Polar Coordinates in Graphing and Hydrogen Atom Orbitals

2. Polar Coordinates One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. A polar coordinate pair Initial ray determines the location of a point.

3. Polar Coordinates To define the Polar Coordinates of a plane we need first to fix a point which will be called the Pole (or the origin) and a half-line starting from the pole. This half-line is called the Polar Axis. Polar Angles P(r, θ) The Polar Angle θ of a point P, P ≠ pole, is the angle between the Polar Axis and the line connecting the point P to the pole. Positive values of the angle indicate angles measured in the counterclockwise direction from the Polar Axis. r θ Polar Axis A positive angle. Polar Coordinates The Polar Coordinates(r,θ) of the point P, P ≠ pole, consist of the distance r of the point P from the Pole and of the Polar Angle θ of the point P. Every (0, θ) represents the pole.

4. More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are:

5. The connection between Polar and Cartesian coordinates From the right angle triangle in the picture one immediately gets the following correspondence between the Cartesian Coordinates (x,y) and the Polar Coordinates (r,θ) assuming the Pole of the Polar Coordinates is the Origin of the Cartesian Coordinates and the Polar Axis is the positive x-axis. (x,y) r y θ x x = r cos(θ) y = r sin(θ) r2 = x2 + y2 tan(θ) = y/x Using these equations one can easily switch between the Cartesian and the Polar Coordinates.

6. 3) The hydrogen atom orbitals Radial components of the hydrogen atom wavefunctions look as follows: R(r) =for n = 1, l = 0 1s orbital R(r) =for n = 2, l = 0 2s orbital R(r) =for n = 2, l = 1, ml = 0 2pz orbital R(r) = n = 3, l = 0 3s orbital R(r) =n = 3, l = 1, ml = 0 3pz orbital Here a0 = , the first orbit radius (0.529Å). Z is the nuclear charge (+1)

7. THE ELECTRONIC STRUCTURE OF THE HYDROGEN ATOM 1) Theoretical description of the hydrogen atom: Schrödinger equation:HY = EY H H is the Hamiltonian operator (kinetic + potential energy of an electron) E – the energy of an electron whose distribution in space is described by Y - wave function YY*is proportional to the probability of finding the electron in a given point of space If Y is normalized, the following holds: The part of space where the probability to find electron is non-zero is called orbital When we plot orbitals, we usually show the smallest volume of space where this probability is 90%. :

8. 2) Solution of the Schrödinger equation for the hydrogen atom. Separation of variables r, q, f For the case of the hydrogen atom the Schrödinger equation can be solved exactly when variablesr, q, fare separated like Y = R(r)Q(q) F(f) Three integer parameters appear in the solution, n (principal quantum number), while solving R-equation l (orbital quantum number), while solvingQ-equation ml (magnetic quantum number), while solvingF-equation n = 1, 2, … ∞; defines energy of an electron

9. 4) Radial components of the hydrogen atom orbitals • The exponential decay of R(r) is slower for greater n • At some distances r where R(r)is equal to 0, we have radial nodes, n-l-1 in total 1s 2s 2p node

10. 5) Radial probability function Radial probability function is defined as [r R(r)]2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: http://www.phy.davidson.edu/StuHome/cabell_f/Radial.html 1s 2p 2s

11. 6) Angular components Angular components of the hydrogen atom wave functions, QF for l = 0, ml = 0; QF = s orbitals for l = 1, ml = 0; QF = pz orbitals

12. 4) Radial components of the hydrogen atom orbitals • The exponential decay of R(r) is slower for greater n • At some distances r where R(r)is equal to 0, we have radial nodes, n-l-1 in total 1s 2s 2p node

13. 5) Radial probability function Radial probability function is defined as [r R(r)]2 – probability of finding an electron in a spherical layer at a distance r from nucleus Radial probability plots for hydrogen atom: http://www.phy.davidson.edu/StuHome/cabell_f/Radial.html 1s 2p 2s

14. Summary • Exact energy and spatial distribution of an electron in the hydrogen atom can be found by solving Shrödinger equation; • Three quantum numbers n, l, ml appear as integer parameters while solving the equation; • Each hydrogen orbital can be characterized by energy (n), shape (l), spatial orientation (ml), number of nodal surfaces (n-1)

15. Orbitals...

16. Electrons are part of what makes an atom an atom But where exactly are the electrons inside an atom? atom

17. Orbitals are areas within atoms where there is a high probablility of finding electrons.

18. Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

19. Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

20. Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

21. Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

22. Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

23. Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

24. Knowing how electrons are arranged in an atom is important because that governs how atoms interact with each other

25. Let’s say you have a room with marbles in it

26. The marbles are not just anywhere in the room. They are inside boxes in the room.

27. You know where the boxes are, and you know the marbles are inside the boxes, but…

28. you don’t know exactly where the marbles are inside the boxes

29. The room is an atom The marbles are electrons The boxes are orbitals

30. The room is an atom The marbles are electrons The boxes are orbitals

31. The room is an atom The marbles are electrons The boxes are orbitals Science has determined where the orbitals are inside an atom, but it is never known precisely where the electrons are inside the orbitals

32. So what are the sizes and shapes of orbitals?

33. The area where an electron can be found, the orbital, is defined mathematically, but we can see it as a specific shape in 3-dimensional space…

34. z y x

35. z y The 3 axes represent 3-dimensional space x

36. z y For this presentation, the nucleus of the atom is at the center of the three axes. x

37. The “1s” orbital is a sphere, centered around the nucleus

40. The 2s orbital is also a sphere.

41. The 2s electrons have a higher energy than the 1s electrons. Therefore, the 2s electrons are generally more distant from the nucleus, making the 2s orbital larger than the 1s orbital.

42. 1s orbital

43. 2s orbital

44. Don’t forget: an orbital is the shape of the space where there is a high probability of finding electrons

45. Don’t forget: an orbital is the shape of the space where there is a high probability of finding electrons The s orbitals are spheres

46. There are three 2p orbitals

47. The three 2p orbitals are oriented perpendicular to each other

48. z This is one 2p orbital (2py) y x

49. z another 2p orbital (2px) y x

50. z the third 2p orbital (2pz) y x