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Get Full Access to An Introduction To Thermal Physics - 1 Edition - Chapter 5 - Problem 44p
Get Full Access to An Introduction To Thermal Physics - 1 Edition - Chapter 5 - Problem 44p

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# Suppose that an unsaturated air mass is rising and cooling

ISBN: 9780201380279 40

## Solution for problem 44P Chapter 5

An Introduction to Thermal Physics | 1st Edition

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Problem 44P

Problem 44P

Suppose that an unsaturated air mass is rising and cooling at the dry adiabatic lapse rate found in Problem 1. If the temperature at ground level is 25° C and the relative humidity there is 50%, at what altitude will this air mass become saturated so that condensation begins and a cloud forms (see below Figure 1)? (Refer to the vapour pressure graph drawn shown below Problem 5.)

Figure 1: Cumulus clouds form when rising air expands adiabaticallyand cools to the dew point the onset of condensation slowsthe cooling, increasing the tendency of the air to rise further.These clouds began to form in late morning, in a sky that was clear onlyan hour before the photo was taken. By mid-afternoon they had developedinto thunderstorms.

Problem 1:

In Problem 2 you calculated the pressure of earth’s atmosphere as a function of altitude, assuming constant temperature. Ordinarily, however, the temperature of the bottommost 10–15 km of the atmosphere (called the troposphere) decreases with increasing altitude, due to heating from the ground (which is warmed by sunlight). If the temperature gradient |dT/dz| exceeds a certain critical value, convection will occur: Warm, low-density air will rise, while cool, high-density air sinks. The decrease of pressure with altitude causes a rising air mass to expand adiabatically and thus to cool. The condition for convection to occur is that the rising air mass must remain warmer than the surrounding air despite this adiabatic cooling.

(a) Show that when an ideal gas expands adiabatically, the temperature and pressure are related by the differential equation

(b) Assume that dT/dz is just at the critical value for convection to begin, so that the vertical forces on a convecting air mass are always approximately in balance. Use the result of Problem 2 (b) to find a formula for dT/dz in this case. The result should be a constant, independent of temperature and pressure, which evaluates to approximately - 10°C/km. This fundamental meteorological quantity is known as the dry adiabatic lapse rate.

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 3.) 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(O)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), 02 (21%), and argon (1%).

Problem 4:

Ordinarily, the partial pressure of water vapour in the air is less than the equilibrium vapour pressure at the ambient temperature; this is why a cup of water will spontaneously evaporate. The ratio of the partial pressure of water vapour to the equilibrium vapour pressure is called the relative humidity. When the relative humidity is 100%, so that water vapour in the atmosphere would be in diffusive equilibrium with a cup of liquid water, we say that the air is saturated. The dew point is the temperature at which the relative humidity would be 100%, for a given partial pressure of water vapour.

(a) Use the vapor pressure equation (below Problem 5) and the data in below Figure 2 to plot a graph of the vapor pressure of water from 0°C to 40° C. Notice that the vapour pressure approximately doubles for every 10° increase in temperature.

(b) The temperature on a certain summer day is 30°C. What is the dew point if the relative humidity is 90%? What if the relative humidity is 40%?

Problem 5:

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 ΔU 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 ΔU 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:

Figure 2: Phase diagram for H2O (not to scale). The table gives the vapor pressure 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).

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