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Chapter 3: Q. 3.28 (page 113)
A liter of air, initially at room temperature and atmospheric pressure, is heated at constant pressure until it doubles in volume. Calculate the increase in its entropy during this process.
The increase in entropy is.
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A bit of computer memory is some physical object that can be in two different states, often interpreted as 0 and 1. A byte is eight bits, a kilobyte is bytes, a megabyte is 1024 kilobytes, and a gigabyte is 1024 megabytes.
(a) Suppose that your computer erases or overwrites one gigabyte of memory, keeping no record of the information that was stored. Explain why this process must create a certain minimum amount of entropy, and calculate how much.
(b) If this entropy is dumped into an environment at room temperature, how much heat must come along with it? Is this amount of heat significant?
An ice cube (mass )is left sitting on the kitchen table, where it gradually melts. The temperature in the kitchen is .
(a) Calculate the change in the entropy of the ice cube as it melts into water at . (Don't worry about the fact that the volume changes somewhat.)
(b) Calculate the change in the entropy of the water (from the melted ice) as its temperature rises from to .
(c) Calculate the change in the entropy of the kitchen as it gives up heat to the melting ice/water.
(d) Calculate the net change in the entropy of the universe during this process. Is the net change positive, negative, or zero? Is this what you would expect?
Use the thermodynamic identity to derive the heat capacity formula
which is occasionally more convenient than the more familiar expression in terms of . Then derive a similar formula for , by first writing in terms of and .
In Problem you computed the entropy of an ideal monatomic gas that lives in a two-dimensional universe. Take partial derivatives with respect to , and N to determine the temperature, pressure, and chemical potential of this gas. (In two dimensions, pressure is defined as force per unit length.) Simplify your results as much as possible, and explain whether they make sense.
Consider an Einstein solid for which both N and q are much greater than . Think of each oscillator as a separate "particle."
(a) Show that the chemical potential is
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(b) Discuss this result in the limits and , concentrating on the question of how much increases when another particle carrying no energy is added to the system. Does the formula make intuitive sense?
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