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Physics 140: Greenhouse Effect
Spring 2008
Homework #3 - Due Wednesday, March 12, 2008

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(1) (2 pts) The atmospheric concentration of CO$_2$ is projected to reach 760 ppm before 2050.
(a) Of all molecules in the atmosphere, what percentage would be CO$_2$ molecules?
(b) In a sample of atmosphere containing 500 million molecules, how many would be CO$_2$ molecules?

Answers:
(a) 100% x 760 / 1,000,000 = 0.076%
(b) 760 ppm x 500 million = 380,000

(2) (3 pts) Consider the thermal radiation from three black spherical objects. Object A has a radius of 1.0 m and is at room temperature, Object B has a radius of 0.5 m and is at room temperature, and Object C has a radius of 1.0 m and is at 400 C.
(a) Rank the three objects by the total power emitted by each as thermal radiation.
(b) Rank the three objects by the wavelength of the radiation each mainly emits.
(c) If Object B emits a power of 10 W, how much does Object A emit (use that the area of a sphere is 4 pi r²)

Answer: remember that thermal radiation has the following properties:
power for a given area increases with temperature
wavelength decreases with temperature
power is proportional to area
(a) A emits more than B because they are at the same temperature and B has less area; C emits more than A because it is at higher temperature. So C > A > B.
(b) A and B emit at the same wavelength because they are at the same temperature.  C emits at a shorter wavelength because it is hotter. So A = B > C.
(c) The area of Object A is 4 times as large as Object B, because its radius is twice that of Object B.  Since power is proportional to area, Object A emits 40 W.

(3) (5 pts)  Energy balance In your house, you like to maintain a constant temperature of 67 degrees F. When it is 20 degrees F outside, the rate of heat loss from your house is 50,000 Btu/hour.
(a) If the rate at which heat is put into your house by your furnace is 40,000 Btu/hour, will the temperature go up, go down, or stay the same?
(b) To keep the temperature constant at 67 degrees F, how many Btu/hour should your furnace put into your house?
(c) Assuming 85% efficiency for your furnace, how many gallons of oil do you have to burn per hour to maintain the constant temperature of 67 degrees F?

Answers:
(a) The temperature will go down, because more heat is being lost than is being put in.
(b) 40,000 Btu/hour -- to keep the temperature constant the amount of heat put in should equal the amount lost.
(c) 40,000 Btu/hour x 1 gallon/139,000 Btu  x 1/0.85 = 0.34 gallons/hour

(4) Relating weather to climate by averaging (7 points): In this problem, we go to weather.com about February in Newark:  the averages for February and  the observation for February 2008 (find the latter by choosing Month and go to Previous Month).
(a) On a graph of temperature vs time, plot together the observed daily high temperatures for 2008 and the average daily high temperature for the 29 days of February.
(b) Repeat (a) for the low temperatures.
(c) What do you learn by comparing the average and the 2008 observed temperature curves?

Answer (c): the weather fluctuates greatly from day to day, while the climate (average temperature over the past 30 years) changes gradually. The weather flucutates around the average.

(5) Urban heat island effect (3 pts) Go to weather.com and estimate the magnitude of the urban heat island effect for Philadelphia, PA by comparing the average temperature in a selected month with that of a nearby rural area. Print out the two graphs or table and attach to the homework, and find the temperature difference.

Answer: You can compare, for example, to West Chester, PA, about 20 miles west of the center of Philadelphia. While the daytime average highs are almost identical, at night in West Chester it ranges from 4 degrees F colder (January) to 8 degrees colder (October). The presumption is that the buildings and roads tend to prevent heat loss at night.
Philadelphia (left), West Chester (right)