Problems

3.1 Supposep(z) = p0e-z/H. Evaluate the following.

3.2 Let p = pRT. Evaluate the following.

3.3 The compressibility of a substance is defined by

where X is the variable being held constant. We can compress the gas isothermally (kt) or adiabatically (k0) (0 is the potential temperature). Calculate both for an ideal gas.

3.4 The coefficient of expansion is defined by

Compute ß for an ideal gas. 3.5 Show that for any gas

dT v KT

(Hint: dV = (dp) dp + (dr)p dT, see the Calculus refresher in this chapter.)

3.6 Find the internal energy of 1 kg of dry air at STP.

3.7 Suppose the atmosphere has its pressure given by p(z) = p0e-z/H with p0 = 1 atm and T(z = 0) = 273 K. Now suppose a 1 kg parcel is lifted adiabatically one scale height H. How much work does the parcel do on the environment in the process? What is the change in its specific internal energy, Au?

3.8 Isobaric process A 1 kg parcel of dry air has temperature 285 K and pressure 1000 hPa. It is heated by contact with the dry ground to a temperature of 295 K. (a) What is Q? (b) What is the change of the parcel's specific internal energy? (c) What is the change in the parcel's specific enthalpy?

3.9 Isothermal process 1 kg of dry air at 300 K and 1000 hPa is expanded isothermally (pretty unusual in the atmosphere) from a volume of 2 m3 to twice that value. (a) What is the work done by the gas in this expansion? (b) What is the heat absorbed? (c) What is the change in enthalpy?

3.10 Isochoric process 1 m3 of dry air at 1000 hPa and 290 K is enclosed in a rigid box. What is its density (kgm-3)? 3 g of liquid water are evaporated into the box. What is the increase of temperature in this box whose volume is held fixed? What are the changes in internal energy and enthalpy? Sketch a diagram of the change in the V-p plane.

3.11 Adiabatic process A parcel of mass 1 kg is lifted adiabatically from 800 hPa, where its temperature is 270 K, to 600 hPa. What is the new temperature? What are the changes in internal energy and enthalpy? Sketch a diagram of the change in the V-p plane.

3.12 Isobaric process A parcel of essentially dry air is held at a fixed altitude where the pressure is 950 hPa. It is heated by infrared radiation being absorbed by some water vapor in the parcel. The heating rate is 20 J kg-1 s-1. What is the rate of change of the temperature, specific enthalpy and specific internal energy? How does the potential temperature change per unit time? Sketch a diagram of the change in the V-p plane.

3.13 An air column is composed of dry air and the density of the air is given by p(z) = p0e-z/H, where p0 = 1.25kgm-3 andH = 10km.

(a) What is the mass of air (kg) lying above 1 m2 ?

(b) How many idealized "air" molecules are above the 1m2?

(c) Then approximately how many "air molecules" are there in the entire Earth's atmosphere?

3.14 The speed of sound in air can be computed from the formula vsound = V1 /pKX, where kx is the compressibility holding the parameter X constant. Compare the sound speeds (taking p = p/RT) when X = kt and X = k0 . The latter fits the data. Do you recall from physics why the adiabatic compressibility gives the correct answer instead of the isothermal compressibility? See Problem 3.3 above.

3.15 Suppose the atmosphere satisfiesp(z) = p0e-z/H and that it is isothermal (T(z) = To). What is the potential temperature 0 as a function of z? Sketch a graph.