The Mathematical Function of a D-Orbital

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A D-orbital is a type of orbital of an atom in which the quantum angular momentum number, L, is equal to two. Find out about the mathematical function of a D-orbital with help from an applied physics professional in this free video clip.

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Video Transcript

Hello, my name is Walter Unglaub, and this is "the mathematical function of a D-orbital." A D-orbital is a type of orbital of an atom in which the quantum angular momentum number, L, is equal to two. This means that the magnetic quantum number, M sub L, is greater than or equal to negative two, and less than or equal to positive two in integer steps. So, if we examine the magnetic number, we see that M sub two can be a number in the set negative two, comma negative one, zero, one, or two. So, we see that there are five different types of D-orbitals. Now, the mathematical function to describe the wave function of an orbital is given by the spherical component of the wave function, which is described by spherical harmonics. The spherical harmonic generally has a function of the angular and magnetic quantum numbers in spherical coordinates, is denoted by capital Y and it depends on the polar angle, theta, and the azimuthal angle, phi. And, so the spherical harmonic is given by negative one raised to the power M, times the square root of two L plus one, divided four pi, times L minus M, factorial, divided L plus M factorial, times the Legendre polynomial, P sub L, M, cosign of the angle, theta, the polar angle, times E raised to the I M phi. So, we see, that in this case, L would be equal to two, but, if we're interested in a particular type of D-orbital, we would have to specify the magnetic quantum number, M sub L, in this case, M sub two. So, anywhere you see the M's, we would have to put one of these values, these allowed quantum values into this function, and then we would have a function that would describe the spherical component of the wave function for the atom, in the case of L equals two, the D-orbital. My name is Walter Unglaub, and this is "the mathematical function for a D-orbital."

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