WEBVTT chemistry/physical-chemistry/hovasapian
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Hello, welcome back to www.educator.com, and welcome back to Physical Chemistry.
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today, we are going to continue our discussion of the hydrogen atom.
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we are going to stop and just take a quick look at where we are, what we have done, just so we do not lose our way.
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We have done a lot of quantum mechanics in the last four lessons.
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I want to take this lesson that just to stop, take a look at what is that we have done because there is a lot of mathematics, a lot of symbolism, where we are, what we had at our disposal.
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just so we do not lose sight of the big picture.
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That is the biggest problem with quantum mechanics is there so much mathematics going on.
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There is so much symbolism going on that you tend to lose your way.
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Ultimately, we are concerned with the wave function.
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once we have that wave function, we do certain things to it.
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we operate on that wave function.
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as pretty much what quantum mechanics comes down to.
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you find the wave function and then you operate on that wave function, by means of a bunch of integrals.
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let us see where we are.
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Let me go ahead and work with blue here.
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we wanted to find this ψ of R θ φ, the wave function for the electron in the hydrogen atom.
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just going back over what we did in the last 4 lessons.
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we wrote the ψ function of R θ φ, the three variables we wrote it as a product function.
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as a product of a function just of R called the radial component and as a product of the R the radial component and S which is a function of the θ and φ is called the angular component.
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we also wrote and we broke this up into a function of 2 variables.
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we wrote S of θ φ as T of θ and F of φ.
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We just broke up into a bunch of functions.
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we found that ψ depends on the three quantum numbers.
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for example, we might have ψ of N LM, we express that as R and L as a function of R.
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I'm going to have to stop saying that, I hope by now that we realize that R is just a function of R, the radius and the S is a spherical harmonic is a function of the θ and the φ.
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R and L, and we have S LM, this is the symbolism for it.
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N ran from 1, 2, 3, and so on.
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L depends on N and it ran from 0, 1, 2, all the way to N -1.
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M depends on L, and it was 0, + or -1, + or – 2, all the way to + or – L.
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the expressions for the R and S were as follows.
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this is what we found.
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we found that R and L, I’m going to go ahead and put the R = -, we have got N – L -1! ÷ 2N × N + L!.
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this one is going to be cubed and this one is going to be raised to ½.
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here we have 2/ A₀ ⁺L + 3/2.
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It is going to be multiplied by R ⁺L.
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there is going to be an exponential term E ⁻R/ N A₀ and × this L + L 2L + 1.
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its argument is 2R/ N × A₀, where this L N + L, 2L + 1, 2R/ N A₀ are called the associated Laguerre polynomials.
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this what we be found when we solved one of those differential equations that we found.
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the partial differential equations that we found for the hydrogen atom.
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this was one expression.
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this was the expression for R.
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this is what we found right here.
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it looks complicated, it is not when you actually put in the different values of N, L, and M.
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it actually condenses and becomes reasonably straightforward to deal with.
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we will see that in just a minute.
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Do not let the general form intimidate you.
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it is just a bunch of numbers that you are putting in.
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that is the R part that we found.
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Let me go back to blue here.
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we found the S L sub M which is a function of θ and φ.
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we found that to be equal to 2L + 1 × L - the absolute value of M!/ 4 π × L + the absolute value of M! ^½.
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P sub L absolute value of M, its argument is cos θ and its exponential E ⁺IM φ, where this P sub L superscript absolute value of M cos θ, these are the associated Legendre functions or Legendre polynomials.
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let us go ahead and write out, stop, and just write out the wave function for a particular collection of N, L, and M.
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let us write ψ 2, 1, 1.
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in this case, N is the first quantum number.
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N is 2, L is 1, and M = 1, this particular wave function.
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let us see what we have got.
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It is going to get a little bit messy but again it is just a bunch of numbers that is in there.
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do not worry about it, we will condense it into something really nice.
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let us go ahead and do this in red.
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We are going to do R of 2, 1, R sub NL.
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R of 2, 1 which is a function of R is going to be - 2 -1 -1!/ 2 × 2 × 2 + 1 !³ ^½.
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we have got 2/ 2 A0¹ + 3/2 R¹ E ⁻R 2 α₀.
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I’m just putting in the N and the L, that is all I'm doing.
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this is going to be L of 2 + 1 and this is going to be 2 × 1 + 1.
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Write N + L, 2 + 1, 2L + 1, 2 × 1 + 1, this is going to be 2R/ N A₀/ N is 2 A₀.
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this part right here, when we look up the particular Laguerre polynomial for this one, it just ends up being -3!.
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Always look it up, it ends up being -3!.
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that means that we get R of 2, 1 is equal to -3!.
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That is a minus so it actually becomes positive.
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Then, we put some things together, we get 1/ 4 × 3!³, this is going to be the ½ power.
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I’m hoping that I’m not forgetting anything here.
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It is a little crazy, tedious, just keeping track of every little number that is really the biggest problem of quantum mechanics I find.
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3/ 2, let me make that a little clearer.
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this is 1/ A₀, this is going to be to the 5/2 RE ^- R/ 2 × A₀.
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there we go, we have the radial portion of the wave function for the 2, 1, 1 state.
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Let us go ahead and do the spherical portion.
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we have S1, 1 θ φ that is going to equal L is 1, M is 1, N is 2.
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we have 2 × 1 + 1/ 4 π × 1 -1 !.
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0! is 1, 1 + 1!, all of that raised to the ½ power × 1, 1, 1 of cos θ E ⁺I φ.
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in this particular case, P1, 1 when we look it is equal to 1 – X² ^½, this is the argument for the X so it becomes 1 - cos² θ¹/2.
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1 - cos² that, remember your Pythagorean identity from trigonometry, it is equal to sin².
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and √ sin² is equal to sin θ.
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P1, 1 is just equal to sin θ.
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Therefore, our S1, 1 of θ φ ends up equaling 3 × 1/ 4 π 2! ^½.
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there are different ways that you simplify these numbers.
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I’m just putting in so we actually see everything.
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sin of θ E ⁺I φ, there you go, that is your spherical harmonic.
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that is the angular component of the hydrogen atomic wave function.
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when we put them together, therefore, we have 4 ψ of 2, 1, 1 which is equal to R of 2, 1 × S1, 1.
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we end up with ψ2, 1, 1 is equal to 3! × 1/ 4 π × 3!³ ^½ × 1/ A₀⁵/2 × 3/ 4 π × 2!.
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This is not π, this is 4 × 3! because it was from the radial component.
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You see that is very easy to lose your way here.
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R of E - R/ 2 Α sub .
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Let me make the minus sign a little bit more clear.
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× sin θ × E ⁺I φ.
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that is a version of the ψ 2, 1, 1.
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just plugging in the numbers, plugging in the N, the L, and the M, to actually get the particular wave function that we are talking about.
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we know that we worked out the wave functions but we also know some things about the angular momentum.
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it is very important.
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Let me go back to blue here.
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we also know some things about the angular momentum of the electron, we know that the magnitude of the total angular momentum is equal to H ̅ × √ L × L + 1.
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you notice the magnitude of the angular momentum depends only on the quantum number L.
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the Z component of the angular momentum that is equal to M × H ̅.
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we call N, the principle quantum number.
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we call L, the angular momentum quantum number.
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we call M, the magnetic quantum number.
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The last bit of information that we have, I’m going to go ahead and write this over here.
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the particular energy which depends only on N, as long as it is not a magnetic field is going to be E²/ 8 × π × the permittivity constant × the Böhr radius × N².
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this is the particular energy associated with a given orbital and it depends only on N, provided the electron, the atom is not in a magnetic field.
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I should go ahead and that is not a problem.
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when the wave functions are listed in books, you are definitely going to find several lists of these hydrogen wave functions in your book in the chapter on the hydrogen atom.
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when the wave functions are listed in books, they are simplified to various degrees.
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what I mean by that is, they are simplified to various degrees.
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and what I mean is that different authors and they tend to have combined certain things into one value but say they take the R/ α₀ and they call it Z, or they call it B, or sigma, or something.
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they simplify it to various degrees, in order for it not to look so complicated.
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when you see them in your book, they might actually look different than what it is that I'm writing here.
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that is just simply because different authors write them in different ways to simplify them to various degrees.
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Do not worry about it, it is the same wave function.
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The wave functions that are listed in books, they are simplified to various degrees.
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do not be concerned that what I have written or will write differs from what you see in your book.
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they are the same wave function.
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They are just written differently with different parameters that combine different things.
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Let me give you an example of a couple of these representations that you might see.
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Let me do this in red.
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one representation, I will just write a couple of the wave functions.
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I will write too many of them.
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one such representation is as follows.
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you might see this in your book.
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you might see where N is equal to 1, where L is equal to 0, and M is equal to 0.
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you will see ψ 1, 0, 0, written as 1/ radical π × Z/ A₀³/2 E ⁻B.
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in this particular case, Z is the atomic number and the reason we put the atomic number here, I know we found them for the hydrogen atom but we actually assume that these orbitals are also valid for all the other atoms of the periodic table.
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you just plug in Z, Z is the atomic number, that is all it is.
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B is that factor so B is going to be equal to Z × R/ Α₀.
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Or the α₀ is the bohr radius and the numerical value Α₀ is 5.292 × 10⁻¹¹ N.
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of course, in the case of the hydrogen Z is just 1, it is the atomic number.
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For the wave function, for N is equal to 2, L is equal to 0, M is equal to 0, you will see this one, you might see this one in your book.
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Ψ 2, 0, 0, is equal to 1/ 32 π under the radical Z/ Α₀³/2 and it is going to be 2 - B E ⁻B/ 2, or again, B is that.
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We are just writing it in a way that you can use it, that is all it is.
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Let us go ahead and do one more for good measure.
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for N = 2, for the L = 1, and M = 0, this is going to be ψ 2, 1, 0, that is equal to 1/ 32 π Z/ α₀³/2.
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this time there is going to be a B × E ⁻B/ 2 and is going to be multiplied by cos θ.
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You might see this one, this one, this one in your book or you might see it differently.
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maybe they chose a different variable for B.
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maybe they did not put the Z in there at all.
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maybe they have a different constant.
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Maybe they combine the constants in different ways.
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I just want to let you know what you see in your book is not necessary what you see here, but they are the same functions.
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when you start integrating units, you are going to get the same number.
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Let us see what we have got.
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let me go back to blue, let me actually go back to black.
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pictorial descriptions of the wave functions.
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these wave functions they are orbitals, that is what they are.
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They are orbitals, the 1S, 2S, 2P, the 3S, the 3P, the PX, PY, PZ.
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That is what these different wave function represent.
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They represent the orbitals.
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They represent places where you may find the electron.
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you remember from the general chemistry that you have already seen pictures of what these orbitals look like.
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I will show you some pictures here, that is not a problem.
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However, one of the things that I want to emphasize is that the pictures that you see are there to help you wrap your mind around where the electron might be in space.
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They are actually not very good representations of that truth of what is going on, where the electron is.
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I would really like if you actually start moving away from pictorial versions, geometric notions of orbitals and start moving towards the mathematical description of these orbitals because that is what it is.
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an orbital is a mathematical description of where a particle might be, how it might be moving.
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we want to actually think in terms of the mathematics.
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you want to use the geometry to help you understand but you do not want to rely on it because it is going to lead you astray, if you rely too much on the geometry.
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it is the algebra, the calculus, the mathematics that is actually describing what is going on.
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pictorial descriptions of the wave functions of the orbitals are difficult and misleading because ψ is a function of three variables.
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what that means is that we actually need a 4 dimensional space in order to represent, in order to graph a wave function.
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we would need a 4 dimensional space, 4 dimensional graph, R, θ, φ, and ψ, the value that you get.
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The pictorial descriptions of wave functions are difficult because ψ is a function of 3 variables.
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In order to graph it, you need a 4 dimensional space.
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We do not have a 4 dimensional space, we have no way of representing the 4 dimensional space on a sheet of paper.
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that is a problem with these things.
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for example, if you have Y = F of X, you have the X and you have the Y variable.
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it was a function of 1 variable but in order to graph it, you need a 2 dimensional plane.
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That is what the graph is.
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if you have a function of 2 variables, X and Y, you have Z = some function of X and Y.
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now you have 3 variables that you need a graph, you need to graph the X, you need to graph the Y, and then you need to graph what Z is when you actually solve the equation Z = XY.
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you need a 3 dimensional.
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in this case, we have 3 independent variables.
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we have 1 dependent variable ψ, therefore we need a 4 dimensional space to graph it.
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we do not have a 4 dimensional space.
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the truth is, we have no real way of graphing this, an actual wave function.
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we can consider them separately.
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Let me go back to blue.
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Therefore, consider the radial and spherical and angular components separately.
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the radial component is easy to visualize because it is just a function of 1 variable R.
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all we have to do is graph in the plane really nice.
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the radial component which is R as a function of R, the single variable is easy to visualize because it depends only on R.
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let us look at the one as orbital, the easiest orbital 1S ψ 1, 0, 0.
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let us look at the 1S orbital.
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that is going to be the R 1, 0 of R.
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1, 0 N is 1, L is 0.
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the function R 1, 0 as a function of R is 2/ α₀³/2 E ⁻R/ α₀.
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This is one of the representations of the radial function for 1, 0.
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the normalization condition is the integral from 0 to infinity of R conjugate × R and then R² DR.
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remember, we are integrating this over a surface of the sphere, so we have to have that DV component.
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Remember, R² sin θ D θ DR D φ, do not forget this R² term.
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Let me mark that down in red.
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Do not forget this R² term when doing your integrations, when integrating in spherical coordinates.
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this part right here, we have R conjugate × R R² DR.
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0 to infinity of R conjugate of 1, 0 × R of 1, 0, R² DR is equal to, here is my R.
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R is a real function so R conjugate is the same as R, so we are just going to have R².
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when I Square that, I get the following.
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I will get 4 / A₀³ × the integral 0 to infinity of E⁻² R/ A₀ R² DR.
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this is the normalization condition for the R 1, 0 function.
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we are just taking a look at the simplest orbital.
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this right here, this integrand including the DR, this is the probability.
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this is the normalization condition.
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this represents the probability.
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this is the probability that the electron in this orbital lies between R and DR.
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this part right here, without the DR that is a probability density.
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we are going back several lessons to when we initially started this.
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this is the probability density × its differential element.
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in this particular case, R gives you the probability that it is between, that it is in that little differential length element.
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we have another page here.
00:33:17.400 --> 00:33:47.300
the probability of the 1S orbital is equal to 4/ A₀³ E⁻² R/ A₀ R² DR.
00:33:47.300 --> 00:33:58.500
in general, the probability is equal to, we are talking just about the radial function here.
00:33:58.500 --> 00:34:02.200
we are not talking about the entire wave function, just the radial function here.
00:34:02.200 --> 00:34:14.700
we have broken it down, = R and L conjugate × R and L R² DR.
00:34:14.700 --> 00:34:24.500
we will leave it like that.
00:34:24.500 --> 00:34:30.800
this is the probability to take the radial, again we are just talking about the radial function here.
00:34:30.800 --> 00:34:42.900
we want to be able to graph some radial dependents, as we move a certain distance R away from the nucleus, where is the electron?
00:34:42.900 --> 00:34:46.200
what are the chances that I'm going to find it here or there?
00:34:46.200 --> 00:34:52.300
this easy to do because it is only a function of one variable R so we can plot it in a 2 dimensional plane.
00:34:52.300 --> 00:34:56.500
this is the probability right here.
00:34:56.500 --> 00:35:14.700
when we plot these, when we plot the probability densities which is just this part.
00:35:14.700 --> 00:35:48.800
when we plot the probability densities vs. R the radius for various values of N and L, here is what we get.
00:35:48.800 --> 00:35:56.400
this is a plot of the 1S orbital, the probability density that is this thing right here.
00:35:56.400 --> 00:36:00.300
this is R² R², the radial functions².
00:36:00.300 --> 00:36:12.100
this right here is just R conjugate × R N sub L, you have that R² term.
00:36:12.100 --> 00:36:15.200
this is the probability density on the Y axis.
00:36:15.200 --> 00:36:18.200
On this X axis, we have R, that is all that is going on here.
00:36:18.200 --> 00:36:23.600
The fact that is ÷ the bohr radius is just the way of standardizing it.
00:36:23.600 --> 00:36:25.000
Do not to worry about that.
00:36:25.000 --> 00:36:29.000
this is the probability density vs. R.
00:36:29.000 --> 00:36:30.600
notice what you have got.
00:36:30.600 --> 00:36:47.400
as you move away from the nucleus, as you get further and further away from it, this measures probability that you will actually find the electron within that far from the nucleus.
00:36:47.400 --> 00:36:52.800
we see about here is what is actually the highest probability.
00:36:52.800 --> 00:37:00.000
The electron is going to be spending most of its time between here and here, that is what this means.
00:37:00.000 --> 00:37:14.900
notice a couple of things, notice we have 0 nodes, and notice when the probability density goes to 0, where you absolutely will not find it.
00:37:14.900 --> 00:37:16.800
and 0 does not count for probability.
00:37:16.800 --> 00:37:20.500
the initial, the nucleus is 00 point, it does not count for a node.
00:37:20.500 --> 00:37:22.700
there is no nodes.
00:37:22.700 --> 00:37:25.800
this is N = 1 and L = 0.
00:37:25.800 --> 00:37:31.100
this is the 1S orbital.
00:37:31.100 --> 00:37:48.000
the 1S orbital as you get further from the nucleus, the electron is going to spend, it is going to be there and then as you get past this point, you are probably not going to find the electron anywhere passed 4 units from the nucleus.
00:37:48.000 --> 00:37:54.200
Let us go to 2S orbital.
00:37:54.200 --> 00:38:01.800
this is a 2S orbital, this is N = 2 and L = 0.
00:38:01.800 --> 00:38:10.200
Notice, you have 1 node right there.
00:38:10.200 --> 00:38:20.200
You have 1 node, what this means is that as you get farther from the nucleus, you will probably find the electron somewhere between here and here or between here and here.
00:38:20.200 --> 00:38:26.700
what this node tells you is that you will never ever find the electron there.
00:38:26.700 --> 00:38:28.100
that is all the node means.
00:38:28.100 --> 00:38:30.000
you are never going to find the electron there.
00:38:30.000 --> 00:38:33.900
It will be closer to the nucleus or farther from the nucleus.
00:38:33.900 --> 00:38:40.300
these high points represent the maximum probability that they will probably be there.
00:38:40.300 --> 00:38:44.200
that is what this represents.
00:38:44.200 --> 00:38:51.200
notice that the 00 mark does not count as a node, that is just the origin.
00:38:51.200 --> 00:38:59.600
For the 3S orbital, we are graphing for radial function, that is all we are doing.
00:38:59.600 --> 00:39:03.800
the 3S orbital here, N = 3, L = 0.
00:39:03.800 --> 00:39:07.800
L = 0 is the S orbital.
00:39:07.800 --> 00:39:14.500
Notice, you have 2 nodes.
00:39:14.500 --> 00:39:16.700
3S orbital has 2 nodes.
00:39:16.700 --> 00:39:18.700
the 1S orbital had no nodes.
00:39:18.700 --> 00:39:21.700
The 2S orbital have 1 node.
00:39:21.700 --> 00:39:24.000
the 3S orbital has 2 nodes, you see a pattern.
00:39:24.000 --> 00:39:29.600
it tells me that the 3S orbital, I can find the electron here or here or here.
00:39:29.600 --> 00:39:35.500
at this point and this point, I'm not going to find electron ever.
00:39:35.500 --> 00:39:45.300
Let us go to the 4S orbital, N is equal 4, L is equal to 0, that is the S orbital.
00:39:45.300 --> 00:39:58.500
we have 1, 2, 3, nodes.
00:39:58.500 --> 00:40:00.700
Let us do some P orbitals.
00:40:00.700 --> 00:40:04.100
we set for various values of N and L.
00:40:04.100 --> 00:40:08.500
here, we took N from 1, 2, 3, 4, we left L at 0.
00:40:08.500 --> 00:40:13.600
we are just looking at different orbitals.
00:40:13.600 --> 00:40:24.300
we have the 2P orbital and again this is just the radial component that we are looking at.
00:40:24.300 --> 00:40:27.600
we are not looking at ψ yet.
00:40:27.600 --> 00:40:37.400
it is R and L and S LM, all we are looking at is the radial component because as a function of 1 variable, we can actually see what happens graphically.
00:40:37.400 --> 00:40:43.800
what we are doing is we are plotting the probability density of the y axis and the radius on the X axis.
00:40:43.800 --> 00:40:49.400
the further we get from the nucleus, what is the most likely place that the electron is going to be.
00:40:49.400 --> 00:40:51.900
that is what this graph represents.
00:40:51.900 --> 00:40:57.500
the 2P orbital, the N is 2 and the L is equal to 1.
00:40:57.500 --> 00:40:59.700
L is 1, that is P orbital.
00:40:59.700 --> 00:41:04.500
Notice, 0 nodes.
00:41:04.500 --> 00:41:10.100
Let us do the 3P orbital.
00:41:10.100 --> 00:41:19.900
Basically, what this says is that in the P orbital, this right here underneath about the orb, that is the most likely place that you are going to find the electron in a P orbital.
00:41:19.900 --> 00:41:24.400
You are not probably going to find it out here, you are not going to find it here.
00:41:24.400 --> 00:41:28.800
You are going to find it somewhere between here and here.
00:41:28.800 --> 00:41:35.200
The 3P orbital, the N is equal to 3 and L is equal to 1.
00:41:35.200 --> 00:41:40.200
Notice, we have 1 node.
00:41:40.200 --> 00:41:44.800
the 3P orbital, you are going to find the electron here or you are going to find the electron somewhere around here.
00:41:44.800 --> 00:41:47.400
you will never find it here, that is what this says.
00:41:47.400 --> 00:41:52.100
you will never find it there.
00:41:52.100 --> 00:41:59.400
Let us look at the 4P orbital.
00:41:59.400 --> 00:42:07.200
4P orbital, N is equal 4, L is equal to 1.
00:42:07.200 --> 00:42:08.500
How many nodes do you have?
00:42:08.500 --> 00:42:15.700
You have 1,2, there are some patterns that we are going to elucidate here.
00:42:15.700 --> 00:42:18.400
I’m going to reiterate this over and over again.
00:42:18.400 --> 00:42:20.600
sorry to keep repeating myself, this was very important.
00:42:20.600 --> 00:42:25.900
there is a lot of mathematics here, a lot of graphs that you are going to be seeing, do not lose your way.
00:42:25.900 --> 00:42:28.200
Do not lose the 4 from the 3.
00:42:28.200 --> 00:42:55.000
Do not forget, we are looking only at the radial function of ψ right now.
00:42:55.000 --> 00:43:09.300
ψ which is a function of R, θ, φ, R of R, and S of θ and φ.
00:43:09.300 --> 00:43:11.300
We are only looking at this one, that is all.
00:43:11.300 --> 00:43:16.300
That is what we are graphing.
00:43:16.300 --> 00:43:23.000
Let us move on to D orbital.
00:43:23.000 --> 00:43:27.200
this is going to be a D orbital.
00:43:27.200 --> 00:43:29.200
this is going to be the 3D orbital.
00:43:29.200 --> 00:43:32.800
N is equal to 3 and L is equal to 2.
00:43:32.800 --> 00:43:36.400
Notice, we have 0 nodes.
00:43:36.400 --> 00:43:43.200
In a D orbital, you are most likely to find the electron between here and here, with the highest probability about right there.
00:43:43.200 --> 00:43:49.000
that is all we are doing, we are graphing probability density vs. distance from the nucleus.
00:43:49.000 --> 00:43:50.700
Let us look at the 4D orbital.
00:43:50.700 --> 00:43:55.800
the 4D, N is equal to 4 and L is equal to 2.
00:43:55.800 --> 00:44:00.800
Notice, we have 1 node.
00:44:00.800 --> 00:44:03.000
Do I have another page here?
00:44:03.000 --> 00:44:06.700
I do not.
00:44:06.700 --> 00:44:21.200
when we examine all of the values of N and all of the values of L to see how many nodes depending on what N is, depending on what L is, whether it is S, P, D orbital, we come up with the following.
00:44:21.200 --> 00:44:24.300
I will write it over here.
00:44:24.300 --> 00:44:55.400
When we examine all the N and L values, and analyze the number of nodes, we find the following.
00:44:55.400 --> 00:45:12.400
we find that the number of nodes is equal to N - L-1.
00:45:12.400 --> 00:45:18.900
For the 4D orbital, N is equal to 4, 4 -2 is 2, 2 -1 is 1.
00:45:18.900 --> 00:45:21.100
The D orbital has 1 node.
00:45:21.100 --> 00:45:25.900
There is going to be one place in the D orbital where the electron will never be found.
00:45:25.900 --> 00:45:28.400
that is all that means.
00:45:28.400 --> 00:45:37.600
these are graφcal representations of probability density vs. distance from the nucleus with graphing the radial function only.
00:45:37.600 --> 00:45:41.600
Let us go ahead and do a quick example here.
00:45:41.600 --> 00:45:51.100
calculate the probability of finding electron in the 2S orbital, the hydrogen atom within 1 bohr radius of the nucleus.
00:45:51.100 --> 00:46:04.000
once again, calculate the probability of finding an electron, the 2S orbital of the hydrogen within 1 bohr radius of the nucleus.
00:46:04.000 --> 00:46:15.700
2S means that N is equal to 2 and it means that L is equal to 0.
00:46:15.700 --> 00:46:22.400
Therefore, we are going to be looking at the R 2, 0 function.
00:46:22.400 --> 00:46:25.800
that is the function that we are going to be working with.
00:46:25.800 --> 00:46:31.900
when we put the 2 and 0 into our particular value of R.
00:46:31.900 --> 00:46:36.700
I will just go ahead and write it out.
00:46:36.700 --> 00:47:08.900
R of 2, 0 of R is going to equal -2 -0 -1!/ 2 × 2 × 2 + 0 !.
00:47:08.900 --> 00:47:15.600
I think there is a cubed there also and a 1/2.
00:47:15.600 --> 00:47:38.800
2/ 2 ×₀ × R⁰ E ⁻R/ 2 Α₀ × 2 - R/ α₀ × – 2!.
00:47:38.800 --> 00:48:05.300
we end up with R of 2, 0 is going to end up equaling 1/ 8 × A 0³ ^ ½ × 4 -2 R/ Α₀ × E ⁻R/ 2 Α₀.
00:48:05.300 --> 00:48:07.800
this is our value of R 2, 0.
00:48:07.800 --> 00:48:10.200
this is our wave function, it is the radial wave function.
00:48:10.200 --> 00:48:19.700
we want to find the probability of finding electron in this orbital within 1 Böhr radius of the nucleus.
00:48:19.700 --> 00:48:41.300
the probability is equal to the integral of 0, 2 A₀ within 1 Böhr radius R 2, 0 conjugate × R 2, 0 R² DR.
00:48:41.300 --> 00:48:43.800
that is the φ integral that we have to solve.
00:48:43.800 --> 00:48:48.300
we have our R, we multiply R by itself.
00:48:48.300 --> 00:48:54.700
in this particular case, R conjugate is the same as R because R 0 is a real function.
00:48:54.700 --> 00:49:03.400
It is not a complex function.
00:49:03.400 --> 00:49:11.800
R conjugate of 2, 0 × R 2, 0 is going to be nothing more than R 2, 0².
00:49:11.800 --> 00:49:20.700
when I square this, when I multiply it all by itself, I get that the probability is equal to,
00:49:20.700 --> 00:49:22.200
this thing is going to come out of the integral.
00:49:22.200 --> 00:49:32.900
It is going to be 1/ 8 Α₀³ the integral 0 to A₀.
00:49:32.900 --> 00:49:41.600
when I square this part, I'm going to get 4 -2 R/ α₀².
00:49:41.600 --> 00:49:53.800
and this is going to be E ⁻R/ 2 α₀ × E⁻² R/ 2 α₀.
00:49:53.800 --> 00:49:56.500
it becomes E⁻² R/ 2 α₀.
00:49:56.500 --> 00:50:04.600
the 2 is cancel so you are just left with α₀, R² DR.
00:50:04.600 --> 00:50:10.500
that is my answer right there.
00:50:10.500 --> 00:50:12.200
the probability is very simple.
00:50:12.200 --> 00:50:14.400
the probability is always the same.
00:50:14.400 --> 00:50:20.000
anytime it ask you for probability, it is always going to be some integral of what ever function you are dealing with.
00:50:20.000 --> 00:50:22.800
the conjugate × the function itself.
00:50:22.800 --> 00:50:27.700
in this case, the function is a real so it is just the function² × some volume element.
00:50:27.700 --> 00:50:35.000
that is all you are doing, with appropriate limits of integration A to B, whatever that happens to be.
00:50:35.000 --> 00:50:38.600
in this case, we want to find it within 1 böhr radius of the nucleus.
00:50:38.600 --> 00:50:44.200
the nucleus is 0, 1 böhr radius is A₀ so this is the probability.
00:50:44.200 --> 00:50:47.000
I have the function, I² it.
00:50:47.000 --> 00:50:51.800
I put it into here and now all I have to do is go ahead and put in a mathematical software
00:50:51.800 --> 00:50:54.800
and let the software solve the integral and give you some number.
00:50:54.800 --> 00:51:02.700
I did not bother to putting this into my software and solving it, because I would leave that to you.
00:51:02.700 --> 00:51:10.800
you know when I teach this course, I actually have the kids just leave it in this form,.
00:51:10.800 --> 00:51:13.300
the software will do the work for you.
00:51:13.300 --> 00:51:15.600
I do not need for them to actually do the integration.
00:51:15.600 --> 00:51:18.100
That is calculus work, that is not altogether that important.
00:51:18.100 --> 00:51:22.300
at this point, it is important to be able to construct the integral.
00:51:22.300 --> 00:51:24.000
software will do the rest.
00:51:24.000 --> 00:51:26.200
so it is up to you if you want to put in your software and see what answer you will get,
00:51:26.200 --> 00:51:29.000
I think if you really interesting for you to take a look at it.
00:51:29.000 --> 00:51:31.500
but other than that, that is your answer.
00:51:31.500 --> 00:51:34.000
Thank you so much for joining us here at www.educator.com.
00:51:34.000 --> 00:51:53.000
We will see you next time, bye.