Solving E Extensive Parameter Relation: cE = E(cS, cV, cN)

[ehrenfest]

[SOLVED] extensive quantity

Homework Statement

My teacher said that E=E(S,V,N) implies that c E = E(cS,cV,cN) where c is some constant. The justification was that E is an extensive parameter. I know what an extensive parameter is (proportional to N), but I do not see how that relation follows.

Homework Equations

The Attempt at a Solution

 

[Mapes]

1. E, S, V, and N are extensive quantities.
2. You have a system with a certain S, V, and N, and you calculate the energy E.
3. I replace the system with one twice as big.
4. You can either double S, V, and N and recalculate or just double E, right? This statement is identical to your equation.

 

[Zeppos10]

the proper definition of an extensive quantity is: if there is a system with extensive quanty X' and another system with X" then the two systems together has property X, with X=X'+X". (the property is additive). This is independent of the existence of a function X(a,b,c) where a b and c are also extensive quantities. For example take the internal energy of systems consisting of mono-atomic ideal gases with for helium U'=n'(3/2)RT' and for neon U"=n"(3/2)RT". (Hence U=U(n,T)). Now if we take the systems together U=U'+U" and n=n'+n", independent of the actual values of T'and T". (and we do not need the constant c)
U (=E) is and extensive quantity independent from the suggested function, that is actually non-existent for most systems.