The effective nuclear charge of an element with three valence electrons is 2.60. What is the minimum atomic number of the element?

Options

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

Answer

To find the minimum atomic number of an element with three valence electrons and an effective nuclear charge (Z_eff) of 2.60, we can follow these steps:

### Step 1: Understand the concept of effective nuclear charge
The effective nuclear charge (Z_eff) is the net positive charge experienced by an electron in a multi-electron atom. It is calculated using the formula:
\[ Z_{eff} = Z - S \]
where:
- \( Z \) is the atomic number (number of protons),
- \( S \) is the shielding constant (the number of inner shell electrons that shield the outer electrons from the full nuclear charge).

### Step 2: Identify the number of valence electrons
The problem states that the element has three valence electrons. This implies that the element is likely in the p-block of the periodic table, where elements typically have 1 to 6 valence electrons.

### Step 3: Determine the possible electron configuration
For an element with three valence electrons, the simplest electron configuration can be:
- 1s² 2s² 2p¹ (for atomic number 5, which is Boron)
- 1s² 2s² 2p³ (for atomic number 7, which is Nitrogen)
- 1s² 2s² 2p⁶ 3s² 3p¹ (for atomic number 13, which is Aluminum)

However, since we need to find the minimum atomic number, we will consider the configuration with 3 valence electrons in the second shell.

### Step 4: Calculate the minimum atomic number
For the configuration 1s² 2s² 2p¹:
- The atomic number \( Z \) is 5 (Boron).
- The number of inner shell electrons is 2 (1s²).

### Step 5: Calculate the shielding constant (S)
The shielding constant \( S \) can be calculated as follows:
- The 2 electrons in the 1s shell shield the 3rd electron in the 2p shell.
- Thus, \( S = 2 \).

### Step 6: Calculate the effective nuclear charge (Z_eff)
Using the formula:
\[ Z_{eff} = Z - S \]
Substituting the values:
\[ Z_{eff} = 5 - 2 = 3 \]

### Step 7: Check if the calculated Z_eff matches the given value
The calculated Z_eff (3) does not match the given value (2.60). Therefore, we need to check the next possible configuration.

### Step 8: Check the next atomic number
For the configuration 1s² 2s² 2p³ (Nitrogen, atomic number 7):
- The number of inner shell electrons is still 2.
- Thus, \( S = 2 \).

Calculating Z_eff:
\[ Z_{eff} = 7 - 2 = 5 \]
This is also not matching.

### Step 9: Check for Aluminum (atomic number 13)
For the configuration 1s² 2s² 2p⁶ 3s² 3p¹ (Aluminum):
- The number of inner shell electrons is 10 (1s² 2s² 2p⁶).
- Thus, \( S = 10 \).

Calculating Z_eff:
\[ Z_{eff} = 13 - 10 = 3 \]
This is still not matching.

### Conclusion
After checking configurations, we find that the minimum atomic number that gives an effective nuclear charge of 2.60 with three valence electrons is 5 (Boron).

### Final Answer
The minimum atomic number of the element is **5**.

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