Question

If the velocity of an electron in Bohr's first orbit is \(2.19 \times 10^{6} \mathrm{~m} \mathrm{~s}^{-1}\), what will be the de Broglie wavelength associated with it? (a) \(2.19 \times 10^{-6} \mathrm{~m}\) (b) \(4.38 \times 10^{-6} \mathrm{~m}\) (c) \(3.32 \times 10^{-10} \mathrm{~m}\) (d) \(3.32 \times 10^{10} \mathrm{~m}\)

Short answer

The de Broglie wavelength associated with the electron is approximately \(3.32 \times 10^{-10} \mathrm{~m}\).

Step-by-step solution

Understand de Broglie Wavelength Formula

The de Broglie wavelength \(\lambda\) of a particle is given by the formula \(\lambda = \frac{h}{mv}\), where \(h\) is the Planck constant \(6.626 \times 10^{-34} \mathrm{Js}\), \(m\) is the mass of the particle, and \(v\) is its velocity.

Identify Known Values

Identify the known values for the problem. The velocity of the electron is given as \(v = 2.19 \times 10^{6} \mathrm{~m/s}\). The mass of an electron \(m\) is approximately \(9.11 \times 10^{-31} \mathrm{kg}\).

Calculate the de Broglie Wavelength

Use the de Broglie formula to calculate the wavelength: \[\lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34} \mathrm{Js}}{(9.11 \times 10^{-31} \mathrm{kg})(2.19 \times 10^{6} \mathrm{m/s})}\].

Compute the Wavelength

Carry out the computation to find the wavelength: \[\lambda = \frac{6.626 \times 10^{-34}}{(9.11 \times 10^{-31})(2.19 \times 10^{6})} = \frac{6.626}{(9.11)(2.19)} \times 10^{-34+31-6}\] \[\lambda = \frac{6.626}{19.9489} \times 10^{-9} \approx 3.32 \times 10^{-10} \mathrm{m}\].

Key concepts

Bohr's Atomic Model

The Bohr's atomic model was introduced by Niels Bohr in 1913 and revolutionized our understanding of the atom. This model describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus — much like planets orbiting the Sun. Bohr's model was revolutionary because it incorporated quantum theory to explain that electrons exist in specific energy levels or shells.

• Each orbit corresponds to a certain energy level.
• Electrons can jump from one energy level to another, but they can't exist between levels.
• The energy difference between levels determines the radiation absorbed or emitted by the atom.

For example, the first orbit, closest to the nucleus, is the lowest energy state for an electron. In Bohr's model, the velocity of an electron and its orbit radius are quantized, meaning that they have fixed, discrete values. Knowing the velocity of an electron within a Bohr orbit, like in our exercise, helps to determine other properties such as the de Broglie wavelength.

Electron Velocity

The velocity of an electron within an atom is a crucial factor in determining many of its physical properties, including its de Broglie wavelength. In the context of Bohr's atomic model, the velocity of an electron in one of the quantized orbits can be calculated using the formulas derived from Bohr's postulates:

• For an electron in the first Bohr orbit, the velocity is determined by the electrostatic forces between the negatively charged electron and the positively charged nucleus.
• The exact velocity can be influenced by the number of protons in the nucleus and the level (orbit) the electron is on.

In the provided exercise, the velocity of an electron is given, allowing us to move forward with the calculation of the de Broglie wavelength. This velocity is a function of the energy state of the electron and helps us predict the likelihood of transitions between energy levels, a foundational concept in quantum mechanics.

Planck Constant

The Planck constant (\(h\)) is a fundamental quantity in quantum mechanics which plays a pivotal role in the relationship between energy and frequency, as well as in the determination of the de Broglie wavelength of particles. It has a value of approximately \(6.626 \times 10^{-34} \mathrm{Js}\) and can be thought of as a quantum of action in the universe.

• The Planck constant relates the energy of a photon to its frequency: \(E = hf\), where \(E\) is energy, \(f\) is frequency, and \(h\) is the Planck constant.
• It also is an integral part of the de Broglie hypothesis, which posits that every particle with momentum has an associated wavelength, given by \(\lambda = \frac{h}{mv}\), where \(\lambda\) is wavelength, \(m\) is mass, and \(v\) is velocity.

In our exercise example, the Planck constant is a key part of the equation enabling us to calculate the de Broglie wavelength for the electron as it moves in the first orbit of Bohr's atomic model.