Randall D. Knight — Work, Heat, and the First Law of Thermodynamics: Problem 69

Some gases in the earth’s atmosphere—notably carbon dioxide and water vapor—keep the earth warmer than it would be otherwise by absorbing some of the infrared radiation that the earth is trying to radiate into space. This is the greenhouse effect, and it’s a natural feature of the earth’s climate. You can make a good estimate of the size of the greenhouse effect. Model the earth as a sphere of uniform temperature \(T\) and emissivity \(e = 1\) radiating energy into outer space at \(T_{\text{space}} \approx 0\text{ K}\). The earth’s temperature is not really uniform, but the uniform-temperature assumption will allow you to find the earth’s average temperature. Assume that nothing absorbs or interferes with this radiation. At the same time, the earth is constantly absorbing energy from the sun. The solar intensity \(I\) just above the earth’s atmosphere is \(1360\text{ W/m}^2\). Seen from the sun, the absorbing area of the earth is a circular disk. Measurements show that \(70\%\) of the solar energy striking the earth is absorbed, \(30\%\) is reflected back into space. Averaged over many days, as the earth rotates, the energy absorbed from the sun balances the energy radiated into space. Now calculate \(T_{\text{earth}}\), what the earth’s average temperature in \(^\circ\text{C}\) would be in the complete absence of infrared-absorbing greenhouse gases. The actual average temperature is a much more pleasant \(15^\circ\text{C}\), which shows that the greenhouse effect is essential to life on earth.