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Chapter 4: Problem 175

State the Heisenberg uncertainty principle and what it implies about the structure of an atom.

Short Answer

Expert verified
The Heisenberg uncertainty principle states that it's impossible to accurately know both the position and momentum of a particle simultaneously, with the relation \(\Delta x \cdot \Delta p_x \geq \frac{\hbar}{2}\). In the context of an atom, this means that we cannot determine the exact position of electrons while knowing their exact momentum. Instead, electrons move in probability distributions called atomic orbitals, which represent regions of high probability for finding electrons around the nucleus. These orbitals have various shapes, such as s-orbitals (spherical) and p-orbitals (dumbbell-shaped), and can accommodate a maximum of two electrons each.

Step by step solution

01

Introduction to the Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle, proposed by Werner Heisenberg in 1927, is a fundamental principle in quantum mechanics. It states that it's impossible to accurately know both the position and momentum of a particle (like an electron) simultaneously. Mathematically, the principle can be written as: \[\Delta x \cdot \Delta p_x \geq \frac{\hbar}{2}\] where \(\Delta x\) is the uncertainty in position, \(\Delta p_x\) is the uncertainty in momentum, and \(\hbar = \frac{h}{2\pi}\), where \(h\) is the Planck's constant.
02

Understanding the Implications of the Principle

The uncertainty principle implies that the more accurately we know the position of a particle like an electron, the less accurately we can know its momentum, and vice versa. In the context of an atom, this means that we cannot determine the exact position of electrons at any given time while knowing their exact momentum. Therefore, we can only describe the probability distribution of finding an electron in a particular region around the nucleus.
03

Applying the Principle to the Structure of an Atom

As a consequence of the Heisenberg uncertainty principle, we need to think of electrons in atoms as moving in "clouds" or probability distributions rather than in fixed circular orbits like planets around the Sun. This leads to the concept of atomic orbitals – regions around the nucleus where the probability of finding an electron is high. Orbitals are represented by wave functions, which can be used to calculate the probability of finding an electron at a particular location.
04

Visualizing Atomic Orbitals

Atomic orbitals come in various shapes and sizes, including s-orbitals, p-orbitals, d-orbitals, and f-orbitals. These shapes represent the probability density of finding an electron within that region. S-orbitals are spherical, while p-orbitals are dumbbell-shaped, and d- and f-orbitals have more complex shapes. Each orbital can accommodate a maximum of two electrons.
05

Conclusion

In conclusion, the Heisenberg uncertainty principle fundamentally limits our ability to simultaneously measure the position and momentum of particles, such as electrons in an atom. This principle leads to the concept of atomic orbitals, which are regions of high probability for finding electrons around the nucleus of an atom. These orbitals help describe the arrangement of electrons in various atoms and help us understand the basis of the periodic table and chemical bonding.

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Most popular questions from this chapter

A student has written what he thinks are some ground-state electron configurations. Which ones have something wrong with them? What is wrong? (a) \(1 s^{2} 2 p^{6} 3 s^{1}\) (b) \(1 s^{2} 2 s^{6} 3 s^{2} 3 p^{6} 4 s^{2} 4 p^{6}\) (c) \(2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{7}\) (d) \(1 s^{2} 2 s^{2} 2 p^{7} 3 s^{3} 3 p^{6}\) (e) \(1 s^{2} 2 s^{2} 2 p^{5} 3 s^{1}\) (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{1} 4 d^{4}\) [This is the transition metal niobium, which violates the four-block electron-assignment rules, so be careful.]

Of the following transitions in the Bohr hydrogen atom, which transition results in the emission of photons with the shortest wavelength? (a) \(n=1 \rightarrow n=6\) (b) \(n=4 \rightarrow n=1\) (c) \(n=9 \rightarrow n=5\) (d) \(n=3 \rightarrow n=6\) (e) \(n=5 \rightarrow n=9\)

Arrange the following atoms in order of increasing atomic size: \(\mathrm{Ne}, \mathrm{As}, \mathrm{Se}, \mathrm{K}, \mathrm{S}, \mathrm{Cl}\)

What is wrong with Bohr's planetary model of atomic electrons according to modern quantum mechanical theory? (Hint: Use Heisenberg's uncertainty principle in your answer.)

Which is the correct ground-state electron configuration for antimony, Sb? (a) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{10} 5 s^{2} 5 d^{3}\) (b) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{10} 5 s^{2} 5 p^{3}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{2} 4 d^{10} 5 p^{4}\) (d) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{2} 5 p^{3}\) (e) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{2} 4 f^{3}\)
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