Question

The radius of which of the following orbits is same as that of the first Bohr's orbit of hydrogen atom? (a) \(\mathrm{He}^{+}(\mathrm{n}=2)\) (b) \(\mathrm{Li}^{2+}(\mathrm{n}=2)\) (c) \(\mathrm{Li}^{2+}(\mathrm{n}=3)\) (d) \(\mathrm{Be}^{3+}(\mathrm{n}=2)\)

Short answer

The radius of the orbit for \( \mathrm{Be}^{3+} \) (\( n=2 \)) matches the first Bohr's orbit of hydrogen.

Step-by-step solution

Understanding Bohr's Orbit

The radius of the nth Bohr orbit for a hydrogen-like atom is given by the formula: \[ r_n = a_0 \frac{n^2}{Z} \]where \( a_0 \) is the Bohr radius (approximately 0.529 Å), \( n \) is the orbit number, and \( Z \) is the atomic number of the element. For hydrogen, \( Z = 1 \), and the first Bohr orbit \( n = 1 \) has a radius \( r = a_0 \). We will find a radius equal to this for one of the given ions.

Calculating Radius for He⁺ (n=2)

For \( \mathrm{He}^{+} \) (Z=2) and \( n=2 \):\[ r_n = a_0 \frac{n^2}{Z} = a_0 \frac{2^2}{2} = 2a_0 \]The radius is 2\( a_0 \), which is not equal to \( a_0 \).

Calculating Radius for Li²⁺ (n=2)

For \( \mathrm{Li}^{2+} \) (Z=3) and \( n=2 \):\[ r_n = a_0 \frac{n^2}{Z} = a_0 \frac{2^2}{3} = \frac{4}{3}a_0 \]The radius is \( \frac{4}{3}a_0 \), which is not equal to \( a_0 \).

Calculating Radius for Li²⁺ (n=3)

For \( \mathrm{Li}^{2+} \) (Z=3) and \( n=3 \):\[ r_n = a_0 \frac{n^2}{Z} = a_0 \frac{3^2}{3} = 3a_0 \]The radius is 3\( a_0 \), which is not equal to \( a_0 \).

Calculating Radius for Be³⁺ (n=2)

For \( \mathrm{Be}^{3+} \) (Z=4) and \( n=2 \):\[ r_n = a_0 \frac{n^2}{Z} = a_0 \frac{2^2}{4} = a_0 \]The radius is \( a_0 \), which is equal to that of the first Bohr orbit of hydrogen.

Key concepts

Hydrogen atom

The hydrogen atom is the simplest and most fundamental atom in chemistry and physics. It consists of only one proton in its nucleus and one electron orbiting around it. This simplicity makes it an ideal model for understanding atomic behavior. The hydrogen atom has played a crucial role in the development of quantum mechanics, particularly through Bohr's model, which explains electron orbits, energy levels, and the spectral lines of hydrogen.
In Bohr's model, any atom consists of electrons orbiting a central nucleus in fixed paths called orbits. This model helps to understand how electrons transition between energy levels and how these transitions result in the absorption or emission of light with specific wavelengths. The hydrogen atom is pivotal in this study as it depicts a clear example of atomic interactions governed by fundamental forces.

Bohr orbit radius

The Bohr orbit radius, commonly represented by \( a_0 \), is a fundamental concept introduced by Niels Bohr to describe the orbiting distance of an electron from the nucleus in an atom. For the hydrogen atom, the Bohr radius defines the size of its first electron shell. This radius is approximately 0.529 angstroms (Å).

• The Bohr radius is a constant that is crucial in atomic physics calculations.
• It serves as a reference for estimating the size of hydrogen-like atoms.
• In hydrogen's first orbit, \( n = 1 \), the electron travels in a path defined by the Bohr radius.

This foundational concept helps in understanding more complex atoms and how their electron shell structures might compare to that of hydrogen.

Atomic number

The atomic number, denoted by \( Z \), is one of the most important concepts in chemistry and atomic physics. It defines the number of protons in the nucleus of an atom and thereby determines the element's identity.

• The atomic number is unique for each element and is listed on the periodic table.
• For hydrogen, \( Z = 1 \), meaning it has only one proton.
• For helium, \( Z = 2 \), and so on, increasing incrementally in the periodic table.

The atomic number also informs us about the chemical properties of an element and its behavior in periodic trends. In Bohr's model, it plays a critical role in determining the electron orbit radii, as seen in the exercise, where the radius changes with different atomic numbers for hydrogen-like atoms.

Hydrogen-like atoms

Hydrogen-like atoms refer to ions or atoms that have a single electron orbiting around a nucleus. These atoms are analogous to hydrogen in structure and are critical to understanding atomic physics using simplified models.
Examples include:

• \( \text{He}^+ \) (a helium ion with one electron)
• \( \text{Li}^{2+} \) (a lithium ion missing two electrons, with only one remaining)
• \( \text{Be}^{3+} \) (a beryllium ion missing three electrons)

Bohr's model applies effectively to these single-electron systems, allowing for calculations of electron orbits and energy levels. Such hydrogen-like atoms follow a similar pattern of behavior, where changes in \( Z \) (the atomic number) influence the size and energy of the electron orbits, just like in hydrogen itself.