P310/510 Notes for Exam I September 14, 2016
Dear P310 and P510’ers,
Our first test is Thursday, September 22, during class in our usual classroom SW220. By then you will be in pretty good shape for the test because you will have done the homework problems. That said, I list here key points that are likely to occur to me when I composed this year's Exam I. You might use this as a kind of check list for your own preparation. Feel free to contact me if you have questions. (, IU Tel: 855-3881, Home Tel: 332-6507) Ben Brabson
Energy Units: There are lots of different units for energy including the standard SI unit, J (Joule). For example, 1 Kcal = 4186 Joules is the mechanical or heat energy necessary to raise the temperature of 1 kg of H2O by 1oC, and 1 Btu = 1055 Joules is the energy to raise the temperature of 1 pound of water by 1oF. A Joule is also the work to lift an object weighing 1 Newton by 1 meter and a jelly doughnut has about 106 Joules of chemical energy. A very convenient energy unit for discussions about individual molecules and atoms is the electron-volt = 1 ev = 1.6 x 10-19 Joules, the work to move an electron through a potential difference of 1 volt.
Work: W = F o d = Fd cos(q) defines what we mean by work where Theta (q is the angle between the vectors F and d. Positive work in Joules is done when F in Newtons is in the same direction as d in meters.
Fgravity = mg where g = 9.8 N/kg on the surface of the Earth. We often round g to 10 N/kg for convenience.
Potential energy, PE, can be written as: PE = mgh , where h = vertical height. You store energy as potential energy either by pulling two attractive objects apart (lifting a book away from the Earth) or by pushing two repulsive objects together (pushing two positive charges together into a nucleus). Only changes in potential energy are important. Where you choose to put zero potential energy is up to you. For gravitation problems zero is often taken to be at floor level. For electrostatic problems considering forces between a couple of charges, Q1 and Q2, zero potential energy is usually taken to be at r = infinity giving U = kQ1Q2/r. Then, a system is a "bound" system when its total energy (Etotal = KE + U) is negative.
Kinetic energy, KE, energy of motion, KE = 1/2mv2,can also be viewed as a way to store energy. A rotating flywheel with essentially all of its mass in the rim also has KE = 1/2mv2 where v is the speed of the edge of the wheel.
Energy conservation for a closed (isolated) system says that: Etotal(now) = Etotal(later). For objects in an approximately constant gravitational field (near the surface of the Earth, for example where g = 9.8 N/kg) ignoring heat and friction, energy conservation says:
mgh1 + 1/2mv12 = mgh2 + 1/2mv22
Power = Energy/time = Work/time in Joules/second or watts. (Though it makes sense to ask how many Joules of energy were consumed in a day, it does not make sense to ask how many watts were consumed in a day. If you find yourself wondering how to convert Joules to watts be sure to carefully figure out why that question doesn't make sense.
Chemical Energy & Bonding: An attractive force between atoms in a molecule. Covalent bonding takes place when shared electrons between positive ions provide the electrostatic attractive interaction. Ionic bonding is an extreme form of covalent bonding where an electron from one atom spends most of its time around a neighboring atom. Then, an attractive force holds the two ionized atoms together. Van der Waals (weak) bonds exist between dipoles, neutral atoms or molecules that are polarized (non-symmetric charge arrangement). The polarization can either be induced or permanent. In each case the potential energy of a bound system is negative. The binding energy is the work required to separate the components of the molecule. [We didn't discuss this explicitly in class. Chapter 29 in Physics, 5th Edition by Doug Giancoli gives a lovely 5-page description of these bonding processes.]
A Production Graph shows the quantity of a resource extracted each year and plotted versus time. A consumption graph does the same for the quantity of a resource consumed each year. The area under such a curve, of course, represents the total quantity produced or consumed. The production curve shapes we discussed were 1.) constant, 2.) linearly increasing/decreasing, 3.) exponential, and 4.) Hubbert (~normal dist.). Presented with such a curve, you should be able to estimate the total resource by making an approximate estimate of the area under the curve.
Exponential behavior occurs when dN ~ Ndt. When discussing population growth, N is the number of people and dN the number of babies arriving in time dt. That is, growth by a certain fraction (dN/N) per time interval (dt) leads to exponential growth because the differential equation, dN/dt = kN has the solution: N = No ekt where k = growth constant. When a production curve follows an exponential (as US oil consumption did from 1890-1970, then the total resource used, QT is given by the integral of the exponential: QT = (No/k)(ekT - 1) and the resource used in time T equals T = (1/k)*ln[(kQT/No) + 1].
T2 = doubling time for an exponential. T2 = (ln 2)/k = 0.693/k, where k is the growth constant. For small values, k is approximately equal to the fractional increase per year or, times 100, the percent interest per year paid by a bank, R. Approximately, T2 = 70 / (%/yr), the so-called "Rule of 70." For large values of R, we must calculate k = ln(1+R/100).
A semilog plot of an exponential function gives a straight line. For this reason extrapolating an exponential is easier done on a semilog plot. As an example, the energy consumption per year in the US has grown essentially exponentially for 140 years. If you plot these numbers directly on semi-log paper they lie along a straight line.
Hubbert model: P = PM exp(-(z)2/2) where PM = QT/(√(2p)*s) is the peak of the distribution, and z = (t - TM)/s. The shape of the distribution is that of a normal distribution with mean, NM, and standard deviation, s. For Hubbert any three parameters are sufficient to determine the entire production model. You might keep in mind why the Hubbert Model shape works for extracted resources like oil, gas, and coal.
Resources and Reserves: For a particular energy commodity, reserves estimate the amount in the ground that can be recovered at competitive prices. A much less well determined number, resources estimate the total amount of a commodity including that part yet to be discovered. A large energy unit used when discussing reserves and resources is a quintillion Btu = 1 Q = 1000 quads = 1018 Btu = 1.055 x 1021 Joules. Roughly, world resources are:
US World
Gas 2 Q 10 Q
Oil 0.6 Q 7-12 Q
Uranium-235 1 Q 2-3 Q
Coal 40 Q 200 Q (the dominant fossil fuel resource)
Fusion (Ocean deuterium: 1010 Q)
Methane Hydrate (rough guess ~400 Q)
Oxidation or burning of a hydrocarbon:
C + O2 ---> CO2 + 95 Kcal/mol of carbon
2H2 + O2 ---> 2H2O + 68 Kcal/mol of hydrogen
Specific chemical energy content of many fuels lies between 20 and 45 MJ/kg. For example, anthracite coal contains ~29 MJ/kg, methanol about 20 MJ/kg, natural gas about 43 MJ/kg...
Photosynthesis: CH2O is simplest of the carbohydrates, Cm(H20)n where m and n are integers. An example of the photosynthesis of this simplest carbohydrate is:
(g)sun + CO2 + H2O ---> C(H20) + O2 Where (g)sun is a photon of energy from the sun.
THERMODYNAMICS:
Definition of temperature: For an ideal monatomic gas, temperature, T is defined by: 3/2kT = <KE>molecule, where k = Boltzmann's constant = 1.38 x 10-23 Joules/K.
Total internal energy, U, is defined by the sum of the kinetic energies of the individual molecules in the gas (assuming no potential energy between molecules). Be careful to distinguish the idea of temperature from the idea of internal energy.
Pressure = Force/Area P[N/m2] = F[N] / A[m2]
Specific Heat: When heating a substance, the quantity of heat needed is proportional to the mass and to the change in temperature. The constant of proportionality is called the specific heat, c.
Q = c m DT, where c = 1 Kcal/(kg*K) for water, for example.
First Law of Thermodynamics: dQ = dU + dW
where dW = p dV is the work done by the system, dQ the heat added to the system and dU the change in the internal energy. For an engine where it must return to its original condition after each cycle, dU = 0 in one cycle, so dQ = dW for a heat engine, dQHot = dQCold + dW. (Be sure to be able to draw the diagram for a heat engine.) This leads immediately to the definition of efficiency for a heat engine:
real efficiency = e = 1 - QCold/QHot
Here, both QCold and QHot are measured in energy units such as Joules or Kcal.
Entropy change is defined by dS = dQ/T = change in entropy, and entropy is a measure of the disorder of a system in units of Joules/K. Adding a small amount of heat energy to a cold system greatly increases its disorder; but adding small amounts of heat energy to a hot system has very little effect on its disorder.
In the language of statistical mechanics, entropy is proportional to the number of states or conditions of a system that are available with a certain amount of heat energy. As you increase the amount of heat energy in a system, you increase the number of available states of the system.
Second Law of Thermodynamics: The entropy of an isolated system must always remain the same or increase. This has as its consequence that the maximum efficiency is given by:
maximum efficiency = emax or effmax = 1 - Tcold/Thot
Coefficients of performance (COP) for heat pumps, AC, and Refrigerators is always the ratio of "the good stuff"/ "what you pay."
Engines:
dU = 0 for a full cycle of an engine
Work done in a cycle = work inside the loop in the pV diagram.
Ideal Gas:
pV = nRT , R = 8.31 J/mol/K = 1.99 cal/mol/K
U = 3/2 NkT = nRT and <1/2mv2> = 3/2 kT (monatomic gas)
dU = 3/2 NkdT = 3/2 nRdT (monatomic gas)
Conduction and convection:
dQ/dt = (kA)dT/dx where Q in Joules is the heat transferred through a thickness dx of
material with coefficient of conductance, k when the temperature difference across the material is dT.
dQ/dt = (A dT)/R where R = R1 + R2 + R3... and R = dx/k for materials and
R = 1/h for dead air layer near various surfaces:
h = 1.8 (DTair)1/4 for vertical surfaces
Electromagnetic Radiation: For periodic wave motion we can write: v = f*l, or velocity = frequency x wavelength. The velocity of light in vacuum is a constant, v ≈ 3.00 x 108 m/s.
Intensity [Watts/m2] = Power[Watts]/Area[m2], For example sun’s intensity at the top of the atmosphere = Io = 1370 Watts/m2.
Stefan's Law: The Power emitted by a black body at temperature T is given by: P = esAT4, where s = 5.67 x 10-8 Watts/m2/K4, and where e is the emissivity of the surface, a number close to 1 for most non-reflective surfaces.
Wein's Law: The peak wavelength emitted by a body at temperature T is given by the equation:
lmax[m]*T[K] = 2.9 x 10-3 [m K]
Ideas from Articles and Readings:
1.) A Measurable Planetary Boundary for the Biosphere, Steven Running, Science, 21Sept,2012.
2.) Scientists help make deserts into solar-energy hubs: Physics Today July 2011, page 21
3.) Driving on Biomass, John Ohlrogge et al, Science 324, pp 1019-1020, 22May 2009.
4.) How Much Coal, Richard Kerr, Science 323, pp.1420-1421, 13 Mar. 2009)
5.) Sandia Laboratory's Stirling Energy System
6.) Garbage Trucks running on Methane from garbage and plasma gasification of garbage
7.) Coastal inundation
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