The charge/size ratio of a cation determines its polarizing power. Which one of the following sequences represents the increasing order of the polarizing power of the cationic species, `K^(+), Ca^(2+), Mg^(2+), Ba^(2+)`?

Options

A. `Ca^(2+) lt, Ca^(2+) lt Be^(2+) lt K^(+)`
B. `Mg^(2+) lt, Be^(2+) lt K^(+) lt Ca^(2+)`
C. `Be^(2+) lt, K^(+) lt Ca^(+) lt Mg^(2+)`
D. `K^(+) lt, Ca^(2+) lt Mg^(2+) lt Be^(2+)`

Explanation

To determine the increasing order of the polarizing power of the cationic species \( K^+, Ca^{2+}, Mg^{2+}, Ba^{2+} \), we need to consider the charge-to-size ratio of each cation. The polarizing power of a cation is influenced by its charge and size, specifically, it is directly proportional to the charge density, which can be defined as:

\[ \text{Charge Density} = \frac{\text{Charge}}{\text{Size}} \]

### Step 1: Identify the charges and sizes of the cations

- **Potassium ion (\( K^+ \))**: Charge = +1, relatively large size.
- **Calcium ion (\( Ca^{2+} \))**: Charge = +2, smaller than \( K^+ \).
- **Magnesium ion (\( Mg^{2+} \))**: Charge = +2, smaller than \( Ca^{2+} \).
- **Barium ion (\( Ba^{2+} \))**: Charge = +2, larger than both \( Ca^{2+} \) and \( Mg^{2+} \).

### Step 2: Compare the polarizing power based on charge density

1. **Potassium (\( K^+ \))**:
- Charge density is low due to its +1 charge and larger size.

2. **Barium (\( Ba^{2+} \))**:
- Charge density is lower than \( Ca^{2+} \) and \( Mg^{2+} \) because although it has a +2 charge, it is larger in size.

3. **Calcium (\( Ca^{2+} \))**:
- Charge density is higher than \( Ba^{2+} \) because it has a +2 charge and a smaller size compared to \( Ba^{2+} \).

4. **Magnesium (\( Mg^{2+} \))**:
- Charge density is the highest due to its +2 charge and the smallest size among the listed cations.

### Step 3: Establish the increasing order of polarizing power

Based on the analysis, we can conclude:
- \( K^+ \) has the least polarizing power.
- \( Ba^{2+} \) has a higher polarizing power than \( K^+ \) but lower than \( Ca^{2+} \) and \( Mg^{2+} \).
- \( Ca^{2+} \) has a higher polarizing power than \( Ba^{2+} \).
- \( Mg^{2+} \) has the highest polarizing power.

Thus, the increasing order of polarizing power is:

\[ K^+ < Ba^{2+} < Ca^{2+} < Mg^{2+} \]

### Final Answer:
The correct sequence representing the increasing order of the polarizing power of the cationic species is:

\[ K^+ < Ba^{2+} < Ca^{2+} < Mg^{2+} \]