What is the Issue with Extensive Properties of Entropy?

[Vectorcrust]

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:

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.

 

[Mapes]

Hi Vectorcrust, welcome to PF!

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.

 

[capandbells]

The U and V logarithms will give you positive ln(2) terms and the logarithm with N inside will give you -ln(2) terms. They will cancel.

 

[Vectorcrust]

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!

 

[Mapes]

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!

 

[Curl]

also note that the formula you posted is not for any "ideal gas", it is for a monoatomic gas. Just keep that in mind.