Why There's Frost on the Pumpkin Walter H. Corkern and L. H. Holmes, Jr. Southeastern Louisiana University, Hammond, LA 70402
Every fall, the weather cools, and the day finally amves when we find frost on the rooftops, on the grass, on the windshields-and, of course, on the pumpkins. This delightful scene is usually first noticed in the early morning, but frost (or ice) can form on a surface whenever certain conditions are met. The air temperature may be above freezing, usually 3-4 'C (35-38 'F) or slightly less. The surface, however, must be 0 'C or less, and this can occur even though the air temperature is above 0 Y-if the surface cook below air temperature. I t is reasonable to assume that obiects lose enerev bv radiation into space. We've all noticedthat on clear 2ght"s the temperature drops much more than on cloudy nights, due to radiation. If the rate a t which an object loses energy through radiation exceeds the rate a t which enerev is transrerred to it by conduction or convection, then-the temperature of its surface drops below the temperature of the air around it. A black-body radiator will radiate an amount of energy Q, per unit time per unit area, given by Stefan's law
where T is in units of Kelvin; o is the Stefan-Boltzman constant; Q, is in J/s.m2 if o = 5.67 x 10.' J/s.m2.K4;and e is emissivity. Values of emissivity, which depends on the nature of the surface and the wavelength of the radiation emitted, are 0-1.00, for complete reflection (nonabsorber) to no reflection (a perfect absorber or black body). A surface coated with lampblack will have e > 0.9, while a polished silver surface will have e = 0.1. Most surfaces have e 2 0.1. If the temperature is 273 K, then
Consider a surface a t 273 Kin contact with still air. Two centimeters away from the surface the air is a t 278 K. The energy Q, conducted per unit time per unit area through the layer of still air can be calculated using Newton's law of cooling
where ATis the difference in temperature (Kelvin);L is the distance separating the region of low temperature from the region of high temperature; k is the thermal conductivity of air (0.0256 J1s.m.K for air near 20 'C). Presented at Louisiana Academy of Sciences, February 8, 1991.
For the situation above, L = 0.02 m, and AT = (278-273)K =5Ksothat
Comparing eqs 5 and 3 shows that the rate of conduction to the surface is much lower than the rate ofradiation from the surface for the conditions usually found on surfaces isolated from the ground in still air under clear skies. Consider glass with a specific heat of 0.84 J/g T and a density of 2600 kg/m3. To cool a square meter piece of the glass (0.005-m thick) by 5 'C over a 5-h period requires the removal of
To freeze a gram of water vapor at 0 'C over a period of 5 h r would require the removal of
Near 273 K, to cool a square meter of glass to 5 'C below ambient temperature and to freeze 100 g of ice on the surface in 5 h would require the loss of
Considering the net loss of energy from a surface with e = 0.1, eqs 3 and 5 indicate that this is possible. If e were larger, whichis likely, the loss becomes evenmore probable. These calculations assume no transfer of energy by convection. When there is no wind, this is fine. When there is wind, it is commonly observed that dew and frost often do not form. The wind may mix the air so that the amount of water vapor in the air near surfaces does not exceed the amount a t saturation. It is more difficult to explain why dew or frost do not form under these conditions when the amount of water vapor in the air exceeds the equilibrium vanor Dressure of water (or ice) in the windv air. D U & ~ the fall of 1990,roof A d windshield temperatures for automobiles ~ a r k e doutside were measured in the earlv morning over several weeks. When no cloud cover or fog was present the temperature on their surface was about 3O C lower than the surrounding air temperature. The data are clearly consistent with the conclusions reached above. Other experiments have been conducted, and the data are available upon request. Volume 68 Number 10 October 1991
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