Find the kinetic energy, potential energy and total energy in
first and second orbit of hydrogen atom if potential energy in first orbit is taken
to be zero.

Explanation

To find the kinetic energy, potential energy, and total energy in the first and second orbits of a hydrogen atom, we will use the following formulas:

1. The energy in the nth orbit (E_n) is given by:
\[ E_n = -\frac{13.6}{n^2} \text{ eV} \]

2. The kinetic energy (K.E.) in the nth orbit is given by:
\[ K.E. = -\frac{E_n}{2} \]

3. The potential energy (P.E.) in the nth orbit is given by:
\[ P.E. = 2E_n \]

Given that the potential energy in the first orbit (n=1) is taken to be zero, we will calculate the energies for both the first (n=1) and second (n=2) orbits.

### Step 1: Calculate the Energy in the First Orbit (n=1)

Using the formula for energy in the nth orbit:
\[ E_1 = -\frac{13.6}{1^2} = -13.6 \text{ eV} \]

### Step 2: Calculate the Kinetic Energy in the First Orbit (n=1)

Using the formula for kinetic energy:
\[ K.E. = -\frac{E_1}{2} = -\left(-\frac{13.6}{2}\right) = 6.8 \text{ eV} \]

### Step 3: Calculate the Potential Energy in the First Orbit (n=1)

Since it is given that the potential energy in the first orbit is zero:
\[ P.E. = 0 \text{ eV} \]

### Step 4: Calculate the Total Energy in the First Orbit (n=1)

The total energy is the sum of kinetic and potential energy:
\[ \text{Total Energy} = K.E. + P.E. = 6.8 + 0 = 6.8 \text{ eV} \]

### Step 5: Calculate the Energy in the Second Orbit (n=2)

Using the formula for energy in the nth orbit:
\[ E_2 = -\frac{13.6}{2^2} = -\frac{13.6}{4} = -3.4 \text{ eV} \]

### Step 6: Calculate the Kinetic Energy in the Second Orbit (n=2)

Using the formula for kinetic energy:
\[ K.E. = -\frac{E_2}{2} = -\left(-\frac{3.4}{2}\right) = 1.7 \text{ eV} \]

### Step 7: Calculate the Potential Energy in the Second Orbit (n=2)

Using the formula for potential energy:
\[ P.E. = 2E_2 = 2 \times -3.4 = -6.8 \text{ eV} \]

### Step 8: Calculate the Total Energy in the Second Orbit (n=2)

The total energy is the sum of kinetic and potential energy:
\[ \text{Total Energy} = K.E. + P.E. = 1.7 + (-6.8) = -5.1 \text{ eV} \]

### Summary of Results:
- For the first orbit (n=1):
- Kinetic Energy: \( 6.8 \text{ eV} \)
- Potential Energy: \( 0 \text{ eV} \)
- Total Energy: \( 6.8 \text{ eV} \)

- For the second orbit (n=2):
- Kinetic Energy: \( 1.7 \text{ eV} \)
- Potential Energy: \( -6.8 \text{ eV} \)
- Total Energy: \( -5.1 \text{ eV} \)