(a) Using the Bohr model, calculate the speed of the electron in a hydrogen atom in the n=1,2, a

(a) Using the Bohr model, calculate the speed of the...

Key Concepts

Bohr Model

The Bohr model is a seminal atomic model that explains the structure of the hydrogen atom by assuming that electrons move in circular orbits around the nucleus with quantized angular momentum. It provides a framework for understanding why electrons occupy only certain discrete energy levels and explains phenomena such as the discrete spectral lines observed in hydrogen's emission spectrum.

Angular Momentum Quantization

Angular momentum quantization is a key principle of the Bohr model stating that the electron’s angular momentum in its orbit must be an integer multiple of a fundamental constant. This restriction leads directly to the derivation of specific radii, energies, and speeds for the allowed electron orbits, forming the basis for calculations of various atomic properties.

Orbital Speed Calculation

In the Bohr model, the orbital speed of the electron is determined by balancing the electrostatic force acting between the nucleus and the electron with the required centripetal force for circular motion. The resulting formula shows that the speed is inversely proportional to the principal quantum number, so electrons in lower energy levels move faster than those in higher levels.

Orbital Period Calculation

The orbital period of an electron in the Bohr model is calculated by dividing the circumference of its circular orbit by its orbital speed. Since both the orbit’s size and the speed depend on the principal quantum number, the period increases for electrons in orbits with larger quantum numbers, reflecting their slower speed and larger path.

Excited State Lifetime and Transition Dynamics

The concept of excited state lifetime in atomic physics refers to the average time an electron remains in an excited energy level before transitioning to a lower level through photon emission. In the context of the Bohr model, by comparing the lifetime of the excited state with the orbital period, one can estimate how many complete orbits the electron makes before the transition occurs, linking the classical orbital picture with quantum decay processes.

Transcript

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

00:08 So the first expression that we have to know is for the velocity of the electron in a given quantum orbit.

00:17 So for a given a quantum level, the velocity of the electron in the board model is 1 over epsilon 0, on which epsilon 0 is the electric constant for vacuum times e squared on which e is the electron charge divided by 2 times the quantum number times planks constant okay so this is one of the expressions that we have to know and the other one is regarding the borr's radius so what is the radius of an electron of the electron's orbit if the electron is in a given quantum number? so this is equal to a 0, which is a constant times n squared.

01:13 So this is all that we have to know.

01:16 Okay? now, let's start solving the exercise.

01:20 So question a asks us to calculate the velocity of the electron for the levels n equal to 1.

01:29 1, 2, and 3.

01:33 So this is pretty straightforward.

01:35 We just apply this equation.

01:39 We have that v1 is going to be 1 over epsilon 0 times electron 2 squared, divided by 2 times h, blanks constant.

01:56 So i'm going to write here in green the values for epsilon 0.

02:02 E and h that i'm going to use.

02:05 So for epsilon zero, the value that we are going to substitute to calculate v is 8 .85 times 10 to the minus 12, kulum squared per jule times meter.

02:26 For the electron charge, i'm using minus 1 .602 times 10 to the minus 19 kilon.

02:39 And finally, for planx constant, i'm using 6 .626 times 10 to the minus 34 jules times second.

02:55 Okay? so if we substitute all these constant values inside the expression for b1, we're going to find that the velocity for the first level is going to be 2 .18 times 10 to the 6 meters per second.

03:22 Now we do the same thing for v2 and v3.

03:26 So v2 1 over epsilon 0 e squared divided by 2 times 2 times 2 times h.

03:41 This is going to be equal to 1 .09 times 10 to the 6 meters per second.

03:55 And v3, 1 over epsilon 0, e squared 2 times 3 times h.

04:07 It's going to be equal to 0 .73 times 10 to the 6 meters per second.

04:17 Okay? now, in question b, the exercise wants us to find what is going to be the orbital period on which of these energy levels.

04:35 So we want to find t of n for n equal to 1, 2 and 3.

04:43 Okay, so if we recall that in boris model, we are assuming that the electron is moving in a circular orbit, we can use the relation, that the velocity is related to the angular frequency of rotation and the radius, so that for vn, you're going to have the vn is.

05:16 So recalling that this is going to be 2 pi times f, and that f is 1 over t, which is a period, we have 2 pi over t of n times the radius of the electrons orbit...

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