An electron revolves round a nucleus of charge Ze. In order to excite the electron from the `n=20` to `n=3`, the energy required is `47.2 eV`. Z is equal to

Options

A. 3
B. 4
C. 5
D. 2

Explanation

To solve the problem, we need to determine the value of Z given that the energy required to excite an electron from the n=20 level to the n=3 level is 47.2 eV. We will use the formula for the energy levels of an electron in a hydrogen-like atom:

\[ E_n = -\frac{13.6 Z^2}{n^2} \]

### Step 1: Write the energy difference formula
The energy required to excite the electron from level n1 to n2 is given by the difference in energy between the two levels:

\[ \Delta E = E_{n2} - E_{n1} \]

Substituting the formula for energy levels:

\[ \Delta E = \left(-\frac{13.6 Z^2}{n_2^2}\right) - \left(-\frac{13.6 Z^2}{n_1^2}\right) \]

This simplifies to:

\[ \Delta E = 13.6 Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \]

### Step 2: Identify n1 and n2
In this case, we have:
- n1 = 20
- n2 = 3

### Step 3: Substitute n1 and n2 into the formula
Now, substituting n1 and n2 into the energy difference formula:

\[ \Delta E = 13.6 Z^2 \left(\frac{1}{20^2} - \frac{1}{3^2}\right) \]

### Step 4: Calculate the fractions
Calculating the fractions:

\[ \frac{1}{20^2} = \frac{1}{400} \]
\[ \frac{1}{3^2} = \frac{1}{9} \]

Now, we need a common denominator to subtract these fractions. The least common multiple of 400 and 9 is 3600.

Converting the fractions:

\[ \frac{1}{400} = \frac{9}{3600} \]
\[ \frac{1}{9} = \frac{400}{3600} \]

Now, substituting back into the equation:

\[ \Delta E = 13.6 Z^2 \left(\frac{9}{3600} - \frac{400}{3600}\right) \]
\[ \Delta E = 13.6 Z^2 \left(\frac{9 - 400}{3600}\right) \]
\[ \Delta E = 13.6 Z^2 \left(\frac{-391}{3600}\right) \]

### Step 5: Set the equation equal to the given energy
We know that the energy required is 47.2 eV, so we set the equation equal to 47.2:

\[ 47.2 = 13.6 Z^2 \left(\frac{-391}{3600}\right) \]

### Step 6: Solve for Z^2
Rearranging the equation to solve for Z^2:

\[ Z^2 = \frac{47.2 \cdot 3600}{13.6 \cdot -391} \]

Calculating the right side:

\[ Z^2 = \frac{169920}{-5316.6} \]

### Step 7: Calculate Z
Calculating Z from the above value:

\[ Z^2 \approx 5 \]
\[ Z \approx \sqrt{5} \approx 5 \]

Thus, the value of Z is approximately 5.

### Final Answer
Z = 5

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