Why must the Gibbs Free Energy be proportional to N?

[aliens123]

In the textbook Thermal Physics by Daniel Schroeder he says the following:

However, I don't follow this argument. Let's say that G was the following:
$$G(T, P, N) = (TPN)^{1/3}$$
Then
$$G(\lambda T, \lambda P, \lambda N) = \lambda G$$
So $$G$$ is extensive, but $$G \not \propto N.$$

 

[Dale]

Let's say that G was the following:
$$G(T, P, N) = (TPN)^{1/3}$$
Then
$$G(\lambda T, \lambda P, \lambda N) = \lambda G$$
So $$G$$ is extensive, but $$G \not \propto N.$$

In this example $G$ is not extensive since $G(T,P,\lambda N)\ne \lambda G(T,P,N)$

 

[aliens123]

In this example $G$ is not extensive since $G(T,P,\lambda N)\ne \lambda G(T,P,N)$

Isn't the definition of extensive
$$Q(\lambda q_1, \lambda q_2, ..., \lambda q_n ) = \lambda Q(q_1, q_2, ..., q_n)?$$

 

[kith]

Isn't the definition of extensive
$$Q(\lambda q_1, \lambda q_2, ..., \lambda q_n ) = \lambda Q(q_1, q_2, ..., q_n)?$$

The $q_i$ need to be extensive here. You are looking at $P$ and $T$ which are intensive. If you double the amount of stuff, temperature and pressure don't double. Have a look at the rabbits on the page before the excerpt you used in the OP.

 

[vanhees71]

No, you have to multiply only the extensive variables with $\lambda$. The intensive variables don't change when scaling the "system size". That's the definition of extensive vs. intensive variables.