Mass defect and binding energy

let's say once you calculate the mass of a helium four nucleus well first would to figure out what's in the nucleus so with an atomic number of two we know there are two protons in the nucleus and subtract the atomic number from the mass number four minus two gives us two neutrons and so if we know the mass of a proton and the mass of a neutron we could easily calculate the expected mass of a helium four nucleus and the mass of a proton in a.m. use atomic mass units is equal to one point zero zero seven two seven six four seven and we have two protons so we need to multiply this number by two so let's go ahead and get out the calculator and let's do that so one point zero zero seven two seven six four seven times two gives us two point zero one four five five two nine four so this is equal to two point zero one four five five two nine four remember these are a.m. use atomic mass units a neutron one neutron has a mass in a.m. use of one point zero zero eight six six four nine zero and we have two neutrons so we have to multiply this number by two so let's go ahead and do that so we have one point zero zero eight six six four nine zero times two and this gives us two point zero one seven three two nine eight so two point zero one seven three two nine eight am use so the mass of a helium nucleus if we add those two numbers together we should get that mass so let's do that math so this number plus our first number two point zero one four five five two nine four it gives us four point zero three one eight eight two seven four so four point zero three one eight eight two seven four our AMU so this is the predicted mass of the helium-4 nucleus so let me go ahead and write this this is the predicted mass the actual mass of a helium four nucleus has been found has been measured to be four point zero zero one five zero six zero eight am use let me go ahead and write this this is the actual mass so the actual mass all right there's a difference there they're not the same number the predicted number is higher than the actual mass so let's calculate the difference between those two numbers so if we subtract the actual from the predicted we can see the difference between those numbers so let's go ahead and do that all right so we have the predicted and we're going to subtract the actual from that so four point zero zero one five zero six zero eight right is going to give us point zero three zero three seven six six six AMU so let's go ahead and write that so zero point zero three zero three seven six six six amu so this is the difference between those two numbers and we call this the mass defect so let me go ahead and write that down here this is called the mass defect the difference between the predicted mass of the nucleus and the actual mass of the nucleus and it looks like we lost some mass here all right and really what's happened is the mass that that mass this mass right here the mass defect was converted into energy when the nucleus was formed so that's pretty interesting and we can calculate how much energy according to Einstein's famous equation which relates energy and mass all right so this is the one that most people know it's e is equal to MC squared right so E is equal to MC squared where ear AFER's to the energy in joules m is mass in kilograms and C is the speed of light which is in meters per second and so you'd be squaring that so be meter squared over second squared all right so let's calculate let's calculate the mass that we're dealing with here so using Einstein's equation we see we need kilograms and we've calculated the mass in AM use and so the first thing we need to do is convert the a.m. use into kilograms and I briefly mentioned in an earlier video the conversion factor between a.m. use and kilograms so M uses just a different measure of mass let's get some more room down here alright so 1 amu is equal to one point six six zero five four times 10 to the negative 27 kilograms and so the first thing we need to do is convert that number so let's go ahead and write it down so zero point zero three zero three seven six six six am use so mathematically how would I convert the am use into kilograms well I need to cancel out the AMU units so my conversion factor is going to be one point six six zero five four times ten to the negative twenty-seven that's how many kilograms we have for every one amu so I can put that in here so there's my conversion factor and notice what happens when we do this our units for am use cancel and it's going to give us kilograms so let's go ahead and do this math so what is this equal to all right so we get out the calculator and we take this number and we multiply it by let's use some parentheses here one point six six zero five four times ten to the negative twenty-seven so let's see what this gives us here so this gives us five point zero four four one seven times ten to the negative twenty nine so let's round it like that so this is going to give us let's put it right here five point zero four for 1/7 times 10 to the negative 29 kilograms all right so now we have the mass in n kilogram so let's get some more room and let's go ahead and calculate the energy all right so this is the mass that was lost when the nucleus was formed and let's so let's figure out how much energy was given off so the total energy right energy is equal to that mass so let's go ahead and plug that in 5 point 0 4 4 1 7 times 10 to the negative 29 times the speed of light all right which is approximately 3 times 10 to the 8th meters per second we'll use a more exact number 2 point 9 9 7 9 2 times 10 to the 8th and we need to square that number all right so let's do let's do our last calculation here so let's let's start with the speed of light so 2 point 9 9 7 9 2 times 10 to the 8 and then we need to square that number so we get this number and we're going to multiply that by our mass 5 point 0 4 4 1 7 times 10 to the negative 29th and so let's see what that gives us 4 point 3 3 4 6 times 10 to the negative 12 so let's write that down so 4 point I think there was a 5 in there 4 point 5 3 3 4 so 4 point 5 3 3 4 6 times 10 to the negative 12 let me just double-check that real fast here so 4 point 5 5 3 4 6 times 10 mega 12 is our answer and the unit should be joules so that's how much energy is given off so here's our final calculations that took us a while to get there so this is the remember this is the energy that's released when the nucleus is formed so let me go ahead and get some more space and let me write this down the energy the energy released right when the nucleus is formed energy released when the nucleus is formed so let's let's draw a picture of what's happening so we were talking about the helium nucleus right which had to approach on so let me go ahead and draw those two protons in here and two neutrons so let me use a neutrons here like that and these came together to form our nucleus alright so these two things come together so we have we have our two positive charges in our nucleus and then we have our neutrons as well and this on the this is supposed to represent our nucleus our helium four nucleus here so energy is released when the nucleus is formed so we could also put in this energy right so this energy is given off so that's the number we just calculated so we spent we spent several minutes getting this number right and this is the energy that's released when the nucleus is formed here and so this is just a nice little picture to think about what's happening so whatever this nucleus formed energy was given off the nucleus is stable because energy is given off here and we could also think about going the opposite direction so if you started with the nucleus and you wanted to break it up into the individual components right so if you took this nucleus here and you applied some energy you could break it up and turn it back into protons and neutrons and that energy that you would have to apply is also equal to this energy all right so this is also called the nuclear binding energy so I'm going write that all right nuclear binding energy so again that is the term for the energy that we just calculated here so you can think about two different ways it's the energy that's released when the nucleus is formed and that's also the amount of energy that's needed to break the nucleus apart and and so the nucleus is stable in this case so we have a stable nucleus right this is a stable nucleus but that a little bit weird because we have these positive charges and we know that positive charges repel right so these two positive charges here are repelling each other right we know that like charges repel and so there must be some other force that's holding our nucleus together and that's called the nuclear strong force and we'll talk more about that in the next video