Page 1 :
ATOMIC STRUCTURE, RADIAL DISTRIBUTION FUNCTION
Page 2 :
Content of Today’s Class…….., , • Wave Function & Atomic Orbitals, • Radial Wave Function, • Radial Probability Density, • Radial Distribution Function
Page 3 :
RADIAL & ANGULAR WAVE FUNCTION, , The solutions to the Schrödinger Wave Equation for H-atom, are called Wave Function (𝚿𝚿), , Product of a radial function and a, spherical harmonic function (Angular function), , Radial Function R(r) - describes how far the electron is away from the, nucleus, Angular Function Y(𝜽𝜽,𝝓𝝓) – where the electron is located around the, nucleus (also gives an orbital its distinctive shape)
Page 4 :
RADIAL & ANGULAR WAVE FUNCTION, , This function 𝚿𝚿 can be used to calculate, the probability of finding any electron of an, atom in any specific region around the, atom's nucleus., , Such a 3-D region around the nucleus, of an atom where the probability of, finding an electron is maximum is, called, , ATOMIC ORBITAL
Page 5 :
WAVE FUNCTION & ATOMIC ORBITALS, Atomic orbital is a mathematical function describing the, location and wave-like behavior of a single electron in an, atom., , Example:, , The wavefunction with n = 1, l = 1 , and m = 0 is called the 1s orbital, and an, electron that is described by this function is said to be “in” the 1s orbital, i.e., have a 1s orbital state.
Page 6 :
VISUALIZING ELECTRON WAVEFUNCTION, Visualizing the variation of an, electronic wavefunction with r,, θ, and φ is important because, the absolute square of the wave, function depicts, , the charge, , distribution (electron probability, density) in an atom or molecule., (This reveals the shape and size, of an orbital)
Page 7 :
VISUALIZING ELECTRON WAVEFUNCTION: Challenge, The wavefunctions for the hydrogen atom depend upon, • Three variables r, θ, φ, • Three quantum numbers n, l, and m, , Examining the behaviour of a function of three, variables in three-dimensional space is challenging., This visualization is made easier by considering the, radial and angular parts separately.
Page 8 :
Graphs of the radial functions, R(r) , for the 1s, 2s, and 2p orbitals, , The 1s function starts, with a high positive, value at the nucleus, and exponentially, decays to essentially, zero, , The radial function for, the 2s orbital goes to, zero and becomes, negative. This, behavior reveals the, presence of a radial, node in the function., , A radial node, occurs when the, radial function, equals zero other, than at r=0 or r=∞
Page 9 :
Radial Probability Density, R𝟐𝟐 (r), But plotting the radial and, , The quantity R(r)∗R(r) gives, , angular parts separately does, , the radial probability density;, , not reveal the shape of an, , i.e., the probability density, , orbital very well. The shape can, , for the electron to be at a, , be, , point located at the distance, , revealed, , better, , in, , PROBABILITY DENSITY PLOT., , a, , r from the nucleus.
Page 11 :
Radial Distribution Function, When the radial probability density for every value, of r is multiplied by the area of the spherical surface, represented by that value of r, we get the radial, distribution function., The radial distribution function gives the probability, density for an electron to be found anywhere on the, surface of a sphere located at a distance r from the, nucleus., Since the area of a spherical surface is 4π𝒓𝒓𝟐𝟐 , the, radial distribution function is given by, P(r) = 4π𝒓𝒓𝟐𝟐 R(r)∗R(r)
Page 12 :
RADIAL DISTRIBUTION FUNCTION: 1s & 2s orbitals, At small values of r, the radial, distribution function is low because, the small surface area for small, radii modulates the high value of, the radial probability density, function near the nucleus. As we, increase r, the surface area, associated with a given value of r, increases, and the 𝑟𝑟 2 term causes, the radial distribution function to, increase even though the radial, probability density is beginning to, decrease. At large values of r , the, exponential decay of the radial, function outweighs the increase, caused by the 𝑟𝑟 2 term and the, radial, distribution, function, decreases.
