Problem 142 - A Complete Resource Book in Chemistry for JEE Main

Question

The ground state electronic wave function of hydrogen atom is proportional to \(\mathrm{e}^{-\operatorname{rin}_{0}}\) where \(\mathrm{r}\) is the distance from the nucleus and \(\mathrm{a}_{0}\) is a constant. Calculate the relative probability of finding the electron inside a region of volume \(1.0 \mathrm{pm}^{3}\), located at the nucleus to a distance of \(\mathrm{a}_{0}\) from the nucleus. Report your answer in single digit ignoring any fraction involved. (Given: \(\ln 0.14=-2\) )

Step-by-Step Solution

Answer

The relative probability is approximately 7.

Step 1: Understand the Problem

We are asked to calculate the relative probability of finding an electron in two specific regions: within the nucleus and at a distance of \( a_0 \) away from the nucleus. The wave function is given as \( \Psi(r) \propto e^{-r/a_0} \), hence the probability density is \( |\Psi(r)|^2 \propto e^{-2r/a_0} \).

Step 2: Determine Probability Densities

We need to calculate the probability density at two points; at the nucleus (\( r = 0 \)) and at a distance \( r = a_0 \). For \( r = 0 \), \( |\Psi(0)|^2 \propto e^{-2 \times 0/a_0} = 1 \). For \( r = a_0 \), \( |\Psi(a_0)|^2 \propto e^{-2a_0/a_0} = e^{-2} \).

Step 3: Calculate Relative Probability

The relative probability between these two points is given by the probability densities ratio. Thus, it is \( \frac{|\Psi(0)|^2}{|\Psi(a_0)|^2} = \frac{1}{e^{-2}} = e^2 \). Since \( e^2 = (e^{-2})^{-1} \) and given \( \ln 0.14 = -2 \), it follows that \( e^2 = \frac{1}{0.14} \).

Step 4: Evaluate and Simplify

Approximately, \( \frac{1}{0.14} \approx 7.14 \), and we are to ignore the fraction and round to a single digit. Therefore, the relative probability is approximately \( 7 \).

Key Concepts

Probability Density

In quantum mechanics, the probability density is a crucial concept used to describe where an electron, or any particle, is likely to be found. For the hydrogen atom, this concept is particularly relevant when considering its simplest state, known as the ground state.

The probability density is related to the wave function of the particle, commonly denoted as \( \Psi(r) \). The probability density, \( |\Psi(r)|^2 \), tells us the likelihood of finding an electron at a particular location \( r \). This is not about finding the electron in one precise spot but rather expresses how densely the probability is distributed across different locations.

• This means where \( |\Psi(r)|^2 \) is large, the electron is likely to be found more often.
• Where \( |\Psi(r)|^2 \) is small, the chances of finding the electron are low.

In the ground state of hydrogen, the mathematical expression for the wave function is particularly simple, allowing us to calculate how the probability density decreases with distance from the nucleus; the farther we go, the less probable it is to find the electron.
Understanding this helps in exams and exercises as you can predict how the electron behaves without directly observing it.

Wave Function

The wave function \( \Psi(r) \) is a fundamental concept in quantum mechanics that describes the quantum state of a particle, such as an electron around a hydrogen atom. This is a complex-valued function depending on the position in space, and it incorporates all the information possible about a system.

In essence, the wave function does not directly tell us the probability of finding an electron at a point — rather, it must be squared to give the probability density. For hydrogen in its ground state, this wave function has an exponential form \( \Psi(r) \propto e^{-r/a_0} \).

• When \( r = 0 \), or at the nucleus, \( \Psi(r) \) reaches its maximum value.
• As \( r \) increases, \( \Psi(r) \) decreases exponentially.

The shape and form of this wave function provide insights into how quantum particles differ from classical particles. Instead of moving in a defined path, their behavior is cloud-like, spread across a region, making the understanding of wave functions key to grasping quantum behavior.

Ground State

The ground state of an atom is the lowest energy state that its electron can occupy. In the case of a hydrogen atom, this is the state where the electron is closest to the nucleus.

The ground state is particularly stable and serves as a reference point, allowing scientists to describe and predict the behavior of electrons under different circumstances.

• In hydrogen, the ground state corresponds to the principal quantum number \( n = 1 \).
• All other states (excited states) have higher energies and are less stable.

For a hydrogen atom in the ground state, the wave function is the simplest, having an exponential pattern. The shape of this wave function leads to specific probability distributions that scientists can utilize to predict where the electron is most likely to be found.
This understanding is vital for students studying chemistry and physics as it forms the basis for interpreting many chemical reactions and quantum phenomena.