The electron energy in hydrogen atom is given by `E=(-21.7 xx 10^(-12))ln^2` ergs. Calculate the energy required to remove an electron completely from the n=2 orbit. What is the longest wavelength in cm of light that can be used to cause this transition?

Explanation

To solve the problem step by step, we will calculate the energy required to remove an electron from the n=2 orbit of a hydrogen atom and then find the longest wavelength of light that can cause this transition.

### Step 1: Calculate the energy of the electron in the n=2 orbit

The energy of an electron in a hydrogen atom is given by the formula:

\[ E_n = \frac{-21.7 \times 10^{-12}}{n^2} \]

For n=2:

\[ E_2 = \frac{-21.7 \times 10^{-12}}{2^2} = \frac{-21.7 \times 10^{-12}}{4} \]

Calculating this gives:

\[ E_2 = -5.425 \times 10^{-12} \text{ ergs} \]

### Step 2: Calculate the energy required to remove the electron completely

The energy required to remove the electron completely from the n=2 orbit is the difference between the energy at infinity (which is 0) and the energy at n=2:

\[ \Delta E = E_{\infty} - E_2 = 0 - (-5.425 \times 10^{-12}) = 5.425 \times 10^{-12} \text{ ergs} \]

### Step 3: Calculate the longest wavelength of light that can cause this transition

Using the equation that relates energy and wavelength:

\[ \Delta E = \frac{h \cdot c}{\lambda} \]

Where:
- \( h \) (Planck's constant) = \( 6.626 \times 10^{-34} \) J·s
- \( c \) (speed of light) = \( 3 \times 10^{10} \) cm/s

Rearranging for wavelength \( \lambda \):

\[ \lambda = \frac{h \cdot c}{\Delta E} \]

Substituting the values:

\[ \lambda = \frac{(6.626 \times 10^{-34}) \cdot (3 \times 10^{10})}{5.425 \times 10^{-12}} \]

Calculating this gives:

\[ \lambda \approx 3.67 \times 10^{-5} \text{ cm} \]

### Final Results

- Energy required to remove the electron from n=2: \( 5.425 \times 10^{-12} \) ergs
- Longest wavelength of light that can cause this transition: \( 3.67 \times 10^{-5} \) cm