# General Extensivity of Entropy

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1. Mar 1, 2016

### J co

1. The problem statement, all variables and given/known data
Which of the following are not extensive functions:

S[1] = (N/V)[S[0]+[C][v] ln(T) + R ln(V)]
S[2] = (N)[S[0]+[C][v] ln(T) + R ln(V/N)]
S[3] = ([N][/2])[S[0]+[C][v] ln(T) + R ln(V/N)]

2. Relevant equations
I'm not really sure how to approach this problem. The definition that I find for extensivity states: "At fixed pressure and temperature, if the amount of a substance N is changed by a factor λ, the volume V is also changed by the same factor"

The corresponding example for the definition shows that if a system S(U,V,N) is combined with an identical system then the result should be 2U,2V,2N.

First - I get hung up on the language of these statements quite often which is pretty discouraging. When the definition says "fixed temperature and pressure" and the forumlae have ln(T), how can I say that the system has a fixed temperature?

Second - Should I be determining the extensivity by doubing S, for instance 2(S[1]), or should I consider if in the formula for the system N and V scale by the same factor?

3. The attempt at a solution
Once again, I'm not sure how to begin here.

If someone could simply point me in the right direction it would be greatly appreciated.

Last edited by a moderator: Mar 1, 2016
2. Mar 1, 2016

### Staff: Mentor

You assume that T and P are held constant, and you double the amount of material N. Then you determine whether the property in question doubles.

Again, you hold T and P constant, and see if the property in question is proportional to N.

A property is considered an extensive property if, at constant temperature and pressure, the value of the property is directly proportional to the amount of the material present. At constant temperature and pressure, the system volume V, the system internal energy U, the system enthalpy H, and the system entropy S are all directly proportional to the amount of material you have.

In the case of S[1], is the ratio of N/V directly proportional to N? Is ln(V) = ln(NkT/P) directly proportional to N?

In the case of S[2], at constant temperature and pressure, is the value of S[2] directly proportional to N?

In the case of S[3], at constant temperature and pressure, is the value of S[3] directly proportional to N?

Chet