Calculate the energy required for the process $He^+(g) \rightarrow He^{2+}(g) + e^-$. The ionization energy for the H atom in the ground state is $2.18 \times 10^{-18}\ J\ atom^{-1}$.

Calculate the energy required for the process He^+(g) arrow He²⁺(g) + e^-. The ionization energy for the H ato

Exercises · 2.34

He^+(g) \rightarrow He^{2+}(g) + e^-

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Concept understanding — Energy Level Quantization

Energy Level Quantization: From Intuition to Precision

Imagine you're climbing a smooth ramp. You can stop at any height — 1 metre, 1.5 metres, 2.1 metres — anywhere you like. That's how we intuitively think about energy in everyday life: continuous, like a slide.

Now imagine a staircase. You can stand on step 1, step 2, or step 3 — but you cannot stand halfway between step 2 and step 3. There's no such place. The steps are discrete, not continuous.

Energy level quantization is the idea that in the microscopic world of atoms and molecules, energy behaves like a staircase, not a ramp. Electrons in an atom cannot have just any energy — they can only occupy specific, allowed energy levels. Everything else is forbidden.

Why does this happen? The core intuition

In the classical world, an electron orbiting a nucleus would continuously radiate energy, spiral inward, and crash — atoms would be unstable. But atoms are stable. Nature solved this problem by imposing a rule: the electron's angular momentum (and therefore its energy) can only take certain discrete values.

Think of a guitar string. It can only vibrate at specific frequencies — its fundamental and harmonics. You can't pluck it to produce a frequency halfway between two harmonics. The string's vibration is quantized by its boundaries. Similarly, an electron bound to a nucleus is confined in space, and that confinement forces its energy to be quantized.

Note

Quantization is not a mysterious extra rule — it emerges naturally whenever a wave (like an electron's matter wave) is confined. Confinement creates standing waves, and standing waves only exist at specific frequencies.

The precise statement

For a bound system (like an electron in an atom), the total energy $E$ of the system can only take certain discrete values:

E = E_1, E_2, E_3, \dots

where each $E_n$ is a specific, fixed number. The integer $n$ (1, 2, 3, …) is called the principal quantum number. The lowest energy level ( $n=1$ ) is the ground state; higher levels ( $n>1$ ) are excited states.

For the hydrogen atom, the allowed energies are given by:

E_n = -\frac{13.6 \text{ eV}}{n^2}

So:

$n=1$ : $E_1 = -13.6$ eV (ground state)
$n=2$ : $E_2 = -3.4$ eV
$n=3$ : $E_3 = -1.51$ eV
and so on, approaching 0 eV as $n \to \infty$ (the ionization limit)

Important

The negative sign means the electron is bound to the nucleus. Zero energy corresponds to the electron being free (ionized). The more negative the energy, the more tightly bound the electron.

How do we know this is real?

The most direct evidence comes from atomic spectra. When an electron jumps from a higher energy level to a lower one, it emits a photon of light with energy exactly equal to the difference:

\Delta E = E_{\text{higher}} - E_{\text{lower}} = hf

where $h$ is Planck's constant and $f$ is the frequency of the emitted light. …

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The energy required to remove the single electron from $He^+$ is found by scaling the hydrogen ground-state ionization energy by $Z^2 = 4$ , giving $8.72 \times 10^{-18}\ \text{J atom}^{-1}$ .

The key insight here is that $He^+$ is a hydrogen-like ion — it has only one electron orbiting a nucleus with atomic number $Z = 2$ . The physics of a single electron bound to a nucleus is identical for any hydrogen-like species, differing only in the nuclear charge.

For the hydrogen atom ( $Z=1$ ), the ionization energy from the ground state ( $n=1$ ) is $2.18 \times 10^{-18}\ \text{J}$ . This is the energy needed to completely remove the electron from the proton's influence.

For any hydrogen-like ion, the energy of an electron in the $n$ th orbit is given by:

E_n = - \frac{Z^2}{n^2} \times (2.18 \times 10^{-18}\ \text{J})

The negative sign means the electron is bound. To ionize (remove the electron to $n = \infty$ , where $E_\infty = 0$ ), we need to supply energy equal to the magnitude of this binding energy.

Let's work through it step by step.

Identify the system. $He^+$ has one electron and a nucleus with $Z = 2$ . It is exactly like a hydrogen atom but with double the nuclear charge.
Recall the scaling law. In the Bohr model (and in full quantum mechanics), the ground-state energy of a hydrogen-like atom scales as $Z^2$ . For hydrogen ( $Z=1$ ), the ground-state energy is $-2.18 \times 10^{-18}\ \text{J}$ . For $He^+$ ( $Z=2$ ), the ground-state energy becomes:

E_1(He^+) = - (2)^2 \times (2.18 \times 10^{-18}) = -4 \times 2.18 \times 10^{-18}\ \text{J}

Calculate the binding energy.

E_1(He^+) = -8.72 \times 10^{-18}\ \text{J}

This is the energy of the electron when it is bound to the $He^{2+}$ nucleus in the $n=1$ orbit. …

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Q.Calculate the energy required for the process He+(g)→He2+(g)+e−He^+(g) \rightarrow He^{2+}(g) + e^-He+(g)→He2+(g)+e−. The ionization energy for the H atom in the ground state is 2.18×10−18 J atom−12.18 \times 10^{-18}\ J\ atom^{-1}2.18×10−18 J atom−1.

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Q.Calculate the energy required for the process $He^+(g) \rightarrow He^{2+}(g) + e^-$ . The ionization energy for the H atom in the ground state is $2.18 \times 10^{-18}\ J\ atom^{-1}$ .

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