Coulomb force vs. Pauli principle in atoms

[Heirot]

Homework Statement

In the absence of the Pauli principle, the Coulomb force would not be saturated, and the energy of a system containing N atoms would vary as N^(5/3).

Show that a system of 2N atoms would have an energy lower than that of two N - atom systems.

Homework Equations

E = c * N^(5/3)

The Attempt at a Solution

E(2N) = c * (2N)^(5/3) = c * 2^(5/3) * N^(5/3)
2 * E(N) = 2 * c N^(5/3)

Since, 2^(5/3) > 2, then obviously E(2N) > 2 E(N), contrary to the what had to be shown. Where's the error in my reasoning?

 

[gabbagabbahey]

Homework Statement

In the absence of the Pauli principle, the Coulomb force would not be saturated, and the energy of a system containing N atoms would vary as N^(5/3).

Show that a system of 2N atoms would have an energy lower than that of two N - atom systems.

Homework Equations

E = c * N^(5/3)

The Attempt at a Solution

E(2N) = c * (2N)^(5/3) = c * 2^(5/3) * N^(5/3)
2 * E(N) = 2 * c N^(5/3)

Since, 2^(5/3) > 2, then obviously E(2N) > 2 E(N), contrary to the what had to be shown. Where's the error in my reasoning?

Is $c$ positive or negative, and why?

 

[Heirot]

I can't see how could c be anything but positive due to Coulomb repulsion of like charges.

 

[zhermes]

I can't see how could c be anything but positive due to Coulomb repulsion of like charges.

True, but this is an atom: it has a large positive nuclei

 

[Heirot]

But then then energy wouldn't depend on N as N^(5/3) but also as N^2