The contour representation of one of the orbitals for the n = 3 shell of a hydrogen atom is shown here. (c) In which of the following ways would you modify this sketch if the value of the magnetic quantum number, ml, were to change? (i) It would be drawn larger, (ii) the number of lobes would change, (iii) the lobes of the orbital would point in a different direction, (iv) there would be no change in the sketch.

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Welcome back everyone. The unequal two shell and L equals one orbital of the hydrogen atom has several contour representations. One of them is shown below which of the following is true. When the magnetic quantum number MSA bell is altered, we have choice one. The orientation of the contour representation will change two more lobes will be drawn for the contour representation. Three, the size of the lobes of the contour representation will be smaller or four. The notable plane of the contour representation will shift. We're going to begin with a focus on the term N equals two from the prompt, which recall is the energy level or shell number of the orbital for this hydrogen atom. And we're going to then think of our first type of quantum number which describes the angular momentum of this orbital. This is going to be represented recalled by the variable L and it's a range of zero to N minus one. And so because we know that we have an N value of two, we have a range of 0 to 2 minus one, which would mean we have an L value equal to zero and an L value equal to one. However, the prompt emphasizes an L value of just one only recall that an L value describes the shape of an orbital. And because we're going with the L value of one, this is referring to the shape of a P shaped orbital recall that a P orbital has a dumbbell shape. And so that is why in this illustration, we've got a contour representation of this dumbbell P orbital shaped with one lobe on each side of this X axis. So we'll label this as a PX orbital that is depicted in this illustration. Take note that this PX orbital is intersecting with its horizontal X axis orientation along the Y and Z planes as well. And so we're going to cover a specific plane in a moment. But next, let's focus on what an L value of zero means even though we're not focused on that type of orbital recall that an L value of zero corresponds to the shape of an orbital, which is going to be S an S orbital. And recall that an S orbital is just spherical. So now let's move on and focus on what the prompt tells us next, which is about a magnetic quantum number. M recall that the magnetic quantum number and we'll use a different color. The magnetic quantum number. MS A Bell describes the orientation of the orbital within the subs shell. And recall that it describes a range of negative L to positive L and So, because we have an L value of one emphasized from the prompt, we're going to have an M sub L range of negative 10 and then positive one. So therefore, this tells us that there are three orientations of the three P orbitals. So we can also include if we'd like a P orbital oriented along the z axis. So we'd have a dumbbell shape on with one lobe on each side of the z axis. And then a third P orbital where we have a dumbbell shape oriented along the y axis. Now let's consider our answer choices. We have choice one, the orientation of the contour representation will change. If we alter the magnetic quantum number, I believe that that would make sense. And this is because as we stated, an MBE or magnetic quantum number describes the orientation of the orbital. And so if we change this magnetic quantum number to a different value, instead of emphasizing the one value we would either have am L value of plus one or negative one for this PX orbital. But it could also change to an ML value of zero, which would change the orientation of this orbital to actually a Pz orbital. So let's place a check mark by choice one because since the magnetic quantum number ties to describing the orientation of the orbital, then that would make sense as an answer here. And a true statement. Now looking at statement two more lobes will be drawn for the contour representation. As we stated, a P orbital should have a dumbbell shape with just two lobes in total making up that shape. So adding any more lobes is going to change the type of orbital that we're dealing with. We're going to rule out choice two as a false statement and then choice three, the size of the lobes of the contour representation will be smaller. Again, the magnetic quantum number MS describes the orientation of the orbital. It has nothing to do with the size of the lobes of the orbital. And so them becoming smaller is not going to be a true statement, this is false. So we're going to rule choice three out as an answer choice. And then we have statement four, the not plane of the contour representation will shift. And this is the other axis that I was going to describe from earlier and we'll cover this now. So we called that the not plane describes the pro probability of finding an electron of equaling absolutely zero. So there are no electrons found in a nodal plane region. And so in order to, to determine the nole plane of the PX orbital, recall that nole planes are always going to be perpendicular to the plane or axis of the orbital that we're focusing on because we have the PX orbital, we need a perpendicular nole plane. And that would be described by the y nole plane YZ nodal plane which we can observe intersex at the origin of this PX orbital, the origin being the center point here. And so notice that this YZ nodal plane cuts this PX orbital so that the lobes are now in half, they're no longer connected in a sense you can imagine. And so in this region, in this yy nole plane, there are zero density of electrons. This does not tie to a correct answer statement four because notice that it uses the word shift. This is an incorrect term to use. We would want to change this to saying that the notable plane of the contour representation will undergo an angular transformation. That would be more correct. And this is because the planes that we have the PX orbital plane and then the yy nole plane are intersecting planes, they are not parallel planes. So if we were to just shift this notable plane, this wising nole plane, it can either only undergo a vertical or horizontal shift. And if the orientation of the PX orbital is now oriented to describe a PZ orbital along the Z axis, then a horizontal or vertical shift of the YZ nodal plane would not make sense for this newly oriented PZ orbital. We would need to orient a notable plane now along the xy axis or not plane in order to cut the PZ orbital in half. So although the notable plane of the contour representation should also be oriented under a differently placed plane of a 90 degree angle to the newly oriented orbital that would essentially describe an angular transformation and not a shift that is an incorrect term. And so for that reason, we're going to rule out choice four. And this means that we are able to now confirm that it is statement one, the orientation of the contour representation as the only true statement and correct answer. For this example, when the magnetic quantum number M sub bell is altered, I hope that this made sense. And let us know if you have any questions.

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Key Concepts

Quantum Numbers

Orbital Shapes and Orientation

Hydrogen Atom Orbitals