Worked example: Calculating the maximum wavelength capable of ionization

- [Instructor] We're told that the first ionization energy of silver is 7.31 times 10 to the fifth joules per mole. What is the longest wavelength of light that is capable of ionizing an atom of silver in the gas phase? All right, now before I even ask you to pause and try to do this on your own, let's just remind ourselves or try to understand what first ionization energy even is. This is the energy required to get the highest energy or outermost electron to escape from the atom. So it's not just going to go to a higher energy level. It's just going to completely escape. You could view it as the infinite energy level. And the reason why we're talking about the longest wavelength of light is, remember, the longer the wavelength, the lower the frequency, and the lower the energy. So this is saying really what's the minimum frequency or the minimum energy that's associated with the longest wavelength of light for an atom of silver? So a couple things to pay attention to. They're giving us the first ionization energy in terms of moles, not per atom. And then we just have to remind ourselves all of our different ways of connecting wavelength, frequency, and energy. Now, given all of this, I encourage you to pause this video and see if you can figure this out. What is the longest wavelength of light that is capable of ionizing an atom of silver in the gas phase? All right, now let's work through this together. So the first thing to do is try to figure out the first ionization energy per atom. And so maybe I'll write it like this. So the energy per atom, the first ionization energy per atom is going to be equal to the ionization energy, 7.31 times 10 to the fifth joules per mole times what if we wanna figure out per atom? This is per mole. Well, how many moles are there per atom? Well, we know that they're this number of atoms per mole, so if we wanna know moles per atom, it's going to be one mole for every 6.022 times 10 to the 23rd atoms. I could write atoms here, and then that would give us joules per atom, but we're just gonna get the answer in terms of joules 'cause the moles are going to cancel out. And so this is going to give us approximately... Let's see, we have three significant figures here. 7.31 times 10 to the fifth divided by 6.022 times 10 to the 23rd power is equal to, and we have three significant figure here, so 1.21, approximately equal to 1.21 times 10 to the -18. 1.21 times 10 to the -18, and the units here are joules. This is joules per atom. So now how do we figure out wavelength? Well, as I alluded to, we might wanna use these equations here. We know that the speed of light is equal to the wavelength of that light times the frequency of the light. This is the lowercase Greek letter nu. This is not a V right over here. So if we wanna solve for wavelength, we just divide both sides by frequency. And so you get the wavelength is equal to the speed of light divided by frequency. But how do you figure out frequency from energy? Well, that's what this top equation gives us. Energy is equal to Planck's constant times frequency. So if you wanna solve for frequency, divide both sides by Planck's constant. So that top equation can be rewritten as frequency, I'll write it here, frequency is equal to energy divided by Planck's constant. And so we could take this and substitute it over here, and we would get that our wavelength is equal to the speed of light divided by energy divided by Planck's constant, or we can just rewrite this as being equal to the speed of light times Planck's constant divided by, try to keep the colors consistent, divided by energy. Well, we know what the speed of light is. It is 2.998, or it's approximately 2.998, times 10 to the eighth meters per second. We're gonna multiply that time Planck's constant, which is 6.626 times 10 to the -34th joule seconds. And then we're going to divide that by the first ionization energy per atom, which we figured out right over here. So we're gonna divide that, this E right over here. This is going to be, we figured it out, 1.21 times 10 to the -18 joules. Now let's make sure all the units work out. So this seconds is going to cancel out with this seconds. This joules is gonna cancel out with this joules. And we're just gonna be left with meters, which makes sense. The wavelength can be measured in meters. And so let's just get our calculator out and calculate out what this is going to be. 2.998 times 10 to the eighth times 6.626 times 10 to the -34. And then I'm gonna divide that by 1.21 times 10 to the -18. I think we deserve a little bit of a drum roll. That gets us that. And let's see. If we have three significant figures is our smallest amount in this calculation, so we're gonna go to these three right over here. And so this is going to be 1.64 times 10 to the negative one, two, three, four, five, six, seven. So this is going to be approximately equal to 1.64 times 10 to the -7 meters. And we're done.