Consider a hydrogen-like ionized atom with atomic number with a single electron. In the emission spectrum of this atom, the photon emitted in the 2 to 1 transition has energy 74.8 higher than the photon emitted in the 3 to 2 transition. The ionization energy of the hydrogen atom is 13.6. The value of Z is __________.

Answer

To solve the problem, we need to determine the atomic number \( Z \) of a hydrogen-like ionized atom based on the energy differences of photon emissions during electron transitions.

### Step-by-Step Solution:

1. **Understanding the Energy Formula**:
The energy of the photon emitted during a transition from a higher energy level \( n_i \) to a lower energy level \( n_f \) in a hydrogen-like atom is given by:
\[ \Delta E = 13.6 Z^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \]

2. **Calculate Energy for the 2 to 1 Transition**:
For the transition from \( n = 2 \) to \( n = 1 \):
\[ \Delta E_{2 \to 1} = 13.6 Z^2 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = 13.6 Z^2 \left( 1 - \frac{1}{4} \right) = 13.6 Z^2 \left( \frac{3}{4} \right) \]
Simplifying, we get:
\[ \Delta E_{2 \to 1} = 10.2 Z^2 \]

3. **Calculate Energy for the 3 to 2 Transition**:
For the transition from \( n = 3 \) to \( n = 2 \):
\[ \Delta E_{3 \to 2} = 13.6 Z^2 \left( \frac{1}{2^2} - \frac{1}{3^2} \right) = 13.6 Z^2 \left( \frac{1}{4} - \frac{1}{9} \right) \]
Finding a common denominator (36):
\[ \frac{1}{4} - \frac{1}{9} = \frac{9}{36} - \frac{4}{36} = \frac{5}{36} \]
Thus, we have:
\[ \Delta E_{3 \to 2} = 13.6 Z^2 \cdot \frac{5}{36} = \frac{68 Z^2}{36} = \frac{17 Z^2}{9} \]

4. **Setting Up the Energy Difference Equation**:
According to the problem, the energy of the photon emitted in the 2 to 1 transition is 74.8 eV higher than that in the 3 to 2 transition:
\[ \Delta E_{2 \to 1} = \Delta E_{3 \to 2} + 74.8 \]
Substituting the expressions we derived:
\[ 10.2 Z^2 = \frac{17 Z^2}{9} + 74.8 \]

5. **Clearing the Equation**:
To eliminate the fraction, multiply the entire equation by 9:
\[ 9 \cdot 10.2 Z^2 = 17 Z^2 + 9 \cdot 74.8 \]
This simplifies to:
\[ 91.8 Z^2 = 17 Z^2 + 673.2 \]
Rearranging gives:
\[ 91.8 Z^2 - 17 Z^2 = 673.2 \]
\[ 74.8 Z^2 = 673.2 \]

6. **Solving for \( Z^2 \)**:
Dividing both sides by 74.8:
\[ Z^2 = \frac{673.2}{74.8} = 9 \]

7. **Finding \( Z \)**:
Taking the square root of both sides:
\[ Z = 3 \]

### Final Answer:
The value of \( Z \) is \( \boxed{3} \).