What is the Issue with Extensive Properties of Entropy?

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In summary, the conversation discusses the concept of entropy as an extensive property and an expression for it derived using statistical mechanics methods for an ideal gas. A question is raised about the effects of doubling the number of particles, volume, and energy on the calculated entropy. It is explained that the final entropy will indeed be doubled due to the cancellation of positive and negative logarithmic terms. It is also noted that the formula provided is specifically for a monoatomic gas, not any ideal gas.
  • #1
Vectorcrust
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Hi to all!

The entropy is known as an extensive property. Here is an expression for the entropy of ideal gas derived by statistical mechanics methods:

sgas2.gif


Imagine that I multiply by 2 the number of particles, the volume of particles and the energy of particles(so the molar volume and molar energy and all other intensive properties are the same). According to this expression I'll never get entropy multiplied by 2 because of N that under ln expression.

Where am I wrong?

Thanks.
 
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  • #2
Hi Vectorcrust, welcome to PF!

Vectorcrust said:
According to this expression I'll never get entropy multiplied by 2 because of N that under ln expression.

Sure you will, if you note that U and V get doubled as well and you work through the logarithms.
 
  • #3
The U and V logarithms will give you postive ln(2) terms and the logarithm with N inside will give you -ln(2) terms. They will cancel.
 
  • #4
Mapes said:
Sure you will, if you note that U and V get doubled as well and you work through the logarithms.


I see it now. Sorry for dumb question. Thanks a lot!
 
  • #5
Vectorcrust said:
I see it now. Sorry for dumb question. Thanks a lot!

Not dumb! Stat. mech. requires a lot of staring at logarithm-filled equations no matter how smart you are. Stick around!
 
  • #6
also note that the formula you posted is not for any "ideal gas", it is for a monoatomic gas. Just keep that in mind.
 

Related to What is the Issue with Extensive Properties of Entropy?

1. What is the definition of entropy?

Entropy is a measure of the disorder or randomness in a system. It is a thermodynamic property that describes the amount of energy that is unavailable for work in a system.

2. How is entropy related to the second law of thermodynamics?

The second law of thermodynamics states that in any natural process, the total entropy of a closed system will always increase over time. This means that the disorder or randomness in a system will tend to increase, leading to a decrease in the amount of available energy.

3. Can entropy be reversed or decreased?

No, the second law of thermodynamics states that the total entropy of a closed system will always increase over time. While it is possible to decrease the entropy of a local system, the overall entropy of the entire system will still increase.

4. How does the extensivity of entropy affect larger systems?

The extensivity of entropy means that the total entropy of a system is directly proportional to its size. This means that larger systems will have a higher entropy and will tend to be more disordered than smaller systems.

5. Can entropy be used to predict the direction of natural processes?

Yes, the second law of thermodynamics and the concept of entropy can be used to predict the direction of natural processes. In general, natural processes will tend to move towards a state of higher entropy and disorder.

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