Effect of altitude on boiling water.
(a) Use the result of the previous problem 1 and the data in below Figure to plot a graph of the vapor pressure of water between 50°C and 100°C. How well can you match the data at the two endpoints?
(b) Reading the graph backwards, estimate the boiling temperature of water at each of the locations for which you determined the pressure in Problem 2. Explain why it takes longer to cook noodles when you’re camping in the mountains.
(c) Show that the dependence of boiling temperature on altitude is very nearly (though not exactly) a linear function, and calculate the slope in degrees Celsius per thousand feet (or in degrees Celsius per kilometer).
Figure: Phase diagram for H2O (not to scale). The table gives the vaporpressure and molar latent heat for the solid-gas transformation (first three entries)and the liquid-gas transformation (remaining entries). Data from Keenan et al.(1978) and Lide (1994).
Problem 1:
The Clausius-Clapeyron below relation is a differential equation that can, in principle, be solved to find the shape of the entire phase-boundary curve. To solve it, however, you have to know how both L and ΔV depend on temperature and pressure. Often, over a reasonably small section of the curve, you can take L to be constant. Moreover, if one of the phases is a gas, you can usually neglect the volume of the condensed phase and just take ΔV to be the volume of the gas, expressed in terms of temperature and pressure using the ideal gas law. Making all these assumptions, solve the differential equation explicitly to obtain the following formula for the phase boundary curve:
P = (constant) × e–L/RT,
This result is called the vapour pressure equation. Caution: Be sure to use this formula only when all the assumptions just listed are valid.
Relation:
Problem 2:
The exponential atmosphere.
(a) Consider a horizontal slab of air whose thickness (height) is dz. If this slab is at rest, the pressure holding it up from below must balance both the pressure from above and the weight of the slab. Use this fact to find an expression for dP/dz, the variation of pressure with altitude, in terms of the density of air.
(b) Use the ideal gas law to write the density of air in terms of pressure, temperature, and the average mass m of the air molecules. (The information needed to calculate m is given in Problem.) Show, then, that the pressure obeys the differential equation
called the barometric equation.
(c) Assuming that the temperature of the atmosphere is independent of height (not a great assumption but not terrible either), solve the barometric equation to obtain the pressure as a function of height: P(z) = P(0)e−mgz/kT. Show also that the density obeys a similar equation.
(d) Estimate the pressure, in atmospheres, at the following locations: Ogden, Utah (4700 ft or 1430 m above sea level); Leadville, Colorado (10,150 ft, 3090 m) ; Mt. Whitney, California (14,500 ft, 4420 m); Mt. Everest, Nepal/Tibet (29,000 ft, 8850 m). (Assume that the pressure at sea level is 1 atm.)
Problem 3:
Calculate the mass of a mole of dry air, which is a mixture of N2 (78% by volume), O2 (21%), and argon (1%).
Nature of Earth’s Atmosphere Earth’s three atmospheres -composition of today’s atmosphere Surface processes and earth-atm. Gas exchanges Density and Pressure Layers of the Atmosphere -temp patterns vertically -nature of troposphere and stratosphere -environmental lapse rate (ELR) Special Features: water vapor and clouds Dust (scatters light, etc.) Jet streams (transports hear etc) Ozone layer -Magnetosphere earth ‘s three atmosphere: Origional atmosphere unknown composition Second atmosphere mainly water vapor and CO2 (from volcanic outgassing) Third atmosphere oxygen righ 20% (after photosynthesis developed)
