Quantum numbers are used to describe the location of an electron within the electron cloud. The first quantum number corresponds to the energy level, which matches with the period of the periodic table in which the element is usually found. The second quantum number describes the electron subshell, corresponding to the s, p, d, and f subshells. The third quantum number tells the orbital within the subshell that holds the electron. Finally, the fourth quantum number describes the spin of the electron as either positive or negative. Each orbital can only carry two electrons.

The first quantum number is a positive integer:

The second quantum number is any integer greater than zero and less than the first quantum number:

The third quantum number is an integer between the value of the second quantum number and its inverse:

The final quantum number can only carry a value of positive or negative one-half:

The answer choice that gives an impossible set of quantum numbers has a negative second quantum number. The first and second quantum numbers will always be greater than or equal to zero.

The ground state electron configuration for calcium is:

The question asks us to give the  value for the valence electrons in calcium. Valence electrons are located in the outermost shell, the shell with the highest principle quantum number. For calcium, the outermost shell is 4s. The  value that corresponds to 4s is . 

The other answer choices are incorrect because they describe the  value for the location of non-valence electrons.

The, principal quantum number, , tells us the energy shell that is represented as the leading coefficient in electronic configuration. We can see that the given orbital has a leading coefficient of .

The angular momentum (azimuthal) quantum number, , tells us the shape of the orbital. Each shape corresponds with a numerical value of  and a letter. The values for  and their corresponding subshells are listed below:

We are given an orbital with a p subshell, indicating that the angular momentum quantum number must be .

All s orbitals are spherical in shape, and the smallest principle quantum number will refer to the smallest orbital. As principal quantum number, , increases, so does the size of the orbital. The correct answer for this question will be the s orbital with the smallest coefficient: .

Answer choices containing p or d orbitals are incorrect because these orbitals are not spherical in shape.

The location of a given electron is described by a set of four quantum numbers. The principle quantum number, , gives the energy level of the electron. It essentially describes the size of the electron shell, and can be any integer value. As this value increases, the size of the orbital increases and is located farther from the atom nucleus. The azimuthal (angular momentum) quantum number, , refers to the subshell of the energy level. Each subshell has a different shape, corresponding to spdf configurations. can be any value from zero to . The magnetic quantum number, , gives the orientation of the subshell given by the azimuthal quantum number. These values correspond to orientations along the x-axis, y-axis, or z-axis. Each orientation constitutes a different orbital and can hold exactly two electrons. The magnetic quantum number can be any integer greater than and less than , with the total number of possible orbitals given by .  is the electron spin quantum number. This number is used to distinguish between the two electrons in a single orbital.

The subshell () can carry a total of five orbitals, corresponding to the values of . Each orbital will hold two electrons, for a total of ten.

The only false statement concerns the angular momentum quantum number, confusing it with the quantum number.

The correct answer is 50 electrons.

The number of electrons a shell can hold can be found using the formula . Given the principle quantum number, we can calculate the number of electrons that can fill that energy level.

The, principal quantum number,  tells us the shell, which is represented as the leading coefficient in electronic configuration. Since we are given , we know that the leading number will be five.

The angular momentum (azimuthal) quantum number, , tells us the shape of the orbital. Each shape corresponds with a value of , and a letter. The values for  and their corresponding orbital shapes are listed below:

We are given the subshell , which tells us we are looking at an orbital.

Together, this gives our answer of .

The location of a given electron is described by a set of four quantum numbers. Of these, the magnetic quantum number () gives the orientation of the orbital in three-dimensional space.

The principle quantum number, , gives the energy level of the electron. It essentially describes the size of the electron shell, and can be any integer value.

The azimuthal (angular momentum) quantum number, , refers to the subshell of the energy level. Each subshell has a different shape, corresponding to spdf configurations. can be any value from zero to .

The magnetic quantum number, , gives the orientation of the subshell given by the azimuthal quantum number. For example, a p orbital has three different orientations based on three different values for the magnetic quantum number. These values correspond to orientations along the x-axis, y-axis, or z-axis. Each orientation constitutes a different orbital and can hold exactly two electrons. The magnetic quantum number can be any integer greater than and less than , with the total number of possible orbitals given by .

 is the electron spin quantum number. This number is used to distinguish between the two electrons in a single orbital.

() is not a quantum number. This is a formula that allows us to determine the possible values of  for every principal quantum number.