� � � � � � � Entropy Change in Entropy Reversible Isobaric Process Ideal Gas in a Reversible Process Free Expansion of an Ideal Gas Microscopic Interpretation of Entropy Entropy and the Second Law of Thermodynamics Homework
✞ ✂ ☎ ✆ ✝ � ✆ � ☎ � ✟ ✁ ✠ ☞ ✞ ✂ ✄ ✁ ✝ ☛ ✁ � ✍ � ✄ � ✂ ✂ ✆ ✁ ✌ ✁ ✂ ✄ ✁ ☎ ✡ Change in Entropy The second law of thermodynamics states that when irreversible (real) processes occur, the disor- der in the system plus the surroundings increases The measure of disorder in a system is called entropy The change in entropy of a system as it moves through an infinitesimal process between two equilibrium states is where is the heat transferred in a reversible process between the same two states The change in entropy of a system as it goes from an initial state to a final state is Entropy is a state variable like internal energy and temperature The change in entropy for an irreversible process can be determined by calculating the change in entropy for a reversible process with the same initial and final states
✎ Example 1 Calculate the change in entropy when 0.235 kg of ice melts at 0 C.
✥ ✄ ✑ ✓ ✡ ✗ ✕ ✗ ✖ ✁ ✝ ✝ ✞ ✂ ✂ ✁ ☛ ✘ ✂ ☞ ✄ ✏ ✑ ✣ ✝ ☛ ✝ ☞ ✤ ✏ ✄ ✂ ✁ ☎ ✆ ✄ ✏ ✑ ✒ ✁ ✝ ✁ ✂ ✄ ✁ ☎ ✆ ✔ ✄ ✏ ✑ ✓ ✁ ✝ ✝ ✖ ✝ ✡ ✔ ✕ Change in Entropy for a Reversible Isobaric Process ✓✚✙ ✛✢✜
✖ ✫ ✑ ✏ ✄ ☞ ✂ ✘ ☛ ✂ ✄ ✂ ✞ ✫ ✁ ✙ ✖ ✬ ✕ ✬ ✡ ✭ ✏ ✩ ✝ ✝ ✁ ✁ ✬ ✛ ✕ ✣ ✫ ☛ ☞ ✝ ✫ ☛ ✥ ✤ ☞ ✫ ☛ ✫ ✜ ✜ ✛ ✙ ✭ ✏ ✩ ✥ ✤ ☞ ✝ ☛ ✝ ✣ ✗ ✗ ✝ ✫ ✪ ✝ ✁ ✬ ✑ ✏ ✄ ★ ☞✧ ✦ ✁ ✝ ✁ ✄ ✪ ✩ ☞✧★ ✦ ✁ ✄ ✝ ✆ ☎ ✁ ✄ ✂ ✫ ✏ ✡ ✫ ✬ ✑ ✏ ✄ ✂ ✁ ✖ ✔ ✕ ✔ ✡ ✫ ✁ ✭ ✭ ✏ ✩ ✝ ✝ ✁ ✬ ✑ ✏ ✄ ✂ ✁ ✝ ☞ Change in Entropy for a Reversible Process with an Ideal Gas Note that it is clear from this result that the change in entropy depends only on the properties of the initial state ( and ) and the properties of the final state ( and )
☛ ✦ ✁ ✫ ✄ ✍ � ✁ ☞✧★ � ✄ ✍ ✝ ☞ ✄ ✝ ✪ ✘ ✄ ☎ � � ✮ ✁ ✄ ✍ � ✁ Free Expansion of an Ideal Gas Consider the example of free expansion shown below in which gas that is confined in the left half of an insulated container is allowed to expand into a larger volume The process is irreversible - it does not occur in reverse with the gas spontaneously collecting itself in the left half of the container The container is insulated, so The walls are rigid, so From the first law of thermo we have , so
✕ ✏ ✖ ✬ ✕ ✬ ✡ ✭ ✄ ✫ ✂ ✁ ✖ ✔ � ✔ ✁ ✫ ✫ ✄ ✫ ✣ ☞ ✙ ✭ ✏ ☞ ✞ ✂ ✘ ☛ ✂ ✄ ✂ ✡ ✫ ✫ ✄ ✁ ✪ ✄ ✮ ✁ ✘ ✆ ✄ ☎ ✁ ✤ � ✍ � ✫ ✏ ✁ ✁ ✭ ✏ ✄ ✝ ✆ ☎ ✄ ✭ ✂ ✁ ✫ ✫ ✁ ✝ ☛ Free Expansion of an Ideal Gas (cont’d) To calculate the change in entropy for free expansion, we need to consider a reversible process that connects the same initial and final states We can use the isothermal expansion of an ideal gas ✛✢✜ ✥✰✯ Example 2: One mole of nitrogen undergoes free expansion to double its volume. What is the change in entropy of the gas?
✥ ✣ � ✿ ❀ ✙ ✛ ✜ ✮ ✥ ☛ ✮ ☞ ✤ ✥ ✄ ✵ ✤ ✴ ✻ ✼ ✵ ✽ ✬ ✖ ✬ ✾ ☞ ✵ ✄ ✜ ✣ ✫ ☛ ✫ ✏ ❁ � ✙ ✛ ✮ ☞ ✄ ✏ ✭ ✛ ❀ ✜ ✣ ✫ ☛ ✫ ☞ ✤ ✙ ✿ ✿ ☞ ❀ ✙ ✣ ✫ ☛ ✫ ✤ ✘ ✥ ✿ ❀ ✙ ✛ ✮ ☛ ✻ ✬ ✕ ✮ � ✬ ✸ ✱ ✫ ✳ ☞ ✫ ✷ ✄ ☞ ✄ ☞ ✴ ✮ � ✱ ✫ ✳ ☞ ✫ ✄ ☞ ✲ ☞ ✫ ✱ ✫ � ✴ ✵ ☛ ☛ ✪ ✪ ☞ ✄ � ✵ ✸ ✮ ✱ ✫ ✳ ✪ ☛ ✫ ✮ ✷ ✄ ☛ ✵ ✲ ☞ ✄ ☛ Microscopic Interpretation of Entropy Consider the free expansion of an ideal gas If we assume that each molecule occupies a volume and that each location of a molecule is equally probable, then the total number of possible locations, or microstates, in the initial volume is The number of possible microstates for molecules in the initial volume is ✲✶✵ Similarly, the number of possible microstates for molecules in the final volume is The probability of a given macrostate is proportional to the number of microstates correspond- ing to the macrostate, and thus the ratio of the probabilities of given initial and final macrostates occurring is ✄✺✹ Taking the natural log and multiplying by Boltzmann’s constant we have ✛❂✜
✫ ✣ ✿ ❃ ✂ ✛ ✮ � ✥ ✤ ☞ � ☛ ✫ ✜ ✙ ✛ ✙ ✭ ✏ ✄ ☞ ✂ ✘ ☛ ✂ � � ❀ Microscopic Interpretation of Entropy (cont’d) Earlier we found that the change in entropy for free expansion of an ideal gas in terms of the macroscopic thermodynamic variables is Comparison of the previous two expressions yields the following connection between entropy and the number of microstates associated with a given macrostate This is the famous Boltzmann’s entropy equation, and is engraved on his tombstone Example 3: One mole of nitrogen undergoes free expansion to double its volume. Use the micro- scopic interpretation of entropy to find the change in entropy of the gas?
✞ ❍ ✂ � ✂ ✄ ✞ ✂ ✩ � ✞ ✂ ■ ✧❏ ❄ ✍ ✄ ✸ ✞ ✷ � ✍ ✞ ✂ ❄ ✍ ✟❅❆ ● ❇❈❉❊ ✁ ❅ ❋ ❅ ❉❊ � Entropy and The Second Law of Thermodynamics The second law of thermo can be stated in terms of entropy as: The entropy of an isolated system never decreases. It can only stay the same or increase. If the system is not isolated, then the change in entropy of the system plus the change in entropy of the environment must be greater than or equal to zero Only (idealized) reversible, cyclic processes have The total entropy of any system plus that of its environment increases as a result of any natural process
� � � Homework Set 9 - Due Fri. Jan. 30 Read Sections 18.6 - 18.8 Answer Questions 18.8 & 18.11 Do Problems 18.22, 18.23, 18.26, 18.28 & 18.33
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