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25.1: The Physics of Global Warming

Difficulty Level: At Grade Created by: CK-12
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The observed global warming on Earth is a manifestation of the Second Law of Thermodynamics. The Earth operates like any heat engine. Input heat from solar radiation and exhaust heat (terrestrial radiation) largely determine the operating temperature (global surface temperature). Over geological periods this heat exchange reaches equilibrium and the temperature is stable. If the input heat increases or the exhaust heat decreases the temperature rises and vice versa. Natural processes over geologic time have changed the input and affected both output heat and temperature. In the present era the quantity of exhaust heat is being rapidly restricted by the greenhouse effect; consequently, the earth's temperature must rise to reach equilibrium. How much higher it must rise depends entirely on human activity.

The input heat -- solar energy received -- is a function of solar activity and oscillations in characteristics of the Earth’s orbit.

The quantity of exhaust heat, terrestrial radiation, is largely a function of the presence of certain gases in the atmosphere that absorb outgoing infrared radiation. This is known as the greenhouse effect. The greenhouse effect is due to the differential absorption of certain wavelengths of solar as compared to terrestrial radiation.

The solar energy reaching the surface of the Earth is concentrated in short wavelengths, which can easily penetrate the greenhouse gases, such as Carbon Dioxide and Methane. The Earth, however, is cooler than the sun and it radiates its heat in the form of energy in the far infrared range. These longer wavelengths are partially absorbed by the greenhouse gases and some of the solar heat is returned to Earth. At a certain temperature these processes are in equilibrium and the surface temperature of the Earth is stable. However, if more greenhouse gases are put in the atmosphere the amount of trapped terrestrial radiation increases, leading to an increase in global temperature.

Currently the heating effect of extra greenhouse gases (since the start of the industrial revolution) is equal to about \begin{align*}1.0 \;\mathrm{W/m}^2\end{align*}. Thus the recent period has recorded parallel increases in concentration of carbon dioxide and average global temperature. As more greenhouse gases are put into the atmosphere the temperature will increase further. There are certain effects of a warmer Earth (discussed below) which could accelerate the process, even if no more greenhouse gases are put into the atmosphere (an unlikely prospect for the foreseeable future).

The Key Concepts (Possible Effects That Can Accelerate Global Warming)

  1. Time Lag: The excess energy warms the ocean very slowly, due to water’s high heat capacity. Even in the unlikely event that no more greenhouse gases are added to the atmosphere the temperature increase already measured will be nearly doubled.
  2. The Effect of Water Vapor: Increasing temperatures will lead to more evaporation and more water vapor in the atmosphere. Water vapor is a greenhouse gas and its increased presence may cause further warming in a positive feedback loop. On the other hand if the water vapor results in more clouds more solar radiation will be reflected, a possible negative feedback.
  3. Albedo is the amount of light reflected by a surface. Sea ice has an albedo of .\begin{align*}85\end{align*}, meaning \begin{align*}85\end{align*}% of light is reflected back from its surface (and leaves the Earth) and \begin{align*}15\end{align*}% is absorbed and stays in the Earth; ice-free water has an albedo of \begin{align*}.07.\end{align*}(\begin{align*}93\end{align*}% of the solar energy is absorbed.) Thus the observed melting of sea ice could amplify the effect of global warming
  4. The melting of the Arctic Permafrost also has an amplifying effect by releasing carbon dioxide and methane that is normally trapped in the tundra.
  5. Warmer oceans are hostile to algae and cytoplankton , which are the most important absorbers of carbon dioxide. The loss of the these two photosynthesizers would remove the most important natural \begin{align*}CO_2\end{align*} sink.
  6. Loss of Rainforests would have a similar effect. Global warming is likely to lead to desertification of the habitats of rainforests. The rainforest is the second most important \begin{align*}CO_2\end{align*} sink.

The Key Concepts (Physical Laws and Observations)

1. The relationship between temperature of a body and its radiation wavelength is given by Wien’s Law: For any body that radiates energy, the wavelength of maximum energy radiated is inversely related to the temperature.

2. The effect of global warming on the solubility of Carbon Dioxide \begin{align*}(CO_2)\end{align*} and methane \begin{align*}(CH_4)\end{align*} is governed by two laws that have opposing effects. Henry’s Law: The solubility of a gas is directly proportional to the partial pressure of that gas. The constant of proportionality is Henry’s Law Constant. This constant of proportionality is temperature dependent and decreases as temperature increases. Therefore as carbon dioxide increases in the atmosphere the partial pressure of \begin{align*}CO_2\end{align*} increases and more of it tends to dissolve in the oceans, but as the temperature increases the constant decreases and less of it tends to dissolve. The net effect at a given temperature will have to be calculated.

3. The Solar Radiation peaks at \begin{align*}610\;\mathrm{nm}\end{align*}; there is \begin{align*}61.2\end{align*}% of solar radiation is in the visible band \begin{align*}(400-750 \;\mathrm{nm})\end{align*} with less than \begin{align*}9\end{align*}% in the uv band and about \begin{align*}30\end{align*} % in the near infra red. Some \begin{align*}99\end{align*}% is radiated between \begin{align*}275\end{align*} and \begin{align*}5000 \;\mathrm{nm}\end{align*}. This band largely is unabsorbed by any atmospheric gases. The most significant of the greenhouse gases are \begin{align*}H_2O\end{align*} and \begin{align*}CO_2\end{align*}. The plot above details the absorbance of various wavelengths of radiation by atmospheric gases in the shortwave region.

4. The Earth’s radiation peaks at \begin{align*}11,000 \;\mathrm{nm}\end{align*}, with an intensity of \begin{align*}.04 \;\mathrm{W/cm}^2\end{align*} . Some \begin{align*}99\end{align*}% is radiated between \begin{align*}40,000 \;\mathrm{nm}\end{align*} and \begin{align*}3000 \;\mathrm{nm}\end{align*} in the longer infrared regions. This band is unabsorbed by nitrogen, oxygen and argon (\begin{align*}99\end{align*}%) of the Earth’s current atmosphere), but partially absorbed by carbon dioxide, methane, water vapor, nitrous oxide and some minor gases. The gases that absorb this band of radiation are called greenhouse gases.

5. Earth Orbital Changes: There are three principal variations in orbit that are collectively known as the Milankovitch Cycles. Atmospheric concentrations of methane closely followed this cycle historically and on a larger time frame so have concentrations of \begin{align*}CO_2\end{align*}.

(a) precession of the rotational axis (period: 23,000 years)

(b) variation in tilt of rotational axis from \begin{align*}21.5^\circ\end{align*} to \begin{align*}24.5^\circ\end{align*} (period: 41,000 years)

(c) eccentricity of the elliptical orbit (period: 100,000 years)

6. Departures from the historical cyclical trend began 8000 years ago with the development of agriculture. This led to a temperature rise of \begin{align*}0.8 \ ^\circ\mathrm{C}\end{align*} above expected trends and concentrations of \begin{align*}CO_2\end{align*} rising \begin{align*}30\end{align*} ppm above expected trends with the concentration of methane \begin{align*}450\end{align*} ppb above natural trends. In the last 100 years of industrialization these departures from normal have accelerated with temperature rising an additional \begin{align*}0.8 \ ^\circ\mathrm{C}\end{align*} and \begin{align*}CO_2\end{align*} concentrations rising to \begin{align*}370\end{align*} ppm, which is 90 ppm higher than the recorded \begin{align*}CO_2\end{align*} concentrations at the warmest points in the interglacial periods. Methane concentrations are at \begin{align*}1750 \;\mathrm{ppb}, 1000 \;\mathrm{ppb}\end{align*} above historical highs. Over \begin{align*}70\end{align*}% of the extra greenhouse gases were added after 1950. \begin{align*}CO_2\end{align*} is emitted whenever anything is burned, from wood to coal to gasoline. Methane is produced by animal husbandry, agriculture, and by incomplete combustion or leakage of natural gas. As more greenhouse gases are put into the atmosphere the temperature will increase further. The co-variation of \begin{align*}CO_2\end{align*} concentrations and temperature has been demonstrated not only by recent observation, but by records of the last 700,000 years from Antarctic ice cores. There are many possible effects and feedback mechanisms that are currently being studied and modeled to better predict possible outcomes of this global trend. Many of these are identified above and in the following sections.

The Key Applications

  1. Changing quantity of \begin{align*}CO_2\end{align*} in oceans will lead to a change in pH of the oceans, changing its suitability as a habitat for some species of oceanic life.
  2. Human health problems are associated with warmer temperatures including a projected 10-fold rise in mosquito populations and the diseases they bring as well as the already documented spread of malaria and dengue fever into areas in which these diseases were hitherto unknown.
  3. Loss of water supply: A large part of human and other animal water supply is supplied from glaciers or melting snow-packs. This dependable supply will be disrupted or curtailed for many people. Especially vulnerable are Southeast Asia and India, which depend on the Himalayas, and much of South America, which depends on the Andes. In the US, California and the West stand to have a curtailed water supply in the summer months as a result of global warming.
  4. Weather changes:
    1. Global Warming seems to cause the North Atlantic Oscillation to become stuck in the positive mode. The effect is to have warmer weather in Alaska, Siberia and western Canada, but colder weather in eastern Canada, Europe, and northeast US.
    2. The same effect likely will lead to dry windy conditions in Europe and North America and dry conditions in much of Africa.
    3. Models show global warming leading to droughts in most of the northern hemisphere, particularly in the grain belts of North America, Europe, and Asia.
    4. At the same time, there is predicted to be increased rain overall, but coming in the form of severe storms and consequent flooding.
    5. The conditions that lead to hurricanes and tornadoes are powered by solar energy. More solar energy in the ocean may lead to more severe hurricanes. There is some evidence to support that this has already occurred. The combination of warm Gulf waters and windy plains cause tornadoes. Both of these conditions will be increased by global warming.
  5. Melting of the land glaciers will lead to rising sea levels. The Greenland ice sheet is moving into irreversible melting, which together with the loss of other land ice raise the ocean levels 8 meters in a century. Thermal expansion of water would add several tens of centimeters to this rising sea level.
  6. Ecosystems under stress: When temperature changes occur over thousands of years, plants and animals adapt and evolve. When they happen over decades, adaptation is not always possible. The first flowering days of \begin{align*}385\end{align*} plant species were on average 4.5 days earlier in 1991-2000 than normal. This can lead to lack of pollination and loss of fruiting. A study in the Netherlands showed that weather changes caused oak buds to leaf sooner, causing winter moth caterpillars to peak in biomass earlier. The birds that depend on the caterpillars to feed their chicks did not delay their egg laying. This led to a mismatch of 13 days between food availability and food needs for these birds.

The Key Equations

1. Wien's Law: \begin{align*}T \lambda_{max} = A\end{align*}; where \begin{align*}A = 2.8978 \;\mathrm{m-K}\end{align*}

2. Henry's Law: \begin{align*}C = kP_{partial}\end{align*}, where k is temperature dependent and gas dependent; \begin{align*}CO_2 @ 20^\circ = 3.91\times10^{-3} \;\mathrm{molal/atm}\end{align*}, \begin{align*}CO_2 @ 25^\circ = 3.12\times10^{-2} \;\mathrm{molal/atm}\end{align*}; \begin{align*}CH_4 @ 20^\circ = 1.52\times10^{-3} \;\mathrm{molal/atm}\end{align*}. The concentration is given in molals (Molal is moles of solute/kg of solvent) The partial pressure is given in atmospheres.

4. Energy imbalance of \begin{align*}12 \;\mathrm{watt/m}^2\end{align*}-year leads to deglaciation that raises sea levels 1 meter.

5. Climate Sensitivity: Energy imbalance of \begin{align*}1 \;\mathrm{W/m}^2 \to.75^\circ C \pm .25^\circ \;\mathrm{C}\end{align*} change in average global temperature

6. Present Energy Imbalance = about \begin{align*}1 \;\mathrm{W/m}^2 (\pm .5 \;\mathrm{W/m}^2)\end{align*}

7. The picture above shows the normal energy balance of the Earth. Note that normally the \begin{align*}342 \;\mathrm{W/m}^2\end{align*} incoming is balanced by \begin{align*}235 \;\mathrm{W/m}^2\end{align*} outgoing \begin{align*}+ 107 \;\mathrm{W/m}^2\end{align*} reflected radiation. At present, the atmospheric window allows only \begin{align*}39 \;\mathrm{W/m}^2\end{align*} out resulting in a total of \begin{align*}234 \;\mathrm{W/m}^2\end{align*} outgoing and an energy surplus of \begin{align*}1 \;\mathrm{W/m}^2\end{align*} that results in temperature increases. (These figures are \begin{align*}\pm .5 \;\mathrm{W/m}^2\end{align*}).

8. \begin{align*}1\;\mathrm{kwh} = .68 \;\mathrm{kg} \ CO_2\end{align*} (EPA estimates)

9. \begin{align*}10,000\;\mathrm{kWh} = 1.4\end{align*} cars off the road = \begin{align*}2.9\end{align*} acres of trees planted (EPA estimates)

Problem Set Chapter 25

  1. One \begin{align*} \;\mathrm{W/m}^2\end{align*} energy imbalance may not seem much. (In the following calculations assume for the sake of significant digits that this is an exact number. It is in fact \begin{align*}\pm 0.5 \;\mathrm{W/m}^2\end{align*})
    1. Calculate the total watts received by Earth. Surface area of a sphere is \begin{align*}4\pi r^2\end{align*}.
    2. Convert to energy in kWh.
    3. How many joules of extra energy are received by Earth in a year?
    4. To estimate the contrasting energy of an atomic bomb, assume \begin{align*}100 \;\mathrm{kg}\end{align*} of \begin{align*}U^{235}\end{align*}, isotopic mass of \begin{align*}235.043924\end{align*}, is split into \begin{align*}Xe^{142}\end{align*}>, isotopic mass of \begin{align*}141.929630\end{align*}, \begin{align*}Sr^{90}\end{align*}, isotopic mass of \begin{align*}89.907738\end{align*} and \begin{align*}3\end{align*} neutrons, each with mass of \begin{align*}1.008665\end{align*}. All masses are given in amu's. First, find the mass difference between reactant and products. Then, converting to kilograms and using \begin{align*}E= \Delta\! \;\mathrm{mc}^2\end{align*}, find the energy in joules of an atomic bomb.
    5. How many atomic bombs would have to be set off to equal the extra energy the Earth receives in one year from global warming?

  2. It is estimated that a \begin{align*}12 \;\mathrm{W/m}^2\end{align*} energy imbalance leads to sufficient melting of land ice to cause the sea levels to rise one meter.
    1. How many joules is that?
    2. What mass of ice is melted? The heat of fusion of water is \begin{align*}3.33 \times 10^5\;\mathrm{J/kg}\end{align*}.
    3. What volume of water is that? \begin{align*}(\rho = 1000 \;\mathrm{kg/m}^3)\end{align*}
    4. From the above result, you should be able to estimate the surface area of the world’s oceans and check the given estimate.
  3. Given the uncertainty of \begin{align*}\pm 0.5 \;\mathrm{W/m}^2\end{align*}, give the high and low estimates of global sea level rise in a century. Draw two new world maps using this data. Draw maps of your state, if it is a coastal state, 100 years from now given these estimates. (Perhaps your inland state will become a coastal state.)
The growth in concentration of greenhouse gases over time (From the Intergovernmental Panel on Climate Change)
year \begin{align*}[CO_2] \;\mathrm{ppm}\end{align*} \begin{align*}[CH_4] \;\mathrm{ppb}\end{align*}
\begin{align*}1940\end{align*} \begin{align*}310\end{align*} \begin{align*}1100\end{align*}
\begin{align*}1960\end{align*} \begin{align*}315\end{align*} \begin{align*}1250\end{align*}
\begin{align*}1980\end{align*} \begin{align*}335\end{align*} \begin{align*}1550\end{align*}
\begin{align*}2000\end{align*} \begin{align*}370\end{align*} \begin{align*}1750\end{align*}
\begin{align*}2020\end{align*} (IPCC* projection) \begin{align*}420\end{align*} \begin{align*}2150\end{align*}
  1. Given the Table (above),
    1. Graph this data with time on the horizontal axis
    2. Determine the rate of increase in the concentrations of the two gases
      1. 1940 – 2000
      2. 1960 – 2000
      3. 1980 – 2000
      4. the instantaneous rates of change in 2000
      5. the instantaneous rates of change projected for 2020
  2. Climate forgings can come from a variety of sources besides methane and carbon dioxide. Determine whether the following are positive feedbacks (contribute to global warming) or negative. You may have to do some research on this.
    1. Black Carbon Soot
    2. Reflective Aerosols
    3. Chlorofluorocarbons
    4. Nitrous Oxide
    5. Ozone
    6. Cloud Droplet Changes

  3. An overlooked area of additional global warming is the traditional cook stove. In one Honduran study, the soot smoke produced from one stove absorbed \begin{align*}65\end{align*}% of terrestrial radiation that then went into warming the atmosphere. There are \begin{align*}400\end{align*} million such cook stoves worldwide, each of which emit \begin{align*}1.5 \;\mathrm{g}\end{align*} of soot per kilogram of wood burned. The average daily use of wood is \begin{align*}7.5 \;\mathrm{kg}\end{align*} per stove. Calculate the mass of soot released through cook stoves per day, per year.

For Problems 7 - 10 use the following tables:

Electricity Emission Rates: (EPA)
State or region \begin{align*}CO_2\end{align*} in \begin{align*} \;\mathrm{kg/ Mwh}\end{align*} \begin{align*}CH_4\end{align*} in \begin{align*} \;\mathrm{kg/Mwh}\end{align*} \begin{align*}N_2 \ O\end{align*} in \begin{align*} \;\mathrm{kg/Mwh}\end{align*}
California \begin{align*}364.8\end{align*} \begin{align*}.00304\end{align*} \begin{align*}.00168\end{align*}
Michigan \begin{align*}740.1\end{align*} \begin{align*}.00662\end{align*} \begin{align*}.0133\end{align*}
New York City \begin{align*}494.3\end{align*} \begin{align*}.00367\end{align*} \begin{align*}.00404\end{align*}
Oregon \begin{align*}304.3\end{align*} \begin{align*}.00149\end{align*} \begin{align*}.00154\end{align*}
Global Warming Potential of Gases Compared to Carbon Dioxide (IPCC):
Greenhouse gas Multiplier
Carbon dioxide \begin{align*}CO_2\end{align*} \begin{align*}1\end{align*}
Methane \begin{align*}CH_4\end{align*} \begin{align*}23\end{align*}
Nitrous Oxide\begin{align*} N_2 \ O\end{align*} \begin{align*}296\end{align*}
A/C refrigerant \begin{align*} \;\mathrm{HFC}-143a\end{align*} \begin{align*}4300\end{align*}
Auto A/C refriger \begin{align*} \;\mathrm{HFC}-134a\end{align*} \begin{align*}1300\end{align*}
\begin{align*}SF_6\end{align*} \begin{align*}22,000\end{align*}
\begin{align*}C_2\ F_6\end{align*} \begin{align*}11,900\end{align*}

  1. A typical household air conditioner draws about \begin{align*}20a\end{align*} from a \begin{align*}240 \;\mathrm{v}\end{align*} line.
    1. If used for \begin{align*}8\end{align*} hours how many kwh does it use?
    2. In the course of a \begin{align*}120\end{align*} day summer how many Mwh is that?
    3. Calculate the mass of carbon dioxide one summer’s use of ac contributes. (Pick a state or region from above.)
    4. Calculate the mass of methane and \begin{align*}N_2O\end{align*} emitted.
    5. Using the global warming multipliers for the latter two gases calculate the global warming potential in equivalent \begin{align*}kg\end{align*} of \begin{align*}CO_2\end{align*} for all \begin{align*}3\end{align*} gases.
  2. If you “shut down” your computer, but the LED light is still on, it consumes about \begin{align*}4 \;\mathrm{w}\end{align*} of power. Suppose you do that for every weekend (60 hours) every week of the year. Repeat the calculations in problem \begin{align*}7\end{align*} to find out the global warming potential in \begin{align*}kg\end{align*} of \begin{align*}CO_2\end{align*}.
  3. In 2006 Natomas High School in California used \begin{align*}1692 \;\mathrm{Mwh}\end{align*} of electricity. Repeating the calculations above, find the \begin{align*}kg\end{align*} of carbon dioxide emitted.
  4. A large car or SUV typically carries \begin{align*}1.0 \;\mathrm{kg}\end{align*} of refrigerant for the \begin{align*}a/c\end{align*}.
    1. If this were released into the atmosphere calculate the equivalent of carbon dioxide released.
    2. Repeat this calculation for a residential air conditioner (capacity is \begin{align*}2.8 \;\mathrm{kg}\end{align*}.), using \begin{align*} \;\mathrm{HFC}-143a\end{align*}.
    3. Your school has a commercial chiller maybe (1000 ton) with a refrigerant capacity of \begin{align*}1225 \;\mathrm{kg}\end{align*}. If it uses \begin{align*}\;\mathrm{HFC}-134a\end{align*} calculate the equivalent of \begin{align*}CO_2\end{align*} emitted, if the chiller is decommissioned.
Emissions of Carbon Dioxide for Different Fuels
Fuel \begin{align*}Kg\end{align*} of carbon dioxide emitted/gallon
Gasoline \begin{align*}8.78\end{align*}
California reformulated gasoline, \begin{align*}5.7\end{align*}% ethanol \begin{align*}8.55\end{align*}
Ethanol \begin{align*}6.10\end{align*}
Diesel #2 \begin{align*}10.05\end{align*}
biodiesel \begin{align*}9.52\end{align*}
Jet fuel \begin{align*}9.47\end{align*}
propane \begin{align*}5.67\end{align*}
Natural gas/gasoline gallon equivalent \begin{align*}6.86\end{align*}
  1. Compare the carbon “footprint” of the following:
    1. a hybrid car \begin{align*}(45 \;\mathrm{mpg})\end{align*} that drives \begin{align*}21,000\end{align*} mile per year in Calif.
    2. an SUV \begin{align*}(17 \;\mathrm{mpg})\end{align*} that drives \begin{align*}21,000\end{align*} miles per year also in Calif.
    3. a mid-size car \begin{align*}(24\;\mathrm{mpg})\end{align*} that uses ethanol and drives \begin{align*}21,000\end{align*} miles per year
    4. a commercial flatbed \begin{align*}(11 \;\mathrm{mpg})\end{align*} that drives \begin{align*}21,000\end{align*} miles per year and uses bio diesel

  2. Research some typical mileages, type of fuel used, and miles covered in a year and determine the carbon footprint for:
    1. a tractor-trailer truck
    2. a commercial airliner
    3. a corporate jet
    4. a bus
    5. Amtrak
  3. Looking at the above problems another way, suppose you want to travel from California to New York find your carbon footprint for the trip using:
    1. Amtrak
    2. a jet plane
    3. a bus
    4. an SUV
    5. a hybrid

    Assume \begin{align*}90\end{align*}% full loads on the commercial transports and \begin{align*}2\end{align*} passengers on the cars. You will have to go on-line to find the loads of the commercial transports.

  4. China is putting two coal-fired electrical plants in operation each week. These plants do not typically use any scrubbing or pollution controls. Research the typical Mwh output, and, using either the table for problem \begin{align*}7\end{align*} (Michigan depends more on coal than the other states listed.) or a more direct source for \begin{align*}CO_2\end{align*} emissions for a coal plant, find the gain in greenhouse gas emissions each year from this source alone. Compare to the results in problem \begin{align*}4\end{align*} and determine if the IPCC is underestimating the problem.

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