Entropy Change in Entropy Reversible Isobaric Process Ideal Gas in - - PowerPoint PPT Presentation

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

• f 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

• f 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
• f 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

• f 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

• ccurring 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
• f 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