Question

The radius of hydrogen atom is \(0.53 \AA\). The radius of \(\mathrm{Li}^{2+}\) is of: (a) \(1.27 \AA\) (b) \(0.17 \AA\) (c) \(0.57 \AA\) (d) \(0.99 \AA\)

Short answer

The radius of \\(\mathrm{Li}^{2+}\) is \\(0.17 \\, ext{Å}\).

Step-by-step solution

Understand the Question

We need to determine the radius of the lithium ion \(\mathrm{Li}^{2+}\) given the radius of a hydrogen atom is \(0.53 \ \text{Å}\). The options include \(1.27\ Å\), \(0.17\ Å\), \(0.57\ Å\), \(0.99\, \ ext{Å}\).

Identify the Formula

The radius of a hydrogen-like ion can be calculated using \(r_n = n^2 \frac{a_0}{Z}\), where \(a_0 = 0.53 \, \ ext{Å}\) is the Bohr radius, \(n\) is the principal quantum number, and \(Z\) is the atomic number.

Apply to \\( \mathrm{Li}^{2+}\\) Ion

For \(\mathrm{Li}^{2+}\), \(Z = 3\) and \(n=1\), so the equation becomes \(r_1 = 1^2 \frac{0.53}{3}\).

Calculate the Radius

Plug the values into the formula: \(r_1 = \frac{0.53}{3}\) which gives \(r_1 \approx 0.177\ Å\).

Match the Values to Choices

Compare the calculated radius \(0.177 \, \ ext{Å}\) to the given choices. The closest value is \(0.17 \, \ ext{Å}\).

Key concepts

Hydrogen-Like Ions

Hydrogen-like ions are atoms or ions that have only one electron orbiting their nucleus, similar to a hydrogen atom. This means they have lost all other electrons and now behave like hydrogen in terms of electronic structure. These ions are typically formed by atoms with more than one electron initially, like lithium. When a lithium atom loses two of its three electrons, it becomes \(\text{Li}^{2+}\), a hydrogen-like ion. Such ions are useful in understanding atomic behavior because their simplicity allows for straightforward application of quantum mechanical models, such as the Bohr model. This makes calculations, like determining the radius of an ion, manageably simple and precise. Their study helps enhance our understanding of atomic structure at a fundamental level.

Bohr Model

The Bohr model is an early atomic model proposed by Niels Bohr in 1913 to describe the structure of atoms. It specifically applies well to hydrogen-like ions as they have a simple one-electron system. The model posits that electrons orbit the nucleus in defined circular paths or shells, with each shell corresponding to a specific energy level, characterized by the principal quantum number, \(n\). For hydrogen-like ions, the radius of an electron's orbit can be calculated using:

• \(r_n = n^2 \frac{a_0}{Z}\)

where \(a_0\) is the Bohr radius (approximately \(0.53 \, \text{Å}\)), \(n\) is the principal quantum number of the orbit, and \(Z\) is the atomic number. While the Bohr model successfully explains some atomic spectral lines, especially for hydrogen-like ions, it does have limitations. Nevertheless, it remains an excellent tool for introducing atomic theory concepts.

Ionic Radii

Ionic radii refer to the effective distance from the nucleus of an ion to its outermost electron. When an atom becomes an ion, its electron cloud changes due to electron gain or loss, affecting its size. For hydrogen-like ions, such as \(\text{Li}^{2+}\), the ionic radius can be calculated precisely using formulas derived from the Bohr model. This is made possible by the fact that these ions only have one electron, simplifying the calculations. Understanding ionic radii is important in predicting the behavior of ions in different chemical contexts and their role in forming compounds. These radii affect properties like ionization energy, electronegativity, and bond length, making the study of ionic radii crucial in both chemistry and materials science.