gases.  Most gases consisting of molecules of two atoms (diatomic gases) and all gases consisting of three or more atoms can absorb and emit infra red-radiation.  The three main gases in the atmosphere, nitrogen (N2), Oxygen (O2), and argon (Ar) do not absorb or emit infra-red.  They are described as being IR-transparent.  Infra-red radiation may be emitted as a result of collisions between molecules.

 

In the atmosphere, the main gases that can absorb and emit infra-red radiation are:

These are called the greenhouse gases.

 

The picture here shows the idea of carbon dioxide molecules interacting with rays of infra-red radiation which are a mixture of short and long wavelengths.  Carbon dioxide is the most common greenhouse gas.

 

The short wavelength IR photons are not absorbed as much as the long wavelength photons.  These are re-radiated randomly (just like the photons from an excited atom).  Some photons will interact with other greenhouse gas molecules and be re-radiated.  Many will increase the internal energy of the molecules in the atmosphere, resulting in an increase in temperature.  The more carbon dioxide there is, the more likely that such events will happen.

 

The internal energy of carbon dioxide can result in the molecules:

The peak infra-red emission is at about 15000 nm.  This is at the boundary of infra-red and microwaves in the electromagnetic spectrum.  There are other peaks at 2000 nm, 3000 nm, and 4000 nm as shown on the graph:

 

Other molecules have different patterns unique to them.

 

Molecules can absorb energy at only specific wavelengths.  The interaction is through the electric field component of the electromagnetic wave with the charge distribution of the molecule.  The excited state lasts from several microseconds to tenths of a second.  This is many times longer than the excitation lifetime of an electron.

 

Do not confuse the excitation of an electron with the excitation of molecules.

 

Question 3

Calculate the photon energy in J and eV for an infra-red photon of wavelength 15000 nm.

Answer

 

The concentration of carbon dioxide has risen markedly since 1960 as shown in the diagram:

 

 Image by Delorme - Wikimedia Commons

 

Most scientists believe that the rise in CO2 levels is due to human activities.  These include:

The level of carbon dioxide has been increasing since the industrial revolution in the eighteenth and nineteenth centuries.  It was only in the latter part of the twentieth century that people realised that the increase in atmospheric CO2 is not sustainable.

 

Among the effects of increased CO2 emission are:

 

A single abnormally hot summer (or abnormally cold winter) is NOT sufficient evidence for global warming. 

 The trends need to be over decades.

 

Nations are coming together to address this issue.  In these tutorials, we will look at some ways that the problems can be overcome, but it will take a long time.  One example is the abolition of petrol and diesel cars by 2040.  However there are problems with electric cars (nothing new - the electric car was around before the petrol car):

 

Some right-wing politicians, mostly in the USA, say that the whole idea of human-made global warming is nonsense. 

 

Solar Energy Transfer

The only way that energy is transferred from the Sun across 150 million km of space to the Earth is by electromagnetic radiation.

 

Stars like the Sun glow in the same way as other glowing objects.  If we turn the voltage up across a light bulb from zero volts up to its normal voltage, we see the filament glow a dull red, then to orange, to yellow to white.  If we look at a spectrum as this happens, we see that:

The light that we see is the resultant of that mixture of colours and other wavelengths.  On this graph, the visible spectrum is to the left, between 300 and 600 nm.  To the right are wavelengths of infra-red radiation.

 

We do not see green stars because even if the peak wavelength were in the green region, 500 nm, there are also red and blue components as well.  Therefore the star appears white, because red, green, and blue make white.

 

 

Black Body Radiation

We look at the temperature of stars by looking at their colours.  A lot of energy is given off as thermal radiation.  Objects that are red hot have a temperature of about 1200 K.  To understand how the colour of an object depends on its temperature, we need to understand the concept of a black body.  A black body is a perfect absorber so that all radiation that falls on it is absorbed.

 

 

 

A black body is NOT the same as a black hole

 

A perfect absorber is a perfect emitter.  Therefore if we heat it up it will emit radiation including visible light.  This is true (to a first approximation) for stars.  Note the following for black bodies:

 

Absorption of Solar Radiation by the Atmosphere

Most radiation that reaches the Earth's surface is made up of visible and infra-red radiation.  The atmosphere filters out the more energetic short-wavelength radiations like ultra-violet radiation.  Much of the radiation is scattered by the atmosphere (which is why the sky appears blue).  Also the energy will be reflected by clouds and haze.  The solar radiation that reaches the ground directly from the Sun is called direct solar radiationIndirect radiation is radiation that has been scattered by the sky and the clouds or reflected from objects.

 

 

The graph has been simplified to remove peaks at different wavelengths that are transmitted from the Sun.  Also the atmosphere absorbs some frequencies more than others. 

 

Question 4

Explain how the UV of the sunlight outside the Earth's atmosphere differs from the UV detected at sea-level

Answer

 

Some ultra-violet radiation does reach the surface of the Earth.  There are three kinds:

 

Ultra-violet Radiation

Wavelength Range / nm

Notes

UV-A

320 - 400

Not absorbed by ozone.  Can penetrate water depending on turbidity.  Causes sunburn.  Clouds absorb it.  It can inhibit photosynthesis by reducing the efficiency of electron transport.  It can cause fluorescence.  See Quantum Physics Tutorial 5 .

UV-B

280 - 320

It can penetrate up to 20 m in seawater, depending on turbidity and chemistry.  Penetrates less far in fresh water.  Photons are more energetic than UV-A, and cause mutations by making Thymine dimers in the DNA.  This can lead to cancer.  It appears to impair photosynthesis in plants.

UV-C

200 - 290

Although UV-C makes up 0.5 % of the radiation, it is the most damaging to living organisms.  It is absorbed readily by ozone in the stratosphere.

 

Question 5

Some years ago, it was discovered that chlorofluorocarbons (very un-reactive molecules used as aerosol propellants and refrigerants) were interacting with ozone in the stratosphere, leading to "the hole in the ozone layer".  Explain why this caused a great deal of alarm, leading to a world wide ban on these substances.

Answer

 

Wien's Law

The peak wavelength is lmax which is the wavelength at which maximum energy is radiated.  This is inversely proportional to the Kelvin temperature.  It is called Wien's  Displacement Law (as the peak is displaced towards shorter wavelengths).  We write it as:

 

lmax T = constant = 0.00289 m K

 

 

Worked example

What is the peak wavelength of a black body emitting radiation at 2000 K?  In what part of the electromagnetic spectrum does this lie?

Answer

lmax = 0.00289 m K 2000 K

lmax = 1.45 x 10-6 m = 1450 nm

This is in the infra-red region.

 

Question 6

The Earth emits radiation with a peak of 10.5 mm.  What temperature does this correspond to?

Answer

 

 

Luminosity of the Sun

 

The area under the graph above is related to the rate at which a black body radiates energy.  The luminosity of a star is the total energy given out per second, so it's the power.  From the graph the luminosity increases rapidly with temperature, which gives rise to Stefan's Law.  Formally this is stated as:

 

The total energy per unit time radiated by a black body is proportional to the fourth power of its absolute temperature.

 

In other words double the temperature and the power goes up sixteen times.  In physics code we write:

 

[P- Power (W); s - Stefan's constant; A - area (m2); T - temperature (K)]

 

The strange looking symbol s is "sigma", a Greek letter lower case 's'.  It Stefan's Constant.

 

s = 5.67 x 10-8 W m-2 K-4

 

We can treat a star as a perfect sphere (A = 4pr2) and a perfect black body.  So for any star, radius r, we can write:

 

 P = 4pr2sT4

 

(Note: in some text books the power may be represented as luminosity with the physics code L)

 

Stars with the same absolute magnitude have the same power output.  We can justify this statement by considering stars P and Q:

 

 

We can equate the two expressions to give:

APsTP4 = AQsTQ4

   

So we can write:

 

So if the temperatures are the same, the areas will be the same.  Therefore the radii will be the same.

 

Question 7

If the Sun has a radius of 6.96 x 108 m and a surface temperature of about 6000 K, what is its total power output?  What is the power per unit area?  What is the peak wavelength? 

Answer

 

The Power of the Sun

On the equator, the average intensity of the Sun's rays is about 1400 W m-2.  In practice, some is absorbed by the atmosphere, and some is reflected as heat, but we will use this in a calculation to work out the power given out by the Sun.

 

 

 

We can work out the power of the Sun by working out the total area of a sphere that has the radius of the Earth's orbit.

 

A = 4pr2 = 4 p (1.50 1011)2 = 2.83 1023 m2

 

Since each square metre receives 1400 W, the total power of the Sun is:

 

2.83 1023 m2 1400 W m-2 = 3.96 1026 W = 4.0 1026 W (to 2 s.f.)

 

You may have noticed that this figure is slightly lower than the answer worked out in Question 7, but some energy is absorbed and reflected, so 1400 W m-2 is slightly too low.

 

 

Rising Sea Levels

Twenty thousand years ago, the British Isles was a peninsula on the north-west coast of Europe.  It was between two large rivers, the Seine and the Rhine.  Doggerland, off the current North Sea coast was a range of low hills, home to herds of animals and the first human inhabitants of this region.  Between the south coast of England and France, a range of chalk hills separated the two great rivers.  The soil in Kent is the same as that in Champagne, and Kentish wine producers make a sparkling wine that is very similar to Champagne.

 

During the ice age, there were very cold winters, but the summers could be benign.  The global mean temperature then was 10 oC, compared to 14 oC today.  The ice-sheets that covered much of Northern Europe started to melt.  About seven thousand years ago (yesterday in geological terms), a massive lake of melt-water built up in what is now the southern North Sea, hemmed in by ice dams.  One of these burst resulting in a massive flood that breached the range of hills between the Rhine and the Seine to form what is now the English Channel.  It also swept away the hill that formed one side of the Solent Valley to separate the Isle of Wight from the mainland. 

 

The sea-level has risen by about 120 m during the melting process which started 19000 years ago and finished 6000 years ago.  The sea-level rose by about 1 metre every year.  Rising sea-levels are therefore nothing new.  The sea is actually retreating on the west coast of Wales.  Harlech Castle was on sea-cliffs when it was built in the Fourteenth Century.  It is now about 3 km inland.  The UK mainland is rising more in the west and sinking to the east.  This is because the land is rising more in the west as it has been relieved of the weight of the ice.

 

Global warming is a major concern as it will cause sea-levels to rise, leading to low-lying lands being flooded.  London could well fall victim in a couple of hundred years. Let us have a look at why the sea levels are rising:

These are intuitive.  Let's consider what happens to a floating iceberg as it melts.  How much does floating ice contribute to the rise in sea-levels?  To answer this, we need to think about Archimedes' Principle and density.  See Materials Tutorial 1 to revise density.

 

 

Archimedes' Principle

Any object that floats in a liquid has an upthrust that is equal to its weight.  Since the weight acts vertically downwards, the upthrust acts vertically upwards.  The idea is shown below:

 

The weight of water displaced is the same as the weight of the object.  If the upthrust is less than the weight, the object will sink.

 

When an object is totally immersed in the water, the volume of water displaced is the volume of the object.  You will have used this idea to find out the density of an irregular object.  See Materials Tutorial 1.

 

Archimedes' principle states:

Any body wholly or partly immersed in a fluid experiences an upthrust equal to the weight of the fluid displaced

 

The equation associated with Archimedes Principle is:

Please see Materials Tutorial 4 for the derivation.

 

The volume of the object is tA.  This volume is the same as the volume of liquid that is displaced.  The mass of liquid displaced is volume density, so the mass of liquid displaced is tAr.  Therefore the weight displaced is tArg.    We can write this in terms of the volume:

 

Fr = Vrg

 

Question 8

A diver is salvaging a spherical cannon ball from the wreck of an ancient ship.  The cannon ball has a diameter of 10 cm and is made of iron of density 7900 kg m-3.  The density of seawater is 1030 kg m-3.

 

What is the force needed to lift the cannon ball:

(a) on land;

(b) under the water?

 

Does the depth of the wreck matter?

 

Acceleration due to gravity = 9.81 m s-2.

Answer

 

So let's model a melting iceberg using a cube of ice from the freezer placed in 200 cm3 water of temperature of 0 oC.

 

 

The ice cube has sides that are 0.030 m.  Therefore the volume is 27 10-6 m3.  The density of water ice is 919 kg m-3.  The density of water is 1000 kg m-3.

 

Question 9

Show that the mass of the water in the ice cube is about 0.025 kg.

What is the weight?

Answer

 

The weight of water displaced will be 0.243 N by Archimedes' Principle.

 

Question 10

What is the volume of water displaced by the ice cube?

What is the level to which the water would rise, assuming there was a volume scale on the beaker.

Answer

 

As the ice melts, less water would be displaced, until eventually the 25 cm3 of water in the cube is incorporated into the water in the beaker.  The number of water molecules in the 27 cmof water ice at a density of 919 kg m-3 is the same as the number of water molecules in 25 cm3 of liquid water at a density of 1000 kg m-3.  Therefore the water level remains the same.

 

From this, we can conclude that melting sea ice would not cause sea-levels to rise.  The ice at the North Pole is sea-ice, so the melting of the sea-ice there would not have a significant effect on sea-levels.

 

Question 11

A glacier from a mountain range feeds ice into the sea.  Every so often it "calves", meaning that a chunk breaks off and floats away as an iceberg.  Would it contribute to the rise sea-levels?

Answer

 

The melting of the Antarctic ice-cap would make the sea-level rise by 61 metres.