Question

The energy change is greatest for a hydrogen atom when its state changes from

Answer 1

The energy transition will be equal to 1.55⋅10−19J.

So, you know your energy levels to be n = 5 and n = 3. Rydberg's equation will allow you calculate the wavelength of the photon emitted by the electron during this transition

1λ=R⋅(1n2final−1n2initial), where

λ - the wavelength of the emitted photon;
R - Rydberg's constant - 1.0974⋅107m−1;
nfinal - the final energy level - in your case equal to 3;
ninitial - the initial energy level - in your case equal to 5.

So, you've got all you need to solve for λ, so

1λ=1.0974⋅107m−1⋅(132−152)

1λ=0.07804⋅107m−1⇒λ=1.28⋅10−6m

Since E=hcλ, to calculate for the energy of this transition you'll have to multiply Rydberg's equation by h⋅c, where

h - Planck's constant - 6.626⋅10−34J⋅s
c - the speed of light - 299,792,458 m/s

So, the transition energy for your particular transition (which is part of the Paschen Series) is

E=6.626⋅10−34J⋅s⋅299,792,458m/s1.28⋅10−6m

E=1.55⋅10−19J