Consider the hydrogen atom. how does the energy difference between adjacent orbit radii change as the principal quantum number increases?

the energy difference between adjacent levels decreases as the quantum number increases

Explanation:

The energy levels of the hydrogen atom are given by the following formula:

where

is a constant

n is the level number

We can write therefore the energy difference between adjacent levels as

We see that this difference decreases as the level number (n) increases. For example, the difference between the levels n=1 and n=2 is

While the difference between the levels n=2 and n=3 is

And so on.

So, the energy difference between adjacent levels decreases as the quantum number increases.