Calculate energy of electron which is moving in the orbit that has its radius , sixteen times the radius of first Bohr orbit for H-atom.

Explanation

To calculate the energy of an electron moving in an orbit that has a radius 16 times the radius of the first Bohr orbit for a hydrogen atom, we can follow these steps:

### Step 1: Understand the relationship between energy and radius
The total energy (E) of an electron in a hydrogen atom is given by the formula:

\[ E = -\frac{Z^2 \cdot k \cdot e^4 \cdot m}{2 \cdot \hbar^2 \cdot n^2} \]

For hydrogen (Z = 1), this simplifies to:

\[ E = -\frac{2.176 \times 10^{-18} \text{ J}}{n^2} \]

Where \( n \) is the principal quantum number. The first Bohr orbit corresponds to \( n = 1 \).

### Step 2: Determine the radius of the first Bohr orbit
The radius of the first Bohr orbit (n=1) for hydrogen is given by:

\[ r_1 = 5.29 \times 10^{-11} \text{ m} \]

### Step 3: Calculate the radius of the new orbit
Given that the radius of the new orbit is 16 times the radius of the first orbit:

\[ r = 16 \cdot r_1 = 16 \cdot 5.29 \times 10^{-11} \text{ m} = 8.464 \times 10^{-10} \text{ m} \]

### Step 4: Determine the principal quantum number for the new orbit
The radius of the nth Bohr orbit is given by:

\[ r_n = n^2 \cdot r_1 \]

Setting \( r_n = 16 \cdot r_1 \):

\[ n^2 = 16 \implies n = 4 \]

### Step 5: Calculate the energy for n = 4
Using the energy formula for \( n = 4 \):

\[ E = -\frac{2.176 \times 10^{-18} \text{ J}}{4^2} = -\frac{2.176 \times 10^{-18} \text{ J}}{16} \]

\[ E = -1.36 \times 10^{-19} \text{ J} \]

### Final Answer
The energy of the electron in the orbit that has a radius 16 times the radius of the first Bohr orbit for the hydrogen atom is:

\[ E = -1.36 \times 10^{-19} \text{ J} \]

---