this video is sponsored by librarians stay tuned for a special offer for arvan ash viewers if you've had any kind of basic chemistry class in your life you're probably familiar with this chart that might have scared you a little bit when you first saw it but it's just a list all the different types of atoms that exist on earth it's simply arranged based on the number of protons in the atom's nucleus and for a neutral atom with no charge it also has the same number of electrons these numbers are the main determinant of its chemical properties if you look more closely at this chart you'll see that it's not just a simple list of atoms in numerical order it is arranged such that certain groups of elements have similar properties but why do these elements have similar properties their properties have to do with the way electrons are arranged around the nucleus of atoms but why should electrons be arranged this way so specifically in certain orbitals and shells it turns out that we have a very powerful tool to understand why and that tool is quantum mechanics it does nothing less than explain the entire periodic table of elements that is it can account for all the atoms found on earth how is this possible that's coming up right now the structure of individual atoms and chemistry is all about electrons how they're arranged and shared between atoms it all boils down to electron shells in the precise way electrons orbit the atom the universe tends towards the lowest energy state this applies to electrons some configurations of electrons are simply more energy efficient than others it so happens that the lowest energy configurations of electrons is when the electron shells are either empty or full what do i mean by shells to explain this concept in a brief way let's look at the oversimplified bohr model of the atom bohr showed that electrons can only exist in certain stable orbits around a nucleus where the angular momentum is proportional to planck's constant these are like shells around the nucleus we can draw the atom where the nucleus is in the center and the rings around it represent the electron orbits or shells but this model of course was an oversimplification erwin schrodinger understood that rather than being confined to an orbit like a planet an electron is a matter wave that forms a probability cloud in 3d space smeared around the nucleus according to quantum mechanics and he formulated the rules to describe this behavior the famous schroedinger equation this equation as it turns out went beyond the bohr model of the atom by more accurately describing every atom that exists in nature in other words the structure of the entire periodic table of the elements can be predicted by schrodinger's equation this was nothing short of astounding the schrodinger equation showed that each shell has a maximum number of electrons it can hold the innermost shell holds a maximum of two electrons the second holds a maximum of eight electrons the next up to 18 then 32 then 50 and so on chemistry is about substances exchanging or sharing electrons in order to fully fill up these shells okay so now the elephant in the room that needs explaining is why are these numbers so special these numbers look random why in the world do atomic interactions care about conforming to these numbers it all comes down to energy systems in the universe tend towards their lowest energy state to minimize energy the electrons fill up the orbits starting from the center and moving out and it turns out that having full or empty orbitals minimizes the energy of the atom to understand why we have to solve the schrodinger equation although this equation looks intimidating and it can be in its essence is just a statement of energy conservation it simply says that total energy is equal to potential energy plus kinetic energy that's it this equation can be solved most easily for the hydrogen atom because it is the simplest atom just one electron orbiting one proton because the core is made of just one proton it's spherically symmetrical if you imagine the proton like a ball it doesn't matter how you rotate it it always looks the same it has symmetry if we consider something like hydrogen 2 also called deuterium which has a proton and neutron in the nucleus it would no longer be spherically symmetrical because if you rotate the nucleus the neutron would point in a different way this spherical symmetry is essential for making the solution simple enough to solve the schrodinger equation exactly now i don't want you to get intimidated by this equation i just want you to get familiar with it so that you can understand where the electron shell numbers are coming from in the schrodinger equation we have a kinetic energy term and a potential energy term that we define because hydrogen has a positively charged nucleus and a negatively charged electron there's an attractive coulomb potential between the electron and the proton this is where the potential energy term comes from there's several constants in this equation that we don't have to solve for as shown here planck's constant epsilon or the vacuum permittivity mass of electron and proton and e the electric charge constant i won't derive this equation in this video because that's not important for understanding the subject we're covering today the key term to look at here is psi this is the term we have to solve for it is the wave function of the hydrogen atom it represents a value that is related to the probability of this atom being in certain quantum states the radius and angles theta and phi represent the spherical coordinates of our system in three-dimensional space now even though this is the simplest atom to solve for it's still quite difficult to do but solving for the wave function of the hydrogen atom gets you the following result this equation is a bit of a mouthful and it takes many pages and hours of work to derive but let's look at some of the components so that you can understand how this plays a role in the shells around the nucleus the main thing i want you to look at are these three extra labels n l and m where did these come from when solving the schrodinger equation you get a set of complicated differential equations in order to solve these you have to specify some quantum values for the atom these are values or quantities that have to be inserted in the equation to solve for psi i have a link to the long solution if you want to see how this is done n just represents the electron shell layer n equals 1 is the ground state or the lowest energy state of the hydrogen atom but hydrogen is not always found in its lowest energy state so it can take on other values l is a quantum number defining the orbital angular momentum and m is the number which specifies the orientation and space of the orbital now here's the key to the significance of this equation when we plug in different values for n l and m in the schrodinger equation for the hydrogen atom it also approximately represents the solution to all the quantum states of electrons for any other atom so this equation allows us to predict how electrons behave for all the elements of the periodic table now there are some rules for choosing the numbers n must be an integer starting with one since it specifies the shell there is no such thing as a 1. 5 shell or 1. 3 shell the shells can only be 1 2 3 and so on depending on the value of n l will be an integer from 0 to n minus 1.
and finally the value of m depends on the value of l this can be integer values between negative l and positive l for example if n equals 1 then we must have that l equals 0 and thus m equals 0. if n equals 2 l can equal 0 or 1 and m can equal negative 1 0 and 1.
