A single electron system has ionization energy `20902.2 kJ//mole`. Find the number of protons in the nucleus of the system.

Answer

To find the number of protons in the nucleus of a single electron system with an ionization energy of 20902.2 kJ/mole, we can follow these steps:

### Step 1: Understand the relationship between ionization energy and atomic number
The ionization energy (IE) of a hydrogen-like atom can be expressed using the formula:
\[ IE = \frac{2.18 \times 10^{-18} Z^2}{n^2} \]
where \( Z \) is the atomic number (number of protons) and \( n \) is the principal quantum number (which is 1 for a single electron system).

### Step 2: Convert ionization energy from kJ/mole to Joules
Given the ionization energy is 20902.2 kJ/mole, we need to convert this to Joules. Since 1 kJ = 1000 J, we have:
\[ IE = 20902.2 \, \text{kJ/mole} \times 1000 \, \text{J/kJ} = 20902200 \, \text{J/mole} \]

### Step 3: Use Avogadro's number
Since the ionization energy provided is per mole, we need to divide by Avogadro's number (\( N_A = 6.022 \times 10^{23} \, \text{mol}^{-1} \)) to find the energy per atom:
\[ IE_{\text{per atom}} = \frac{20902200 \, \text{J}}{6.022 \times 10^{23}} \approx 3.47 \times 10^{-19} \, \text{J} \]

### Step 4: Substitute values into the ionization energy formula
Now we can substitute \( IE_{\text{per atom}} \) into the ionization energy formula:
\[ 3.47 \times 10^{-19} = \frac{2.18 \times 10^{-18} Z^2}{1^2} \]

### Step 5: Solve for \( Z^2 \)
Rearranging the equation gives:
\[ Z^2 = \frac{3.47 \times 10^{-19}}{2.18 \times 10^{-18}} \approx 0.159 \]

### Step 6: Calculate \( Z \)
Taking the square root of both sides:
\[ Z \approx \sqrt{0.159} \approx 0.398 \]

### Step 7: Round to the nearest whole number
Since \( Z \) must be a whole number (as it represents the number of protons), we round \( Z \) to the nearest whole number, which is approximately 4.

### Conclusion
Thus, the number of protons in the nucleus of the system is:
\[ \text{Number of protons} = Z = 4 \]

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