Problem 38 - A Complete Resource Book in Chemistry for JEE Main

Question

The most probable radius (in \(\mathrm{pm}\) ) for finding the electron in \(\mathrm{He}^{+}\)is (a) \(105.8\) (b) \(52.9\) (c) \(26.5\) (d) \(0.0\)

Step-by-Step Solution

Answer

The most probable radius for \( \mathrm{He}^{+} \) is \( 26.5 \, \mathrm{pm} \), which is option (c).

Step 1: Recall the formula for Bohr radius

The most probable radius for finding an electron in a hydrogen-like ion is given by the formula: \[ r_{n} = a_0 \frac{n^2}{Z} \] where \( r_{n} \) is the radius of orbit, \( a_0 \) is the Bohr radius in picometers (\(52.9 \, \mathrm{pm}\)), \( n \) is the principal quantum number, and \( Z \) is the atomic number of the nucleus.

Step 2: Determine the values for the variables

For \( \mathrm{He}^{+} \), \( Z = 2 \) because the atomic number of helium is 2. For the ground state, \( n = 1 \). Therefore, we will substitute \( Z = 2 \) and \( n = 1 \) into the formula.

Step 3: Substitute into the formula and calculate

Substitute the known values into the formula: \[ r_{1} = 52.9 \, \mathrm{pm} \times \frac{1^2}{2} \] This simplifies to \[ r_{1} = 52.9 \, \mathrm{pm} \times 0.5 = 26.45 \, \mathrm{pm} \].

Step 4: Round to the closest value from the options

The calculated most probable radius is \( 26.45 \, \mathrm{pm} \). Round this to one decimal place, which is \( 26.5 \, \mathrm{pm} \). This matches option (c).

Key Concepts

Hydrogen-Like Ion

A hydrogen-like ion is a term used to describe any ion with only one electron orbiting around the nucleus, similar to a hydrogen atom. Imagine helium losing one of its electrons, resulting in a

• positively charged helium ion,
• symbolized as \( \mathrm{He}^{+} \).

Hydrogen-like ions can include ions of other elements as long as they have a single electron. These ions have similar energy levels and spectral lines as hydrogen. When calculating properties such as radii or energy levels, the atomic number \( Z \) of the nucleus becomes quite significant. It represents the positive charge in the nucleus that the lone electron feels.
The more protons in the nucleus (higher \( Z \)), the stronger the attraction it has on the single electron, affecting the electron's orbit considerably. This is crucial in understanding how different elements behave in their hydrogen-like ion form.
Focusing on helium, we substitute \( Z = 2 \) into our formulas because helium has two protons in its nucleus.

Bohr Radius

The Bohr radius \( a_0 \) is a physical constant that holds great significance in atomic physics, especially in the Bohr model of the atom. It represents the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. For hydrogen, this distance is approximately

• 52.9 picometers (
• \( \mathrm{pm} \))

This forms the cornerstone for understanding how electron orbits work in the Bohr model for hydrogen-like ions.
When referring to hydrogen-like ions, we adapt the Bohr radius formula to account for differences in nuclear charge, using the formula \( r_{n} = a_0 \frac{n^2}{Z} \). The variable \( a_0 \) remains constant at 52.9 \( \mathrm{pm} \) regardless of the ion or atomic number. However, by altering \( n \) and \( Z \), we can determine how an electron's distance from the nucleus changes. This method provides a simple yet powerful way to estimate electron positions in atomic structures.Understanding the Bohr radius helps us grasp more complex concepts about atomic interactions and behaviors.

Principal Quantum Number

The principal quantum number \( n \) plays an essential role in determining an electron's position and energy level in an atom. It can take any positive integer value (1, 2, 3,...). In the earliest models of atoms, electrons were thought to orbit the nucleus in a manner similar to planets orbiting the sun. Here,

• \( n = 1 \) describes the ground state
• where \( n > 1 \) indicates excited states or higher energy levels.

The principal quantum number correlates with the energy and average distance of the electron from the nucleus. Higher numbers mean that electrons are further from the nucleus, inhabiting orbitals with greater potential energy.
In the context of helium \( \mathrm{He}^{+} \), we use \( n = 1 \) for the ground state. This means the electron is in the lowest orbital closest to the nucleus. By plugging \( n \) values into the Bohr model equations, we can predict and confirm the nature of electron orbits, giving us insight into the fundamental structure of atoms and ions.