The absorption of energy excites electrons to higher energy levels, from a lower  to a higher one. Since electron shells grow increasingly closer in energy and  increases, the highest gaps occur between lower level shells. Thus, in this question, the largest gap between any two principle quantum number occurs between the first energy level and the third energy level.

2d is a non-existent energy shell because its principle quantum number, , does not exceed its orbital angular momentum quantum number, . This is easily observable on the periodic table, where it is shown that the d-sub shell is only available for energy levels greater than or equal to three. All other answer choices obey the rule that the orbital angular momentum quantum number is less than the principle quantum number. 

The correct answer is that 4n is larger in size than 2n.  

When it comes to quantum numbers, n refers only to size or the atom, or electrons' distance from the nucleus; n is the principal quantum number.  

l is expressed as  and is therefore dependent on the n value; l describes the shape of the orbital (spherical, dumbbell, etc.).  

m_l values range from  to  and describe the orientation of the orbital.  

m_s values are either  or  and represent electron spin.  

Electrons of an atom are located within electronic orbitals around a nucleus. The electrons of each atoms have their own specific energy level called principal energy level. When electrons are excited by absorbing energy the electrons can jump to a higher energy level. When an electron drops back to a lower energy level, the atom emits energy. Therefore, when neon atoms in a neon sign emit light, the electrons are moving from a higher principal energy level to a lower principal energy level. 

To solve this problem we need to use the following equation:

where, 

 = speed of light = . The speed of light is constant. 

 = wavelength (units = )

 = frequency (units =  or )

Now we can plug the wavelength and the speed of light into the equation above and solve for frequency. 

For this question, we're asked to identify an answer choice containing a combination of quantum numbers that cannot exist.

Let's go over what each of the quantum numbers mean. But first, it's important to note that each combination of the following four quantum numbers describes a single electron in an atom within a given atomic orbital. The first three refer specifically to the atomic orbital, while the fourth refers to the electrons within that orbital.

The primary quantum number, , describes the size of the atomic orbital for a given atom. This value takes on whole number integers such as . As the value of  increases, so too does the energy levels of the electrons in that energy level.

The azimuthal quantum number, , describes the shape of the atomic orbital for a given atom. The values that this quantum number can take are restricted by the value of the primary quantum number. The azimuthal quantum number can take on values from . Hence, if the primary quantum number is , then the azimuthal quantum number can only be . If the primary quantum number is , then the azimuthal quantum number can take on either a value of  or . Each of these values of the azimuthal quantum number corresponds to a certain orbital shape in the atom. A value of  corresponds to an s orbital. A value of  corresponds to a p orbital. A value of  corresponds to a d orbital, and a value of  corresponds to an f orbital.

The magnetic quantum, , describes the orientation of the atomic orbital in the atom. Depending on the type of orbital, the orientation of the electrons in that orbital can be situated in different ways. For example, an s orbital can only be situated one way. A p orbital can be situated three ways, and a d orbital can be situated five ways. The value of  is restricted by the value of the azimuthal quantum number. The value of  can range from . For example, if the azimuthal quantum number is , then the magnetic quantum number can take on any of the values .

Lastly, the magnetic spin quantum number, , does not describe atomic orbitals within an atom, but rather describes the electrons within those orbitals. In any given orbital, there can only be two electrons, and both of them must have opposite spins. To denote this, we can give the electron a value of  or .

Based on the information given above, we can decide which answer choice breaks these rules.

The above set of quantum numbers cannot exist. Because the primary quantum number is equal to one, the azimuthal quantum number is restricted to only a value of zero. In this case, since it has a value of one, this is wrong. Hence, this is the correct answer choice.

Quantum numbers describe the orbitals in which electrons are found. There are three types of quantum numbers: the principal quantum number (), the angular quantum number (), and the magnetic quantum number (). 

The principal quantum number describes the size of the orbital. This number cannot be zero, and orbital sizes increase with increasing numbers. Thus, orbitals for which  are larger than orbitals for which . 

The angular quantum number describes the shape of the orbital. If the shape of the orbital is spherical, . If the shape is polar, . If the shape is cloverleaf, . As the value of the angular quantum number increases, the orbital shape becomes more complex. The angular quantum number can be any integer between  and .

Because orbitals of different shapes can be oriented in space in different ways, a third quantum number comes into play. The magnetic quantum number describes the orientation in space of a particular orbital. This value can be any integer between  and . 

This question refers to quantum numbers, which describe the distribution of electrons in an atom. There are three quantum numbers: , , and . 

The principal quantum number, , describes the size of the orbital in which the electron lies. The value of  can be any integer except zero. In this case, .

The angular quantum number, , describes the shape of the orbital. The value of  can be any integer from zero to . If , l can be , , , or .

The magnetic quantum number, , describes the orientation in space of the orbital. The value of  can be any integer from  to . For example, if ,  can be , , , , or .