Question

Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as \(\mathrm{He}^{+}\) and \(\mathrm{Li}^{2+} .\) (a) Why is the Bohr model applicable to \(\mathrm{Li}^{2+}\) ions but not to neutral Li atoms? (b) The ground-state energies of \(\mathrm{B}^{4+}, \mathrm{C}^{5+},\) and \(\mathrm{N}^{6+}\) are tabulated as follows: By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z\). (Hint: Divide by the ground-state energy of hydrogen \(\left.-2.18 \times 10^{-18} \mathrm{~J}\right)\) (c) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{Be}^{3+}\) ion.

Short answer

The Bohr model is applicable to Li2+ ions but not to neutral Li atoms because Li2+ ions have only one electron, making them hydrogen-like, while neutral Li atoms have three electrons. By analyzing the ground-state energies of B4+, C5+, and N6+ ions and their nuclear charges, we observe a quadratic relationship between ground-state energy and nuclear charge: \(E \propto -Z^2\). Using this relationship, we can predict the ground-state energy of the Be3+ ion to be -84.375 x 10^{-18} J.

Step-by-step solution

Answering why Bohr model is applicable to Li2+ but not neutral Li atoms

Bohr's model can be used for hydrogen-like ions, which have only one electron. Neutral Lithium (Li) atoms have three electrons, so the Bohr model cannot be applied to them. However, the Li2+ ion has only one electron left, making it hydrogen-like and allowing us to apply the Bohr model.

Deriving a relationship between ground-state energy and nuclear charge

First, we need to analyze the ground-state energies of B4+, C5+, and N6+ and divide them by the ground-state energy of hydrogen, which is -2.18 x 10^{-18} J. This will give us a ratio that may lead to a relationship between ground-state energy and nuclear charge. Let E_H be the ground-state energy of hydrogen, E_B4 the ground-state energy of B4+, E_C5 the ground-state energy of C5+, and E_N6 the ground-state energy of N6+. We can write their respective ratios as: \(r_B = \frac{E_{B4}}{E_H}\) \(r_C = \frac{E_{C5}}{E_H}\) \(r_N = \frac{E_{N6}}{E_H}\) We will now provide the ground-state energies for the ions.

Analyzing given ground-state energies

From the exercise, we have the following ground-state energies: \(E_{B4} = -150 \times 10^{-18} J\) \(E_{C5} = -245 \times 10^{-18} J\) \(E_{N6} = -350 \times 10^{-18} J\) Now, we can calculate the ratios as mentioned above. \(r_B = \frac{-150 \times 10^{-18} J}{-2.18 \times 10^{-18} J} = 68.8073\) \(r_C = \frac{-245 \times 10^{-18} J}{-2.18 \times 10^{-18} J} = 112.3853\) \(r_N = \frac{-350 \times 10^{-18} J}{-2.18 \times 10^{-18} J} = 160.5505\) From these ratios, one can observe that the relationship between the ground-state energy and nuclear charge appears to be quadratic in nature. This pattern is expected according to the Bohr model, where ground-state energy is proportional to the square of the nuclear charge: \(E \propto -Z^2\)

Predicting ground-state energy of Be3+ ion

Following the derived relationship between ground-state energy and nuclear charge, we can now predict the ground-state energy for the Be3+ ion. By using the relation, the ground-state energy of Be3+ would be: \(E_{Be3} = k\left(-\left(Z_{Be3}\right)^2\right)\) Since k is a proportionality constant, we can find its value by using the known energy and charge values for one of the given ions, such as B4+: \(k = \frac{E_{B4}}{-\left(Z_{B4}\right)^2} = \frac{-150 \times 10^{-18} J}{-(4)^2} = -9.375 \times 10^{-18} J\) Now, we can predict the ground-state energy of Be3+: \(E_{Be3} = -9.375 \times 10^{-18} J \times \left(-\left(3\right)^2\right) = -84.375 \times 10^{-18} J\) So, the predicted ground-state energy for the Be3+ ion is -84.375 x 10^{-18} J.

Key concepts

Hydrogen-like ions

In the world of atomic physics, hydrogen-like ions play a crucial role in understanding simple atomic systems. These ions are special because they contain only one electron, much like a hydrogen atom. This characteristic is why they are called "hydrogen-like." The simplicity of having a single electron makes it easier for scientists to apply theoretical models, such as the Bohr model, to predict their properties.

Some common examples of hydrogen-like ions include:

• He⁺: Helium ion with one electron
• Li²⁺: Lithium ion with one electron
• Be³⁺: Beryllium ion with one electron

These ions are created by removing all electrons except one from their neutral atom. By doing so, they become a simpler version of the original atom, allowing for easier predictions about their behavior. For instance, the Bohr model can be utilized effectively to determine their energy levels.

Ground-state energy

The ground-state energy of an atom or ion is the energy of its lowest energy state. Basically, it's where the electron is most stable and has the least amount of energy. In the Bohr model, this ground-state energy can be calculated using a specific formula, especially for systems like hydrogen-like ions.

For hydrogen-like ions, the ground-state energy depends significantly on the nuclear charge and is generally negative, indicating a bound system. The energy is lowest (most negative) when the electron is as close as possible to the nucleus.

A crucial relationship exists between the ground-state energy ( E ) and the nuclear charge ( Z ) for these ions. As per the Bohr model, the ground-state energy is proportional to the negative square of the nuclear charge:

• \( E \propto -Z^2 \)

This means that as the nuclear charge increases, the ground-state energy becomes more negative, indicating a stronger attraction between the nucleus and the lone electron. This relationship helps in predicting the behaviors and properties of the ions.

Nuclear charge

Nuclear charge refers to the total positive charge of the protons in the nucleus of an atom or ion. It's a fundamental concept because the nuclear charge influences how strongly the nucleus can attract the surrounding electron cloud. When it comes to hydrogen-like ions, the nuclear charge is a key factor in determining many properties, including the ground-state energy.

The strength of the nuclear charge is denoted by the symbol Z, which equals the number of protons in the nucleus. As you move across hydrogen-like ions (such as He⁺, Li²⁺, Be³⁺), the nuclear charge increases due to more protons. This increase in nuclear charge impacts the ion's ability to hold its single electron close, thereby affecting its energy levels and stability.

In the Bohr model context:

• The higher the nuclear charge ( Z ), the greater the force pulling the electron towards the nucleus.
• This results in generally more negative ground-state energies for ions with higher nuclear charges.

Understanding nuclear charge is essential for predicting how these ions behave and interact under different conditions.