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Protons can be accelerated to speeds near that of light in particle accelerators. Estimate the wavelength (in nm) of such a proton moving at $2.90 \times 10^{8} \mathrm{m} / \mathrm{s}$ (Mass of a proton $\left.=1.673 \times 10^{-27} \mathrm{kg} .\right)$

The correct answer is $1.36 \times 10^{-6} \mathrm{nm}$ Sorry for that typo, I cannot correct the image for some reason...

Chemistry 101

Chapter 7

Quantum Theory and the Electronic Structure of Atoms

Electronic Structure

Carleton College

Drexel University

University of Toronto

Lectures

04:49

In chemistry and physics, electronic structure is the way the electrons of an atom are arranged in relationship to the nucleus. It is determined by the subshells the electrons are bound to, which are in turn determined by the principal quantum number ("n") and azimuthal quantum number ("l"). The electrons within an atom are attracted to the protons in the nucleus of that atom. The number of electrons bound to the nucleus is equal to the number of protons in the nucleus, which is called the atomic number ("Z"). The electrons are attracted to the nucleus by this mutual attraction and are bound to the nucleus. The electrons within an atom are attracted to each other and this attraction determines the electron configuration. The electron configuration is described by the term symbol, which is the letter used to identify each subshell.

16:45

In physics, the wave–particle duality is the concept that every object or process, no matter how large or how small, behaves as both a wave and a particle. The wave–particle duality is one of the central concepts in quantum mechanics.

03:59

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01:36

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Okay. This question is asking us to estimate the wavelength of a particle moving very fast. So using a altered version of Deborah Clues equation, we consider it if length is equal to plank's constant divided by square root two times the mass of the particle times the kinetic energy of the particle. Okay, we can find the kinetic energy which is equal to 1/2 massive the particle kinds of velocity square. We know that the velocity is in the mass so we can find the kinetic energy. Just 1/2 1.6, seven three, 73 times 10 to the negative. 27th kilograms did multiplied by speed, which is 2.9 times 10 to the eighth meters per second. That is going to be squared. Just that So kinetic energy comes out to be once we plug it into the calculator. Thank. And she is 7.3 times 10 to the negative 11th jewels. And since we got jewels as our variable, we know that that's at least we know we got we got the units, correct? Because Jules is, ah, units of energy. And let me go ahead and cancel out Ah, when we go and do this kilogram times meters per second funds meter squared over. Second squared is also known as Jules. So when we go ahead and plug the kinetic energy mass and plank's constant into the equation, we get that place. Constant is 6.63 times 10 to the negative 34th Jules per second square root of to times the mass of the particle, which is 1.673 times 10 to the negative. 27th kilograms times the kinetic energy. This is still under the square, which is seven 0.3 times 10 to the negative 11th jewels. So when we plug all of that into the calculator, we get that the wavelength for this particle, which traveling almost a fast is the speed of light, is going to be 1.36 times 10 to the negative 15th meters. Now this question asked for the wavelength in nanometers. So we'll me so that whether we convert that we know that for every meter there are 10 to the Ninth Nana Meters so we can go ahead and say z meters s a. For every one year there's 10 to the ninth nanometers. So when we multiply that together, we get it to be 1.36 times 10 to the negative six nanometers. And that is an insanely small wavelength. His nana meters is already a really small wavelength. And this is times 10 to the negative sixth nanometers. So that has a Thanh of energy that has a really high frequency. So this is the answer for the wavelength of a particle travelling at almost the speed of light with a very, very, very small mass. Thank you very much.

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