Question

How much energy must be supplied to a hydrogen atom to cause a transition from the ground state to the \(n=4\) state?

Short answer

Answer: The energy required for the transition is 12.75 eV.

Step-by-step solution

Energy level formula for hydrogen atom

The formula to calculate the energy levels of the hydrogen atom is: \(E_n = -\frac{13.6 eV}{n^2}\), where \(E_n\) is the energy of the nth energy level and n is the principal quantum number.

Calculate the energy of the ground state (n=1)

Using the energy level formula, we can calculate the energy of the ground state (n=1): \(E_1 = -\frac{13.6 eV}{1^2} = -13.6 eV\)

Calculate the energy of the n=4 state

Now, we can calculate the energy of the n=4 state using the same formula: \(E_4 = -\frac{13.6 eV}{4^2} = -\frac{13.6 eV}{16} = -0.85 eV\)

Find the energy difference between the two states

To find the energy required for the transition, we need to calculate the energy difference between the two states: \(\Delta E = E_4 - E_1 = (-0.85 eV) - (-13.6 eV) = 12.75 eV\)

Final answer

The energy that must be supplied to cause a transition from the ground state to the n=4 state in a hydrogen atom is 12.75 eV.