Problems

4.1 A parcel is lifted adiabatically from z = 0 to z = H, what is its change in entropy?

4.2 Compute the change in entropy for an ideal dry gas of mass M which is heated at constant volume from T1 to T2. Take M = 1 kg, T1 = 300 K and T2 = 310 K.

4.3 A parcel is lifted isothermally from pressure p0 to p1. Find its change in potential temperature. Take po = 1000 hPa and pi = 500 hPa, To = 300 K.

4.4 A 1 kg parcel at 500 hPa and 250 K is heated with 500 J of radiation heating. What is the change in its enthalpy? What is the change in its entropy? Its potential temperature?

4.5 A quantity 18 g of water is (a) heated from 273 K to 373 K, (b) evaporated to gas form, and (c) heated to 473 K. All steps are performed at constant pressure. Compute the change in entropy for steps (a), (b), and (c). Note: the heat capacity for water vapor is « 2kJK-1 kg-1.

Use these data in the next two problems: 2 kg of an ideal gas (dry air) is at temperature 300 K, p = 1000 hPa.

Step 1: the volume is increased adiabatically until it is doubled. Step 2: the pressure is held constant and the volume is decreased to its original value. Step 3: the volume is held constant and the temperature is increased until the original state is recovered.

4.6 (a) Sketch the process (steps 1, 2 and 3) in the V-p plane.

(b) What are the volume and temperature at the end of step 1?

(c) What is the change of enthalpy AH, internal energy A U, and entropy AS, during step 1?

(d) How much work is done by the system in step 1?

4.7 Continuing Problem 4.6.

(a) How much work is performed in step 2?

(b) What is the total amount of work in all three steps?

(c) What is the entropy change AS in step 2?

(d) What are the total changes in U, H, S during all three steps?

4.8 A dry air parcel has a mass of 1 kg. It undergoes a process that is depicted in the V-p plane in Figure 4.6. Calculate the change of entropy, enthalpy and internal energy for this air parcel.

Figure 4.7 Diagram for Problem 4.9.

4.9 Find the change of internal energy and enthalpy for the cyclic process shown in Figure 4.7. Starting from point A describe how the temperature changes during this cyclic process.

4.10 Show that the work performed by a system during a reversible isothermal cycle is always zero.

4.11 Show that for the Carnot cycle of an ideal gas holds (4.74)

Hint: Divide one of the four equations for work done along the different legs of the cycle by another one.

4.12 A tropical storm can be approximated as a Carnot cycle. Air is heated nearly isothermally as it flows along the sea surface (typically 27 °C). The air is lifted adiabatically in the eye wall to a height above the tropopause where it begins to cool due to loss of infrared radiation to space. The temperature where this occurs is about —73 °C. Finally the air descends to the surface adiabatically. Calculate the thermodynamic efficiency of this "heat engine."2

4.13 Show that the work done in a (reversible) Carnot cycle is the product of the entropy difference between the two adiabats and the temperature difference between the two isotherms (see Figure 4.4). The result holds for any system, not just an ideal gas.

4.14 Air is expanded isothermally at 300 K from a pressure of 1000 hPa to 800 hPa. What is the change in specific Gibbs energy?

2 Kerry Emanuel (2005) explains this simple model at beginner's level in Chapter 10 of his book Devine Wind.

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