Arrange the following ions in order of their decreasing ionic radii.
`Li^(o+), K^(o+), Mg^(2+), Al^(3+)`

Explanation

To arrange the ions \( \text{Li}^+, \text{K}^+, \text{Mg}^{2+}, \text{Al}^{3+} \) in order of their decreasing ionic radii, we can follow these steps:

### Step 1: Understand the Concept of Ionic Radii
Ionic radii refer to the size of an ion, which is determined by the distance between the nucleus and the outermost electron shell. The ionic radius can be influenced by the charge of the ion and the number of electrons.

### Step 2: Calculate the Z/E Ratio for Each Ion
The Z/E ratio is the ratio of the number of protons (Z) to the number of electrons (E) in the ion. A higher Z/E ratio indicates a smaller ionic radius, while a lower Z/E ratio indicates a larger ionic radius.

1. **For \( \text{Li}^+ \)**:
- Z (number of protons) = 3
- E (number of electrons) = 2 (since it has a +1 charge)
- \( \frac{Z}{E} = \frac{3}{2} = 1.5 \)

2. **For \( \text{K}^+ \)**:
- Z = 19
- E = 18 (since it has a +1 charge)
- \( \frac{Z}{E} = \frac{19}{18} \approx 1.055 \)

3. **For \( \text{Mg}^{2+} \)**:
- Z = 12
- E = 10 (since it has a +2 charge)
- \( \frac{Z}{E} = \frac{12}{10} = 1.2 \)

4. **For \( \text{Al}^{3+} \)**:
- Z = 13
- E = 10 (since it has a +3 charge)
- \( \frac{Z}{E} = \frac{13}{10} = 1.3 \)

### Step 3: Compare the Z/E Ratios
Now we can compare the Z/E ratios we calculated:
- \( \text{Li}^+: 1.5 \)
- \( \text{K}^+: 1.055 \)
- \( \text{Mg}^{2+}: 1.2 \)
- \( \text{Al}^{3+}: 1.3 \)

### Step 4: Determine the Order of Ionic Radii
Since ionic radius is inversely proportional to the Z/E ratio, we can arrange the ions in order of decreasing ionic radii based on their Z/E ratios:
- The ion with the lowest Z/E ratio has the largest ionic radius.
- Therefore, the order from largest to smallest ionic radius is:
- \( \text{K}^+ \) (largest)
- \( \text{Mg}^{2+} \)
- \( \text{Al}^{3+} \)
- \( \text{Li}^+ \) (smallest)

### Final Answer
The order of the ions in decreasing ionic radii is:
\[ \text{K}^+ > \text{Mg}^{2+} > \text{Al}^{3+} > \text{Li}^+ \]

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