Question

How does the length of the z-component of the net dipole moment of ammonia relate to the actual length of the dipole moment vector?

Answer 1

Well, the length is identical. The dipole moment of ammonia is along its `z` axis, so the `x` and `y` components are zero, and only the `z` component contributes to the length.

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The principal axis of rotation (`C_n`) requires the smallest angle of rotation `360^@/n` before coinciding with the original molecule.

For instance, the `C_n` axis of water requires a `180^@` rotation to bring it back to an orientation indistinguishable to how it was before. So, we call it `C_2` and point it along the `z` axis on the `yz` plane:

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If you want to visualize this more, click on the image and try out the rotation yourself.

For ammonia, the `z` axis points vertically through the nitrogen atom, like the handle of an umbrella with the `"H"` atoms as the umbrella head.

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The dipole moment of ammonia points exactly along the `z` axis. So, we write the dipole moment vector `vecmu` as:

`vecmu = << mu_x, mu_y, mu_z >>`

`= << 0, 0, 1.48 >>`

The length of the vector is given by the vector magnitude:

`color(blue)(|| vecmu ||) = sqrt(mu_x^2 + mu_y^2 + mu_z^2)`

`= sqrt(0^2 + 0^2 + ("1.48 D")^2)`

`=` `color(blue)("1.48 D")`