An atom moving at its root mean square velocity at 100 . ^∘ C has a wavelength of 2.31 ×10^-11

An atom moving at its root mean square velocity at 100 . ^∘...

Key Concepts

De Broglie Wavelength

The de Broglie wavelength is the characteristic wavelength associated with a particle due to its wave-like nature. It is determined by the relationship ? = h/p, where ? is the wavelength, h is Planck's constant, and p is the momentum of the particle. This concept underpins the quantum mechanical idea that particles such as atoms can exhibit both particle and wave properties.

Root Mean Square Velocity

The root mean square (rms) velocity is a statistical measure of the speeds of particles in a gas. It is derived from kinetic molecular theory and is calculated using the temperature and molar mass of the gas. The rms speed provides an estimate of the average kinetic energy and momentum of the particles, which is essential in understanding molecular motion and related phenomena.

Kinetic Molecular Theory

Kinetic molecular theory explains the macroscopic properties of gases by considering their molecular composition and motion. It establishes that gas particles are in constant, random motion and that their collisions are elastic. This theory provides the basis for deriving relationships such as the dependence of rms velocity on temperature and molar mass, linking thermodynamics with microscopic behavior.

Atomic Mass Determination

Atomic mass determination in this context involves using experimental data, like the de Broglie wavelength coupled with rms velocity, to back-calculate the mass of an individual atom. This approach combines quantum mechanical and kinetic theory concepts to identify unknown elements by correlating measurable physical properties to the intrinsic mass of the atom.

Transcript

00:01 The root means speed, or rms, is the speed of a molecule or particle at any given temperature.

00:08 And it can be found by relating the kinetic energy to the temperature, using the following equation.

00:16 1 .5 mv squared equals 3 halves k times t.

00:28 The first part is the kinetic energy, and the v is what we know, the root means, speed.

00:45 Mass is the mass of the particle.

00:51 K is a constant called boltzmann's constant, and it's equal to 1 .38 times 10 to the minus 23rd joules for kelvin.

01:07 And finally, t is our temperature, and the temperature must be in kelvin.

01:14 So in this case, we know that it's 100 degrees celsius or 273 plus 100 or 373.

01:28 So with the first equation, we're trying to identify what the particle is.

01:33 The second thing that information that we know is the wavelength.

01:38 And since the wavelength isn't in this relationship, we look at a second relationship called debroglié's wavelength, which tells us that you can relate the wavelength to the mass of any particle by dividing a constant, planx constant, times mass, times velocity.

01:55 So the velocity is the same as up here.

01:59 Lambda is the wavelength, which is given to us as 2 .3 .1 times 10 to the minus 11 meters.

02:19 H is planck's constant, or 6 .63 times 10 to the minus 34, joules times seconds.

02:32 Mass is the same thing and velocity is the same thing as above.

02:37 So we have to use these two equations to find something that will help us to identify what the substance is.

02:43 And most likely the variable that will help us do that is the mass.

02:48 So we're going to look at how we can use these two equations to solve for mass.

02:57 The first step is to take our de brogley equation and see if we can find a way to replace this v with the v from our other equation.

03:25 So algebraically, i'm going to solve this equation for v and substitute it into this equation.

03:32 So i divide both sides by a half, m, or v squared equals 3 kt over m.

03:49 And then solving for v, v is just the square root of 3 kt, over m...

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