relative humidity-r-bruce-martin


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relative humidity-r-bruce-martin

  1. 1. Chemistry Everyday for EveryoneRelative HumidityR. Bruce MartinChemistry Department, University of Virginia, Charlottesville, VA 22903; Daily, the weather report tells us the percent relative vapor pressure rising with increasing temperature. The curvehumidity. What does this mean? A hot day accompanied by is constructed from tables in handbooks for water saturationhigh relative humidity causes us to experience a muggy feeling. vapor pressure versus Celsius temperature.A sudden switch to a low relative humidity environment The relative humidity, usually expressed as a percentage,results in dryness of lips and mucous membranes. Informal polls is the ratio of the water vapor pressure in the atmosphere toof students indicate that many do not grasp the concept of the saturation vapor pressure at the same temperature. Inrelative humidity and claim that they have not had it in other words, the relative humidity is the amount of waterschool. The absence of material about relative humidity from vapor in the atmosphere divided by the maximum amountmost middle and high school science texts supports this claim. the air may hold at that temperature (multiplied by 100). AtHere is a science topic about which students hear constantly 27.4 °C a vapor pressure of 13.7 mmHg corresponds to 50%but which is given little coverage in school science courses. humidity. The three lower curves in Figure 1 show water In this short article I define and describe relative humidity vapor pressure at the indicated relative humidities. These threeat several levels, discussing the concepts of water vapor pressure curves were constructed by multiplying the saturation vaporin the atmosphere and dew point. There is an opportunity pressure at each temperature by the corresponding decreas-for an instructive application of the ideal gas equation. I finish ing fractional relative humidity.with a new plot directly illustrating the dependence of relative Relative humidity is often measured by a hygrometer––humidity on temperature. I derive the relationship for this an instrument in which a hair under tension slightly altersplot from application of an equation suitable for physical its length owing to humidity changes. The instrument dial ischemistry students, but the results of this original plot are of calibrated to read relative humidity directly from the hair length.general interest and access. Relative humidity may also be determined by comparing the temperatures of dry-bulb and wet-bulb thermometers, theRelative Humidity temperature of the latter being lowered by evaporative cooling of water from a cloth surrounding the bulb. The lower the For any given temperature there is a maximum or satu- relative humidity the greater the cooling effect. The relativeration amount of water vapor that the atmosphere can hold. humidity is then read from a table containing the dry bulbThis amount may be expressed in mmHg of a mercury ba- temperature and the depression of the wet-bulb thermometer.rometer or in atmospheres, where 760 mmHg corresponds The area above the curve for 100% humidity in Figureto one atmosphere. The saturation vapor pressure of water 1 corresponds to an atmosphere saturated with water air increases with temperature as a greater fraction of The dew point is the temperature at which condensation ofmolecules attain the energy to escape the liquid phase. At the water vapor occurs owing to saturation at a given waterfreezing point of water (0 °C) the saturation vapor pressure content. Reading off the curve labeled 100 in Figure 1, weis 4.58 mmHg and it rises to 760 mmHg at the boiling point see that for a water vapor pressure of 27.4 mmHg the dewof water (100 °C). Near room temperature the saturation point is 27.4 °C; dew forms below this temperature for thevapor pressure in mmHg is almost numerically equivalent to specified vapor pressure.the Celsius degrees, becoming equal at a warm room tem- How much mass of water vapor are we talking about?perature for 27.4 mmHg at 27.4 °C (81 °F). The uppermost We may use the ideal gas equation to estimate the mass ofcurve for 100% humidity in Figure 1 shows the saturation water vapor in the atmosphere at a given water vapor pressure. For the vapor pressure of 27.4 mmHg at 27.4 °C (= 27.4 + 273.1 = 300.5 K) we obtain in moles per liter n = P = 27.4 mmHg/760(mmHg/atm) = 0.00146 mol/L V RT 0.0821(L atm/mol K) 300.5 K We multiply this result by the molecular weight of water to obtain 0.0263 gram/liter and hence 26.3 g/m3, or almost one ounce of water vapor in a cubic meter. In a fair-sized room of 100 cubic meters there would be 2.6 kg or 2.6 liters (al- most three quarts) of water vapor. These values are maximal for 100% relative humidity at 27.4 °C; for 50% relative humidity there would be half as much water vapor. For a given amount of water vapor in the atmosphere the relative humidity increases with a decrease in temperature. This point may be illustrated in Figure 1 by selecting anyFigure 1. Absolute water vapor pressure vs Celsius temperature for absolute vapor pressure on the ordinate axis and moving hori-four relative humidity percentages. zontally to the left, passing through the curves for increasing • Vol. 76 No. 8 August 1999 • Journal of Chemical Education 1081
  2. 2. Chemistry Everyday for Everyonerelative humidity as the temperature decreases. For example,for a fixed 20 mmHg vapor pressure a drop in temperaturefrom 34 °C to 27 °C increases the relative humidity from50% to 75%. We may better illustrate the same point byplotting relative humidity on the ordinate axis versus Celsiustemperature for several fixed water vapor pressures. To do so,we need to develop an equation that describes vapor pressure asa function of temperature. The next two quantitative paragraphsmay be skipped in favor of the qualitative paragraphs thatend the article.Quantitative Discussion The Clausius–Clapeyron equation, derived in everythermodynamics textbook, may be used to describe thedependence of vapor pressure on temperature. Figure 2. Relative humidity percentage vs Celsius temperature for d log P ∆H v 10 fixed water vapor pressures. The absolute vapor pressures in = 2 mmHg for the 10 isobaric curves are indicated across the top of dT 2.3RT the figure.The derivative of the common logarithm of the vapor pressurewith respect to the absolute or Kelvin temperature is equal tothe heat of water vaporization divided by the natural logarithm temperature. For each isobaric curve, the water vapor contentof 10 (ln 10 = 2.303) times the gas constant (R = 1.987 cal/ is held constant at a value below saturation. For the leftmost(K-mol)) times the square of the Kelvin temperature. Many curve in Figure 2 the value is set at 4.58 mmHg, the waterare familiar with the heat of vaporization of water as 539 cal/g vapor saturation value at 0 °C. The succeeding curves to theat the boiling point of water, 100 °C. At lower temperatures right refer to ever greater fixed water vapor contents as indi-the heat is greater, as the specific heat of liquid water, 1.00 cated across the top of the, is about twice that of water vapor, 0.45 cal/(deg-g). Above 50% relative humidity the curves in Figure 2 areOver our temperature range of interest, 0–50 °C (32–122 °F), relatively steep. For a typical central curve, at 50% humidity,handbook steam table values for the heat of vaporization of a 12 °C (22 °F) drop in temperature sends the curve throughwater in cal/g are accurately represented by ∆Hv = 596.4 – the top of the figure above 100% humidity. When such cooling0.550t = 596.4 – 0.550(T – 273.15) = 746.6 – 0.550T, where occurs in nature, the atmosphere becomes saturated witht is the Celsius and T the Kelvin temperature. The 0.550 factor water vapor, condensation occurs, and dew appears.arises from the fit and is equal to the difference between the Saturation occurs at the top of Figure 2 at 100% humidity.specific heats of liquid and vapor. Multiplication of the heat The temperature at which a constant vapor pressure curveexpression by the molecular weight of water (18.015) converts intersects the top of Figure 2 at 100% humidity is the dewto a molar basis, and its substitution into the Clausius– point, the temperature below which dew forms. For example,Clapeyron equation followed by integration yields for the third curve at 12 mmHg constant vapor pressure intersects the top of Figure 2 at 14 °C, which is the dew point log Ps = 2939/T – 4.987 log T + 23.573 for that amount of water vapor. This point may also beAn excellent nonlinear least-squares fit of 51 handbook vapor located on the curve for 100% relative humidity in Figure 1.pressures from 0 to 50 °C with a fixed 0.550 factor yields One way to use Figure 2 is to read the temperature and∆H v = 596.2 cal/g at 0 °C and the constant of integration in relative humidity during a spring or fall school day andthe last term of the equation in mmHg. This last equation estimate the absolute vapor pressure. Then assume that theaccurately represents the saturation vapor pressure, Ps , for absolute vapor pressure is constant and trace an isobar to 100%water vapor from 0 to 50 °C (32–122 °F). relative humidity at the top to predict the temperature drop The fractional relative humidity is given by r = P/Ps , required to produce dew. The next day check the overnightwhere the saturation vapor pressure Ps is given by 10 to the temperature drop and any presence of morning dew.power of all three terms on the right side of the above equation. Figures 1 and 2 portray complementary aspects of theFigure 2 shows curves of relative humidity or 100r as a same phenomenon. In Figure 1 each curve corresponds to afunction of Celsius temperature for several fixed or absolute constant relative humidity and the plot shows the variationvalues of the vapor pressure, P. This completes the quantitative of absolute vapor pressure with Celsius temperature. Indescription and we return to a more qualitative discussion of Figure 2 each isobaric curve represents a constant less-than-the results. saturation vapor pressure and the plot shows the dependence of relative humidity on the Celsius temperature. The newQualitative Discussion second figure directly indicates the variation of relative humid- The drapery of curves in the new Figure 2 shows the ity with temperature for a fixed absolute water content ofrelative humidity percentage as a function of the Celsius the atmosphere.1082 Journal of Chemical Education • Vol. 76 No. 8 August 1999 •