# 4.7 Entropy on a microscopic scale  (Page 4/10)

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Check Your Understanding A quantity of heat Q is absorbed from a reservoir at a temperature ${T}_{\text{h}}$ by a cooler reservoir at a temperature ${T}_{\text{c}}.$ What is the entropy change of the hot reservoir, the cold reservoir, and the universe?

$\text{−}Q\text{/}{T}_{\text{h}}$ ; $Q\text{/}{T}_{\text{c}}$ ; and $Q\left({T}_{\text{h}}-{T}_{\text{c}}\right)\text{/}\left({T}_{\text{h}}{T}_{\text{c}}\right)$

Check Your Understanding A 50-g copper piece at a temperature of $20\phantom{\rule{0.2em}{0ex}}\text{°C}$ is placed into a large insulated vat of water at $100\phantom{\rule{0.2em}{0ex}}\text{°C}$ . (a) What is the entropy change of the copper piece when it reaches thermal equilibrium with the water? (b) What is the entropy change of the water? (c) What is the entropy change of the universe?

a. 4.71 J/K; b. −4.18 J/K; c. 0.53 J/K

View this site to learn about entropy and microstates. Start with a large barrier in the middle and 1000 molecules in only the left chamber. What is the total entropy of the system? Now remove the barrier and let the molecules travel from the left to the right hand side? What is the total entropy of the system now? Lastly, add heat and note what happens to the temperature. Did this increase entropy of the system?

## Summary

• Entropy can be related to how disordered a system is—the more it is disordered, the higher is its entropy. In any irreversible process, the universe becomes more disordered.
• According to the third law of thermodynamics, absolute zero temperature is unreachable.

## Key equations

 Result of energy conservation $W={Q}_{\text{h}}-{Q}_{\text{c}}$ Efficiency of a heat engine $e=\frac{W}{{Q}_{\text{h}}}=1-\frac{{Q}_{\text{c}}}{{Q}_{\text{h}}}$ Coefficient of performance of a refrigerator ${K}_{\text{R}}=\frac{{Q}_{\text{c}}}{W}=\frac{{Q}_{\text{c}}}{{Q}_{\text{h}}-{Q}_{\text{c}}}$ Coefficient of performance of a heat pump ${K}_{\text{P}}=\frac{{Q}_{\text{h}}}{W}=\frac{{Q}_{\text{h}}}{{Q}_{\text{h}}-{Q}_{\text{c}}}$ Resulting efficiency of a Carnot cycle $e=1-\frac{{T}_{\text{c}}}{{T}_{\text{h}}}$ Performance coefficient of a reversible refrigerator ${K}_{\text{R}}=\frac{{T}_{\text{c}}}{{T}_{\text{h}}-{T}_{\text{c}}}$ Performance coefficient of a reversible heat pump ${K}_{\text{P}}=\frac{{T}_{\text{h}}}{{T}_{\text{h}}-{T}_{\text{c}}}$ Entropy of a system undergoing a reversible process at a constant temperature $\text{Δ}S=\frac{Q}{T}$ Change of entropy of a system under a reversible process $\text{Δ}S={S}_{B}-{S}_{A}={\int }_{A}^{B}dQ\text{/}T$ Entropy of a system undergoing any complete reversible cyclic process $\oint dS=\oint \frac{dQ}{T}=0$ Change of entropy of a closed system under an irreversible process $\text{Δ}S\ge 0$ Change in entropy of the system along an isotherm $\underset{T\to 0}{\text{lim}}{\left(\text{Δ}S\right)}_{T}=0$

## Conceptual questions

Are the entropy changes of the systems in the following processes positive or negative? (a) water vapor that condenses on a cold surface; (b) gas in a container that leaks into the surrounding atmosphere; (c) an ice cube that melts in a glass of lukewarm water; (d) the lukewarm water of part (c); (e) a real heat engine performing a cycle; (f) food cooled in a refrigerator.

Discuss the entropy changes in the systems of Question 21.10 in terms of disorder.

Entropy is a function of disorder, so all the answers apply here as well.

## Problems

A copper rod of cross-sectional area $5.0\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ and length 5.0 m conducts heat from a heat reservoir at 373 K to one at 273 K. What is the time rate of change of the universe’s entropy for this process?

$3.78\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{W/K}$

Fifty grams of water at $20\phantom{\rule{0.2em}{0ex}}\text{°C}$ is heated until it becomes vapor at $100\phantom{\rule{0.2em}{0ex}}\text{°C}$ . Calculate the change in entropy of the water in this process.

Fifty grams of water at $0\phantom{\rule{0.2em}{0ex}}\text{°C}$ are changed into vapor at $100\phantom{\rule{0.2em}{0ex}}\text{°C}$ . What is the change in entropy of the water in this process?

430 J/K

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