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The arrows that he draws in class is called the spin magnetic quantum number and they are telling us which way the electrons in the orbitals are spinning, which is either up and down. As for how you would determine whether or not the arrows are spinning up or down there will always be spin up and spin down electron in one orbital because the electrons like to pair up, so you would have them going opposite directions.
The arrows will fit the amount of orbitals and each represents an electron. For example, if we potentially have five electrons in the d subshell then we would have five arrows pointing up. Due to the Pauli Exclusion Principle, all of the arrows will face up and spin positively until we have enough to add another and all of the orbitals in the subshell have already been filled.
To further explain why we fill in all the arrows in a parallel spin first is that the electrons must first fill in this lower energy state before having an opposite spinning electron which will be a higher energy level.
The number of arrows is equal to the number of electrons an atom has. The first arrow you draw will always be pointing upwards, then the second one can either be up or down depending on the number of electrons.
The Pauli exclusion principle says that there can be no more than 2 e- per orbital; that's why the arrows are grouped in pairs with opposite spins. If you're filling out diagrams like that, you'd start by adding electrons with a maximum of 2 e- for each orbital, and adding up spin electrons first if the subshell has multiple orbitals.
you know the direction of the spin indicated by the 1/2 or -1/2 number given. 1/2 is spin up and -1/2 is spin down. If there are 2 electrons in one orbital they will have opposite spins because they are paired.
Hund's rule asserts that due to electron repulsion, electrons in the same subshell occupy different orbitals with parallel spin. Parallel spin indicates that electrons have the same spin whereas 2 electrons with opposite spin are paired.
TimVintsDis3C, I think you are asking is why there are sub shells that have multiple arrows pointing up in several orbitals by itself rather than an up and a down in the same orbital. This is due to Hund's rule which claims that all electrons in a sub shell must first fill the parallel spin of the sub shell before doing the opposite spin.
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