The second law of thermodynamics

General

Problem:

Give two statements of the second law of thermodynamics that are not obviously identical in conceptual content but are in fact equivalent.

Solution:

Problem:

Determine the work that can be obtained from the one cycle of the ideal Carnot machine in which the working substance is a photon gas.  The energy density of this gas is given by the u = σT4 and the pressure is P = ⅓u.

Solution:

Problem:

What is the efficiency of a Carnot cycle operating between the same high (TH) and low (TC) temperatures as the ideal gas in this circular cycle?

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Solution:


Heat engines

Problem:

An ideal heat engine is powered by two reservoirs of equal heat capacity C, which is temperature independent.  As the engine works, the reservoirs gradually equilibrate. 
(a)  Find the overall efficiency of the engine from the starting point where the reservoirs are at temperatures T1 = 90 oC and T2 = 30 oC to the moment of complete equilibration.
(b)  Now assume that a real heat engine is powered by the same reservoir with initial temperatures T1 = 90 oC and T2 = 30 oC.  Let C = 105 J/oC.  The real engine's overall efficiency is 20% of the overall efficiency of the ideal engine.  Find the change in entropy of the system once the reservoirs have equilibrated.

Problem:

A stretched rubber band contracts when heated under constant tension.  Its temperature increases when stretched adiabatically.  The equation of state for an idealized rubber band is J = αLT, where J is the tension in the rubber band, L is its length, T is the absolute temperature and α is a constant. 
For reversible processes we have for the rubber band TdS = dQrev = cLdT – JdL. 
The heat capacity of the band at constant length is cL = constant.

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Consider a heat engine that uses a rubber band in the three-step cycle shown. 
Start with a stretched rubber band of length L0 , tension J0, and temperature TA.
Take the band through a sequence of reversible processes.
A --> B:
The rubber band is stretched under constant tension J0 to a length 2L0 while in contact with a heat reservoir of temperature TB
B --> C:
While in contact with a heat reservoir of temperature TC the tension of the rubber band is increased from J0 to 2J0 at constant length 2L0 .
C --> A:
While in contact with a heat reservoir of temperature TA, tension and length decrease linearly from (2J0, 2L0) to (J0, L0).
(a)  Find the ratios TB/TA and TC/TA.  What is the ratio Thot/Tcold?
(b)  Find the work done by the heat engine as it moves through one cycle A --> B --> C --> A.
(c)  During one cycle A --> B --> C --> A, how much heat is extracted from the hot reservoir and how much heat is dumped into the cold reservoir?
(d)  Find the efficiency of this rubber-band heat engine and compare it to the efficiency of a Carnot engine operating between the same temperatures.

Solution:

Problem:

A Carnot heat engine has the following entropy-temperature diagram.

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(a)  Describe the cycle.  For each segment identify the process, say whether work is done by the working system or on it and whether heat is added to the system or extracted from it.
(b)  How much work is done by the system?

Solution:

Problem:

The operation of a gasoline engine is (roughly) similar to the Otto cycle.  A S-V diagram is shown.

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A → B:  Gas compressed adiabatically.
B → C:  Gas heated isochorically (constant volume; corresponds to combustion of gasoline).
C → D:  Gas expanded adiabatically (power stroke).
D → A:  Gas cooled isochorically.
Compute the efficiency of the Otto cycle for an ideal gas (with temperature-independent heat capacities) as a function of the compression ratio VA/VB, and the heat capacity of the gas, CV .

Solution:


Refrigerators, Air conditioners, Heat pumps

Problem:

A refrigerator uses 10 W of electrical power when it is closed to keep the interior temperature stable.  Use reasonable estimates for any relevant temperatures to find an upper bound on the rate at which heat is entering the refrigerator due to imperfect thermal insulation.

Solution:

Problem:

An electric freezer is turned on inside a tent for a long time.  It is 0 oC outside the tent, +1 oC inside the tent, and -13 oC inside the freezer.  What would be the equilibrium temperature inside the tent if another freezer is turned on inside the tent?  The outside temperature remains the same.  The freezers are identical and follow the Carnot cycle.

Solution:

Problem:

A nursery uses natural gas heating to keep the greenhouses at 30 oC all year.  An engineer points out that the water at the bottom of a nearby lake is at a constant temperature of 5 oC, and that he can build an ideal heat pump that will work at maximum possible efficiency to pump heat from this lake water into the greenhouses.  He claims that the nursery will come out ahead with his system, even though it uses electricity instead of natural gas at three times the cost per Joule.  Is he right?  Neglecting capital and maintenance costs by what factor would their energy bill change?

Solution: