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 determine 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 can be calculated using the formula:
\[ Z_{eff} = Z - S \]
where:
- \( Z \) is the atomic number (total number of protons),
- \( S \) is the shielding constant (the extent to which inner electrons shield the outer electrons from the nucleus).

### Step 2: Identify the Number of Valence Electrons
The problem states that the element has three valence electrons. This means that the outermost shell of the element contains three electrons.

### Step 3: Determine Possible Electron Configurations
We need to find a suitable electron configuration that accommodates three valence electrons. The possible configurations based on the principal quantum number (n) are:
- For \( n = 1 \): The maximum number of electrons is 2 (1s²).
- For \( n = 2 \): The configuration can be 1s² 2s² 2p¹, which totals 5 electrons.

### Step 4: Calculate the Atomic Number
From the configuration 1s² 2s² 2p¹, we can see that the atomic number \( Z \) is 5 (2 from 1s and 3 from 2s and 2p).

### Step 5: Calculate the Shielding Constant (S)
For the configuration 1s² 2s² 2p¹:
- The 1s electrons shield the 2s and 2p electrons. The shielding constant can be estimated as follows:
- Electrons in the 1s orbital contribute to the shielding of the 2s and 2p electrons.
- The shielding constant \( S \) can be calculated as:
\[ S = 2 \times 0.35 + 2 \times 0.85 \]
- Here, 0.35 is the shielding contribution for each electron in the same shell, and 0.85 is for the electrons in the inner shell.

### Step 6: Substitute Values into the Z_eff Formula
Now substituting the values into the Z_eff formula:
\[ Z_{eff} = Z - S \]
\[ Z_{eff} = 5 - (2 \times 0.35 + 2 \times 0.85) \]
\[ Z_{eff} = 5 - (0.7 + 1.7) \]
\[ Z_{eff} = 5 - 2.4 \]
\[ Z_{eff} = 2.6 \]

### Conclusion
The effective nuclear charge calculated matches the given Z_eff of 2.60, confirming that the atomic number of the element is indeed 5.

Thus, the minimum atomic number of the element is **5**.

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