What energy in eV will be required to excite the ground state electron of a hydrogen atom to h = 3 state ? Ground state energy = -13.6 eV.

Answer

To find the energy required to excite the ground state electron of a hydrogen atom from the ground state to the n = 3 state, we can follow these steps:

### Step 1: Identify the ground state energy
The ground state energy (E1) of a hydrogen atom is given as:
\[ E_1 = -13.6 \, \text{eV} \]

### Step 2: Calculate the energy of the n = 3 state
The energy of an electron in the nth state of a hydrogen atom can be calculated using the formula:
\[ E_n = \frac{E_1}{n^2} \]
For n = 3:
\[ E_3 = \frac{E_1}{3^2} = \frac{-13.6 \, \text{eV}}{9} \]
Calculating this gives:
\[ E_3 = -1.51 \, \text{eV} \]

### Step 3: Determine the energy required for the transition
The energy required to excite the electron from the ground state (E1) to the n = 3 state (E3) is given by:
\[ \text{Energy required} = E_3 - E_1 \]
Substituting the values we have:
\[ \text{Energy required} = (-1.51 \, \text{eV}) - (-13.6 \, \text{eV}) \]
This simplifies to:
\[ \text{Energy required} = -1.51 + 13.6 \]
Calculating this gives:
\[ \text{Energy required} = 12.09 \, \text{eV} \]

### Final Answer
The energy required to excite the ground state electron of a hydrogen atom to the n = 3 state is:
\[ \text{Energy required} = 12.09 \, \text{eV} \]

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