Question

Plot a graph of the square roots of the ionization energies versus the nuclear charge for the two series \(\mathrm{Li}, \mathrm{Be}^{+}, \mathrm{B}^{2+}, \mathrm{C}^{3+},\) and \(\mathrm{Na}, \mathrm{Mg}^{+}, \mathrm{Al}^{2+}, \mathrm{Si}^{3+} .\) Explain the observed relationship with the aid of Bohr's expression for the binding energy of an electron in a one electron atom.

Short answer

First identify the ionization energies for each ion in both series and calculate their square roots. Identify the nuclear charge for each ion. Using this information, plot square root of ionization energy against nuclear charge for both series. The trend in this plot can be explained by Bohr's expression for the binding energy in a hydrogen-like atom.

Step-by-step solution

Identify Ionization Energies

The ionization energies for ions in the series \(\mathrm{Li}, \mathrm{Be}^{+}, \mathrm{B}^{2+}, \mathrm{C}^{3+}\) and \(\mathrm{Na}, \mathrm{Mg}^{+}, \mathrm{Al}^{2+}, \mathrm{Si}^{3+}\) are identified. These would need to be researched or provided by a periodic table or a reliable database.

Calculate Square Roots of Ionization Energies

Calculate the square root of the ionization energy for each ion in the series. Remember that the unit of ionization energy is typically electronvolts (\(eV\)) or joules (\(J\)). Make sure to keep the units consistent!

Identify Nuclear Charges

Identify the nuclear charge for each ion in the series. The nuclear charge corresponds to the atomic number, which is the number of protons in the nucleus of an atom. Recall that for a net positive ion, the nuclear charge is higher than the number of electrons. For instance, the \(Be^{+}\) ion has a nuclear charge of 4.

Plot the Graph

With the above-collected information, plot the graph with the square root of the ionization energy on the y-axis and the nuclear charge on the x-axis. Two separate series are to be plotted: one for ions \(\mathrm{Li}, \mathrm{Be}^{+}, \mathrm{B}^{2+}, \mathrm{C}^{3+}\), and one for \(\mathrm{Na}, \mathrm{Mg}^{+}, \mathrm{Al}^{2+}, \mathrm{Si}^{3+}\).

Analyze and Explain the Graph Using Bohr’s Theory

Analyze the obtained plots and use Bohr's expression for the binding energy of an electron to explain the observed relationship. According to Bohr’s model, the binding energy of an electron in a one-electron atom is given by \(E = -\frac{13.6eV}{n^2} \times Z^2\), where \(E\) is the energy, \(n\) is the quantum number (principle shell), and \(Z\) is the atomic (nuclear) number. This equation suggests that the energy required to remove an electron (ionization energy) from an atom is directly proportional to the square of the atomic number, which can explain the trend observed in the graph.

Key concepts

Ionization Energy

Ionization energy is a measure of the amount of energy needed to remove an electron from an atom or ion. It tells us how strongly an electron is held within an atom. In the series from Li to Si, we see each element or ion in various charge states, such as Be, Be⁺, and so on. Each increment in charge generally increases the ionization energy because the electron removal becomes more challenging. Sanatom's hold on its valence electrons is stronger, requiring more energy to detach them.

The energy is typically measured in electronvolts (eV) or joules (J), and these values can be found in a periodic table or reliable chemical database. Once ionization energies for each ion are collected, their square roots are taken for plotting.

• This plot shows that ionization energy generally increases as nuclear charge increases.
• The relationship is visible in a graph where higher nuclear charges result in higher ionization energies.

Understanding ionization energy helps us appreciate how atoms interact with each other and react in chemical processes.

Nuclear Charge

Nuclear charge is essentially the total charge of all the protons in an atom's nucleus. It's represented by the atomic number (Z), which is the count of protons. The nuclear charge determines the overall positive charge of the nucleus.

As you move from Li to C³⁺ and Na to Si³⁺, the nuclear charge increases. Each step adds protons in the nucleus, making the positive charge stronger, thereby attracting the electron cloud more firmly.

• For example, Be has a nuclear charge of 4, while Be⁺ retains this charge, even if it has one fewer electron.
• Higher nuclear charge correlates with greater energy required to remove an electron, tying back to ionization energy.

Nuclear charge plays a pivotal role in determining the strength of interaction between the nucleus and electrons. It's increased significance is visible in the relation: the higher the nuclear charge, the tighter electrons are bound to the nucleus, which elevates the ionization energy.

Binding Energy

Binding energy in Bohr's Model is the energy required to remove an electron from an atom. Bohr's theory simplifies the atom into a central nucleus and electron(s) orbiting around it. In this model, Bohr proposed that the binding energy of an electron in a one-electron atom aligns with the formula:\[ E = -\frac{13.6 \text{ eV}}{n^2} \times Z^2 \]where \( E \) is the energy, \( n \) is the quantum number, and \( Z \) is the atomic number.

This equation indicates that binding energy increases with the square of the nuclear charge. That means, as \( Z \) rises, the result is a greater energy need to remove an electron.

• The negative sign signifies that the energy is bound-to-nucleus; thus, it must be positive to detach.
• Bohr’s equation shows why larger nuclear charges (like C³⁺ over Li) demand higher ionization energies.

Bohr's model gives a fundamental understanding of atomic structure and interactions, especially for hydrogen-like systems. His binding energy formula clarifies the direct, quadratic connection between nuclear charge and ionization energy. This relationship underpins the better electron binding for higher charged nuclei in atoms or ions.