Question

(a) Calculate the energies of an electron in the hydrogen atom for \(n=1\) and for \(n=\infty .\) How much energy does it require to move the electron out of the atom completely (from \(n=1\) to \(n=\infty),\) according to Bohr? Put your answer in \(\mathrm{kJ} / \mathrm{mol}\). (b) The energy for the process \(\mathrm{H}+\) energy \(\rightarrow \mathrm{H}^{+}+\mathrm{e}^{-}\) is called the ionization energy of hydrogen. The experimentally determined value for the ionization energy of hydrogen is \(1310 \mathrm{~kJ} / \mathrm{mol}\). How does this compare to your calculation?

Short answer

The energy levels of an electron in the hydrogen atom for $n=1$ and $n=\infty$ are -13.6 eV and 0 respectively. The energy required to move the electron out of the atom completely (from $n=1$ to $n=\infty$) is 13.6 eV, which is equivalent to 1310 kJ/mol. This calculated ionization energy of hydrogen matches the given experimental value of 1310 kJ/mol.

Step-by-step solution

Understanding Bohr's formula for energy levels

According to Bohr's model, the energy of an electron in the hydrogen atom at a given energy level (n) can be calculated using the formula: \[E_n=-\frac{2\pi^2e^4m_eZ^2}{h^2n^2}\] Where \(E_n\) is the energy of the electron at level n, e is the elementary charge, \(m_e\) is the mass of electron, Z is the atomic number (1 for hydrogen), h is the Planck's constant, and n is the energy level.

Calculate the energy for n=1

We will plug in the known values (elementary charge, Planck's constant, electron mass, and atomic number) into the formula and calculate the energy for n=1: \[E_1=-\frac{2\pi^2e^4m_e(1)^2}{h^2(1)^2}\] \[E_1=-13.6 \text{ eV}\]

Calculate the energy for n=∞

Now we calculate the energy for n=∞: \[E_{\infty}=-\frac{2\pi^2e^4m_e(1)^2}{h^2(\infty)^2}\] As n approaches infinity, the energy level will be zero: \[E_{\infty}=0\]

Calculate the energy difference between n=1 and n=∞

The energy required to ionize the electron completely (from n=1 to n=∞) is the difference between the energies for n=1 and n=∞: \[\Delta E=E_{\infty} - E_1\] \[\Delta E=0 - (-13.6 \text{ eV})\] \[\Delta E=13.6 \text{ eV}\]

Convert energy to kJ/mol

To convert the energy difference to kJ/mol, first, multiply by the electron charge to obtain joules: \[\Delta E=13.6 \text{ eV} * 1.6 \times 10^{-19} \text{ J/eV} = 2.176 \times 10^{-18} \text{ J}\] Next, multiply by Avogadro's number to convert from energy per atom to energy per mol: \[\Delta E=2.176 \times 10^{-18} \text{ J/atom} * 6.022 \times 10^{23} \text{ atoms/mol}=1310 \text{ kJ/mol}\]

Compare calculated value with experimental value

According to our calculation, the ionization energy of hydrogen is 1310 kJ/mol, which is the same as the experimental value (1310 kJ/mol) given in the exercise. Therefore, our calculated value of the ionization energy of hydrogen is consistent with the experimental data.

Key concepts

Hydrogen Atom

The hydrogen atom is the simplest atom, consisting of just one electron orbiting a single proton in the nucleus. In Bohr's model, this system can be visualized like a mini solar system with the electron in a circular orbit around the nucleus. This model helps us understand how electrons interact with their nucleus by using specific orbits or energy levels. Bohr's model was one of the first models to incorporate quantum theory, indicating that an electron could only occupy certain allowed orbits. These orbits correspond to specific energy levels characterized by the quantum number \(n\). For the hydrogen atom, when \(n = 1\), the atom is at its lowest energy state, known as the ground state.

• The energy levels are quantized, meaning they can take only certain distinct values.
• The electron transitions between these levels by absorbing or emitting energy.

This model is not entirely accurate for complex atoms, but it is a crucial stepping stone in understanding atomic structure and electron behavior.

Ionization Energy

Ionization energy is the amount of energy required to remove an electron from an atom in its gaseous state. For hydrogen, this means moving the electron from the ground state to a point where it is no longer bound to the nucleus (\(n = \infty\)). In Bohr's model, ionization energy directly relates to the energy difference between the lowest energy level (where the electron is initially) and the point at which the electron is free:\[\Delta E = E_{\infty} - E_1\]The experimentally determined ionization energy for hydrogen is \(1310 \text{ kJ/mol}\). This exact value is also derived theoretically using Bohr's formula, showing the power of Bohr's model in predicting atomic behaviors.

• Ionization energy offers insight into an element's reactivity; elements with lower ionization energy are more reactive.
• This property also affects an element’s placement and behavior in the periodic table.

Energy Levels

In Bohr's model of the hydrogen atom, each orbit or energy level is associated with a principal quantum number \(n\). The energy associated with each level is given by Bohr's formula:\[E_n = -\frac{2\pi^2e^4m_eZ^2}{h^2n^2}\]Here, the energy becomes less negative as \(n\) increases, indicating that the electron is less tightly bound to the nucleus at higher energy levels.

• When \(n = 1\), the energy is \(-13.6 \text{ eV}\), which is the most negative and hence, the electron is most strongly bound.
• For \(n = \infty\), the energy is \(0\), meaning the electron has been removed from the atom.

In the context of electron transitions:- An electron moving to a higher energy level absorbs energy equal to the difference in energy between the initial and final states.- Conversely, when it falls to a lower energy level, it emits energy.These shifts explain why atoms absorb and emit light at specific wavelengths, a principle underpinning the spectra observed in spectroscopy.