The first ionization energy of sodium is 496 kJ/mol. Use Coulomb's law to estimate the average distance between the sodium nucleus and the 3s electron. How does this distance compare to the atomic radius of sodium? Explain the difference.

Video transcript

Welcome back everyone to another video. The first ionization energy of potassium is 419 kilojoules per mole. Use Kams Law to estimate the average distance between the potassium nucleus and the four S electron. How does the distance compare to the atomic radius of potassium? Explain the difference. And we're given four answer choices. A 332 Peters B 288 Peters C 365 Peters and the 254 Peters. Each answer also has an explanation. We're going to read the, the correct explanation when we get the answer. So first of all, what we're going to do here is just solve this problem using the Cool Arms law just as the problem suggests. So what we're going to do here is just write down the law uh since we're looking for the potential energy, let's use the energy form of the law which states that energy is equal to Boltzmann's constant or basically one divided by or by epsilon not. And then we are multiplying that by the magnitude of the first charge, the magnitude of the second charge of interest in dividing that by a radius R, that's our formula which we can rearrange for radius since this is what we are looking for. So radius would simply be equal to the Boltzmann's constant multiplied by charge, one charge two divided by the potential energy. E we have our formula. We can essentially think about our next steps. Well, essentially we can just plug in the data. So first of all, we can say that we have one divided by now we will leave for pi. And now for epsilon knot, we're going to use the tables and identify that it is equal to 8.85 multiplied by sensitive power of negative 12. OK. The units would be cool arms squared divided by joule multiplied by meter. OK. So we have our constant, we can just use the multiplication sign here, we can slightly extend our fraction. And now for the second part, we have two charges. Well, we have an interaction between a proton and an electron. So first of all, we are going to use 1.602 multiplied by 10. The power of negative 19th of a coon, that's the charge of a proton. And the charge of an electron would be negative that we are multiplying by the negative value which is negative 1.602 multiplied by 10, the power of negative 19th of a Coolum. And now we also have to divide that by the energy value since we are looking at only one atom, we're going to use the ionization energy it was given to us, right? And we also want to remember that the energy should be equal to the negative value of the ionization energy of the first ionization energy. Because we are removing the first electron from our shell. Now, why? Well, basically we have to recall that ionization energy is the energy required to remove an electron from a gaseous atom, right? That's why we are taking the negative value here. Now let's get our energy. So first of all, we're going to use that negative value. So negative 419 kilojoules per mole, we're going to turn that into joules. So let's multiply by 10 of the third, we have joules per mole. We want to get joules per atom. And for that purpose, we simply want to multiply by one divided by avocados number because essentially we're dividing the energy per mole right by avocado's number to get the energy per atom. So simple speaking, we're dividing by 6.022 multiplied by 10 to the power of 23rd mods the negative first. If we perform the calculation, we get our answer in meters, which would be 3.32 multiplied by 10, the power of negative 10th of a meter, we can easily convert that into P meters knowing that one petter is equal to sensitive power of negative 12 of a meter. And this gives us 332 PICO meter. So we got our answer. Let's see, which is the answer choice here. So we clearly noticed that the correct answer to this problem would be option a 332 Peters. And now let's think about the explanation. Well, essentially the calculated distance is larger than the actual atomic radius of 243 Peters. This is what we get from the data tables due to the shielding effect of inner electrons reducing the effective nuclear charge experienced by the three S electron which is reflected in the ionization energy. So that would be our explanation to this problem. Let's label the correct answer as option A and conclude the video. Thank you for watching.