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Chapter 20: Problem 37
An Otto engine has a maximum efficiency of \(20.0 \%\) find the compression ratio. Assume that the gas is diatomic.
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Consider a system consisting of rolling a six-sided die. What happens to the entropy of the system if an additional die is added? Does it double? What happens to the entropy if the number of dice is three?
A heat engine cycle often used in refrigeration, is the Brayton cycle, which involves an adiabatic compression. followed by an isobaric expansion, an adiabatic expansion and finally an isobaric compression. The system begins at temperature \(T_{1}\) and transitions to temperatures \(T_{2}, T_{3},\) and \(T_{4}\) after respective parts of the cycle. a) Sketch this cycle on a \(p V\) -diagram. b) Show that the efficiency of the overall cycle is given by \(\epsilon=1-\left(T_{A}-T_{1}\right) /\left(T_{3}-T_{2}\right)\)
One of your friends begins to talk about how unfortunate the Second Law of Thermodynamics is, how sad it is that entropy must always increase, leading to the irreversible degradation of useful energy into heat and the decay of all things. Is there any counterargument you could give that would suggest that the Second Law is in fact a blessing?
The entropy of a macroscopic state is given by \(S=k_{B} \ln w\) where \(k_{\mathrm{B}}\) is the Boltzmann constant and \(w\) is the number of possible microscopic states. Calculate the change in entropy when \(n\) moles of an ideal gas undergo free expansion to fill the entire volume of a box after a barrier between the two halves of the box is removed.
Suppose a Brayton engine (see Problem 20.26 ) is run as a refrigerator. In this case, the cycle begins at temperature \(T_{1}\), and the gas is isobarically expanded until it reaches temperature \(T_{4}\). Then the gas is adiabatically compressed, until its temperature is \(T_{3}\). It is then isobarically compressed, and the temperature changes to \(T_{2}\). Finally, it is adiabatically expanded until it returns to temperature \(T_{1}\) - a) Sketch this cycle on a \(p V\) -diagram. b) Show that the coefficient of performance of the engine is given by \(K=\left(T_{4}-T_{1}\right) /\left(T_{3}-T_{2}-T_{4}+T_{1}\right) .\)
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