Chapter 12: Q7CP (page 488)
There was transfer of energy of 5000 J into a system due to a temperature difference, and the entropy increased by 10 J/K. What was the approximate temperature of the system, assuming that the temperature didn’t change very much?
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Young’s modulus for copper is measured by stretching a copper wire to be about . The density of copper is about , and the mass of a mole is .Starting from a very low temperature, use these data to estimate roughly the temperature T at which we expect the specific heat for copper to approach 3 kB . Compare your estimate with the data shown on a graph in this chapter.
For a certain metal the stiffness of the interatomic bond and the mass of one atom are such that the spacing of the quantum oscillator energy levels is J. A nanoparticle of this metal consisting of atoms has a total thermal energy of role="math" localid="1657867970311" J. (a) What is the entropy of this nanoparticle? (b) The temperature of the nanoparticle is 87 K. Next we add J to the nanoparticle. By how much does the entropy increase?
This question follows the entire chain of reasoning involved in determining the specific heat of an Einstein solid. Start with two metal blocks, one consisting of one mole of aluminum (27 g) and the other of one mole of lead (207 g), both initially at a temperature very near absolute zero (0 K). From measurements of Young’s modulus one finds that the effective stiffness of the interatomic bond modeled as a spring is for aluminum and 5 N/m for lead. (a) Is the number of quantized oscillators in the aluminum block greater, smaller, or the same as the number in the lead block? (b) What is the initial entropy of each block? (c) In which metal is the energy spacing of the quantized harmonic oscillators larger? (d) If we add 1 J of energy to each block, which metal now has the larger number of energy quanta? (e) In which block is the number of possible ways of arranging this of energy greater? (f) Which block now has the larger entropy? (g) Which block experienced a greater entropy change? (h) Which block experienced the larger temperature change? (i) Which metal has the larger specific heat at low temperatures? (j) Does your conclusion agree with the actual data given in Figure 12.33? (The numerical data are given in a table accompanying Problem P64.)
There was transfer of energy of 5000 J into a system due to a temperature difference, and the entropy increased by 10 J/K. What was the approximate temperature of the system, assuming that the temperature didn’t change very much?.
The reasoning developed for counting microstates applies to many other situations involving probability. For example, if you flip a coin 5 times, how many different sequences of 3 heads and 2 tails are possible? Answer: 10 different sequences, such as HTHHT or TTHHH. In contrast, how many different sequences of 5 heads and 0 tails are possible? Obviously only one, HHHHH, and our equation gives , using the standard definition that 0! is defined to equal 1.
If the coin is equally likely on a single throw to come up heads or tails, any specific sequence like HTHHT or HHHHH is equally likely. However, there is only one way to get HHHHH, while there are 10 ways to get 3 heads and 2 tails, so this is 10times more probable than getting all heads. Use the expressionto calculate the number of ways to get 0 heads, 1 head, 2 heads, 3 heads, 4 heads, or 5 heads in a sequence of 5 coin tosses. Make a graph of the number of ways vs. the number of heads.
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