Question

The ionization of the hydrogen atom can be calculated from Bohr's equation for the electron energy. $$ E=-(N R h c)\left(Z^{2} / n^{2}\right) $$ where \(N R h c=1312 \mathrm{kJ} / \mathrm{mol}\) and \(Z\) is the atomic number. Let us use this approach to calculate a possible ionization energy for helium. First, assume the electrons of the He experience the full \(2+\) nuclear charge. This gives us the upper limit for the ionization energy. Next, assume one electron of He completely screens the nuclear charge from the other electrons, so \(Z=1 .\) This gives us a lower limit to the ionization energy. Compare these calculated values for the upper and lower limits to the experimental value of \(2372.3 \mathrm{kJ} / \mathrm{mol} .\) What does this tell us about the ability of one electron to screen the nuclear charge?

Short answer

The ability of one electron to screen the nuclear charge is partial, not complete.

Step-by-step solution

Identify Known Values

We know the constant \(NRhc = 1312 \ \mathrm{kJ/mol}\) and the experimental ionization energy of helium is \(2372.3 \ \mathrm{kJ/mol}\). He is the second element in the periodic table, so \(Z = 2\).

Calculate Upper Limit of Ionization Energy

For the upper limit, assume each electron experiences the full nuclear charge, \(Z = 2\), and \(n = 1\) (since we are considering the ionization of the innermost electron in helium). Substitute these values into the Bohr's equation: \[E = -(NRhc)\left(\frac{Z^2}{n^2}\right) = -(1312)\left(\frac{2^2}{1^2}\right) = -5248 \ \mathrm{kJ/mol}\] The energy required to remove the electron (ionization energy) is the positive value of this energy: \(5248 \ \mathrm{kJ/mol}\).

Calculate Lower Limit of Ionization Energy

For the lower limit, assume one electron completely screens the nuclear charge of the other, so effectively \(Z = 1\) (one electron experiencing the nuclear charge), and \(n = 1\). Substitute values into Bohr's equation: \[E = -(NRhc)\left(\frac{Z^2}{n^2}\right) = -(1312)\left(\frac{1^2}{1^2}\right) = -1312 \ \mathrm{kJ/mol}\] The energy required to remove the electron (ionization energy) is the positive value of this energy: \(1312 \ \mathrm{kJ/mol}\).

Analyze and Compare with Experimental Value

The calculated upper limit of the ionization energy is \(5248 \ \mathrm{kJ/mol}\) and the lower limit is \(1312 \ \mathrm{kJ/mol}\). The experimental value is \(2372.3 \ \mathrm{kJ/mol}\), which lies between the calculated limits, suggesting that while one electron does screen the nuclear charge to some extent, it does not do so completely, allowing for the experimental value to fall between these calculated extremes.

Key concepts

Ionization Energy

Ionization energy refers to the amount of energy required to remove an electron from an atom or ion. This energy quantifies the tendency of an atom to resist t the loss of its outermost electron, showcasing its stability. Understanding ionization energy is crucial because:

• It determines chemical reactivity. Atoms with high ionization energy, like noble gases, tend to be less reactive.
• It helps in predicting the bond formation between atoms. Lower ionization energy usually indicates a higher likelihood of forming positive ions (cations).

In the context of this problem, ionization energy for hydrogen and helium is vital. For helium, calculating theoretical upper and lower bounds of ionization strength provides insight into electron interactions with the nucleus. By analyzing these values alongside experimental data, one can understand the screening effect's role in modifying real-world energy requirements. The step-by-step solution uses Bohr's formula to calculate these theoretical values, which we compare to the experimental data.

Hydrogen Atom

The hydrogen atom is the simplest and smallest atom, consisting of only one proton and one electron. This simplicity makes it a fundamental model in quantum mechanics, explored extensively by Bohr's model. Bohr's model describes:

• Discrete energy levels or orbits where the electron can exist without radiating energy.
• Movement of electrons between orbits, correlating with absorption or emission of specific quantized energy.
• Application in calculating energy states via Bohr's equation, as seen in this exercise.

With the hydrogen atom as a base model, Bohr extended his theories to multi-electron atoms, hinting at complexities like electron shielding and sub-level formations. Understanding the hydrogen atom aids in grasping more intricate atoms, like helium, where additional interactions, such as the screening effect, come into play.

Atomic Number

The atomic number, denoted as \(Z\), represents the number of protons present in the nucleus of an atom. This is a defining characteristic of elements, determining the element's identity in the periodic table. Here are key points about atomic number:

• It defines the electric charge of the nucleus, hence influencing the pull on electrons.
• It's crucial for calculating ionization energy through Bohr's equation, as seen where \(Z\) influences the energy value for ionization.
• Understanding \(Z\) enables insights into how atoms of different elements behave chemically and physically.

For helium, with an atomic number of 2, the "nuclear charge" plays a pivotal role in ionization energy calculations, determining how tightly electrons are held. This was illustrated in our upper-limit calculations, assuming each electron feels the full nuclear charge.

Screening Effect

The screening effect, also known as the shielding effect, refers to the reduction in effective nuclear charge on the electron cloud due to inner-shell electrons. This concept is crucial in multi-electron atoms and significantly affects chemical behavior and ionization energies. Core ideas behind the screening effect include:

• Inner electrons acting as a shield between the nucleus and outer electrons, reducing the net positive charge felt by outer electrons.
• Influencing ionization energies by effectively lowering the attractive force on valence electrons, making them easier to remove.

In the problem, the screening effect is analyzed by assuming that one electron in helium might completely shield the nuclear charge from the other, reducing its effective \(Z\) value from 2 to 1. Calculating the lower limit of ionization energy under this assumption helps illustrate partial shielding influence. The resulting experimental value, which lies between calculated limits, underscores the incomplete screening. This makes understanding the screening effect essential for accurately predicting electronic structure and reactivity in heavier atoms.