An electron is in a bound state of a hydrogen atom. The...

An electron is in a bound state of a hydrogen atom. The energy state of the atom is labeled with principal quantum number $n .$ In the Bohr model description of this bound state, the electron has linear momentum $p=6.65 \times 10^{-25} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s} .$ In the Bohr model description. what are (a) the kinetic energy of the electron, (b) the angular momentum of the electron, and (c) the quantum number $n ?$

Key Concepts

Bohr Model

The Bohr model is a quantum mechanical model of the hydrogen atom that proposes electrons move in fixed, circular orbits around the nucleus. Each orbit corresponds to a specific energy level, and transitions between these levels involve the absorption or emission of quantized amounts of energy. This model introduces the concept of quantized orbits to explain the discrete spectral lines observed in hydrogen’s emission spectrum.

Angular Momentum Quantization

In the Bohr model, the angular momentum of an electron is quantized, meaning it can only take discrete values. This is expressed as L = n?, where L is the angular momentum, n is the principal quantum number, and ? is the reduced Planck constant. This quantization condition is key to determining the allowed electron orbits and is fundamental in connecting classical circular motion with quantum behavior.

Kinetic Energy Calculation

The kinetic energy of an electron, even in a quantum system like the hydrogen atom, can be calculated using the classical formula K = p²/(2m), where p represents the electron's linear momentum and m its mass. This relation provides a bridge between the measurable momentum of an electron and its energy, enabling the calculation of kinetic energy from the momentum information provided by the model.

Principal Quantum Number

The principal quantum number, denoted as n, is a key parameter in the Bohr model that defines the energy level of an electron in an atom. It not only determines the electron's energy but also its orbital radius and angular momentum. Increasing values of n correspond to higher energy levels and larger orbital radii, making this quantum number fundamental in describing the structure of atomic systems.

Transcript

00:01 To serve this exercise, we have to recall just a few expressions from the bor model of the hydrogen atom.

00:07 So, the expression from a board model that we have to remember is this expression for the orbital angular momentum.

00:18 And also, it will be useful for this exercise to remember the definition of kinetic energy as the linear momentum square divided by 2m.

00:30 Okay? now, in this problem, suppose that we have an electron, and this electron is bounded, is in the hydrogen atom, in a given quantum number that we don't know, but if we use bars model, we can measure the electron's linear momentum to be equal to 6 .65 times 10, to the minus 25 kilograms meter per second.

01:10 Now, knowing this, the exercise wants us to find what is going to be the electrons kinetic energy for the borrars model.

01:19 So this is the linear momentum for borr's model.

01:29 And knowing the definition that i gave you here, we can just substitute k, we can just substitute p and use m as the mass of the electron and then find what is the kinetic energy.

01:44 So, k is p squared, so 6 .65 times 10 to the minus 25 kilograms meter per second squared, divided by 2 times the electron's mass.

02:07 So the value that i'm going to use for the electrons mass is 9 .109 times 10 to the minus 31 kilograms.

02:20 Okay? so when we do this calculation, we're going to find that the kinetic energy is 2 .42 times 10 to the minus 19.

02:37 Now questions b and c wants us to find the angular, the orbital angular momentum of the electron, and its quantum number, okay, given the information that we found in question, that the exercise gave us and what we found on question a.

03:05 So i'm going to start by trying to find what is because i'm going to use this expression for the angular momentum.

03:13 And to use the expression, we have to know what is the value of n.

03:21 So to find n, i'm going to use that for bars model, the velocity of the electron for a given quantum number is one over the electric constant in vacuum times the electrons charge squared divided by two pi times blanks constant.

03:48 And v is going to be pretty straightforward for us to find, given that we know the kinetic energy already of the electron.

03:58 So, even better, we can use the definition for linear momentum.

04:13 So from the definition of linear momentum, we have the following...

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