The shapes of s orbitals

We saw in Section 9.4c that in certain cases a wavefunction can be separated into factors that depend on different coordinates and that the Schrodinger equation separates into simpler versions for each variable. Application of this separation of variables procedure to the hydrogen atom leads to a Schrodinger equation that separates into one equation for the electron moving around the nucleus (the analog of the particle on a sphere) and an equation for the radial dependence. The wavefunction is written as 

The factor R r) is a function of the distance r from the nucleus and is known as the radial wavefunction. Its form depends on the values of n and I but is 

In this case the angular wavefunction, Yoo = l/(47t), is a constant, independent of the angles S and (j). You should recall that in Section 9.2 we anticipated that a wavefunction for an electron in the ground state of a hydrogen atom has a wavefunction proportional to e T eqn 9.33 is its precise form. The constant Uq is called the Bohr radius (because it occurred in the equations based on an early model of the structure of the hydrogen atom proposed by the Danish physicist Niels Bohr) and has the value 52.92 pm. 

The amplitude of a Is orbital depends only on the radius, r, of the point of interest and is independent of angle (the latitude and longitude of the point). Therefore, the orbital has the same amplitude at all points at the same distance from the nucleus regardless of direction. Because, according to the Born interpretation (Section 9.2b), the probability density of the electron is proportional to the square of the wavefunction, we now know that the electron will be found with the same probability in any direction (for a given distance from the nucleus). We summarize this angular independence by saying that a Is orbital is spherically symmetrical. Because the same factor Y occurs in all orbitals with / = 0, all s orbitals have the same spherical symmetry (but different radial dependences). 

The wavefunction in eqn 9.33 decays exponentially toward zero from a maximum value at the nucleus (Fig. 9.43). It follows that the most probable point at which the electron will be found is at the nucleus itself. A method of depicting the probabihty of finding the electron at each point in space is to represent y/ by the density of shading in a diagram (Fig. 9.44). A simpler procedure is to show only the boundary surface, the shape that captures about 90 per cent of the electron probabihty. For the Is orbital, the boundary surface is a sphere centered on the nucleus (Fig. 9.45). 

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O How do the shapes of s orbitals in different energy levels of a hydrogen atom compare How do the sizes of these orbitals compare ... 

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