The radioactive isotope 100Tc decays to form the stable iso-tope 100Mo. (b) Only one of the pathways is observed. Calculate the energy released by both pathways, and explain why only one is observed. Relevant masses are: 100Tc = 99.907 657, 100Mo = 99.907 48, electron = 0.000 548 6.

Video transcript

Hello. In this problem, we are told the new Clyde from M 2 53 undergoes radioactive decay in which the stable new Clyde and studying to 53 is formed. There are two possible nuclear decay pathways for this process. Were asked to determine the energy in each of the reactions and then identify the reaction that will most likely occur. We asked to use the mask for from 253. I'm studying to 53 and that of an electron. Let's look at the I stopped notation for premium to 53 that for Einstein E um to 53 we see that there will be no change in the mass number represented by the capital letter A Our atomic number represented by the capital C will decrease by one. We want to consider radioactive decay with a Atomic number changing by one. one possibility is positron emission. So we have then a positron which is sometimes referred to as a positive electron. It has the same mass but opposite charge of electron. This results in no change to mass number. Our atomic number would decrease by one. So this is possible data decay we would have a loss of a high energy electron. This would result in no change in the mass number. The atomic number would increase by one. So this isn't a possibility and then we have electron capture so are capturing an electron. This would result no change in the mass number. The atomic number would decrease by one. So this is another possibility. So begin by considering positron emission in this case then our For me, I'm 2 53 Will decay form Einstein EM and emit a positron. First thing we wanna do is calculate our change in mass. So that just be the difference between the mass of our reactions and that of our products. Our reactant then includes Birmingham in nucleus which has a mass equal to that of a from him 2 53. Adam minus the mass of its electrons which is 100. And we have then the Einsteinian nucleus which has mass equal to And Einstein and Adam Minour massive its electrons which are 99. So finding our change in mass which is again reacting sinus products we have then the mass of the fermi um nucleus which is equal to that of the fermi um adam this mask the 100 electrons. And then we have massive and instant Einsteinian nucleus which is equal to that of adam -99 electrons. And then we have an additional positron which has a mass equal to that of electron on the product side. Start changing mass. Then sympathized too. The mass of for me and Adam minus the mass of a stunning madame. And our electrons simplified to -2 electrons. So our mask then of a fermi madam we were told was 253. 81 Atomic Mass Units. That of an Einstein am is 253.0848- atomic mass units. And the mass of two electrons is each electron is 0.0005486 commit mass units. So our change of mass then works out to negative 0.000737 atomic mass units. Our next step then we want to calculate changing energy. Using the relationship between mass and energy. So we're gonna convert our change in mass from that of the mass. In terms of atomic mass units per atom, two g per mole. One atomic mass unit has a mass of 1.66054 times 10 to the minus 24 g and one mole Contains 6.022 times 10 to the 23rd atoms. So atomic mass units cancels. Adams cancels and left with grams per mole. So changing masses negative 0. g per mole. And now calculate our change in energy. Our change in energy then is equal to our change in mass times the speed of light, which is three times 10 to the eight meters per second. We're gonna square that. We will convert our grams two kg and they make use of the fact that one jewel is equal to one kg meter squared per second squared. We will calculate or convert our jewels to kill jules. So our units of grams cancels meters squared cancels second squared cancels kilograms cancels and jewels cancels. So we're left with killed jules for mole. So are changing energy Is equal to negative 6.63 times 10 to the seven joules per mole. So we have then our change in mass or positron emission is negative in sign and our change in energy is also negative and signed. Next we'll consider then electron capture an electron capture For me, I'm 2 53. Then we'll capture an electron the form and stony um 2 53. So the first thing we wanna do is again calculate the changing mass. So you have the mass of our reactant minus the mass of our products. So our reactant then we have the for me um nucleus which adds a mass equal to that of the fermi madam minus the mass of 100 electrons. And we've now captured an electron. So we have the mass of an additional electron on the reacting side. And then we're gonna subtract off. Then the mass of a Einstein E um adam nucleus. And so that's the mass of a Estonia madam. When is the mass of 99 electrons. So we see that our number of electrons will cancel start changing mass then is the mass of our and for me Madam minus the mass of in Einsteinian adam we were told the massive for me madam is 253. atomic mass units. And that oven Einsteinian madam is 253.8 Or - one Atomic Mass Units. Start changing mass, works out to 0.000360 Atomic mass units. And now calculating the changing energy. Can I convert our changing mass from Atomic mass units for Adam two g per mole? When atomic mass unit has a massive 1.6605, 4 times 10 to the -24g and one mole, maintain six point oh 22 times 10 to the 23rd. Adams. R Adams cancels our atomic mass units, cancels and we're left with grams per mole. So our change in mass works out to 3.60 times 10 to the -4 g per mole. Our changing energy that is equal to our change in mass times the speed of light squared, convert our grams to kilograms. Make use of the fact that on jewel is equal to a kilogram meter squared or second squared. And convert our jewels to kill jules. Our units of grams cancels meters squared, cancels, second squared cancels kilogram cancels and jewels cancels. So you're left with killed joules per mole. This works out to 3.24 times 10 to the to kill jules per mole. So our changing energy with a electron capture is equal to positive value comparing these results to again, positron emission. We call that for the positron emission, we had the change in mass equal to negative 0.737 atomic mass units, and the change in energy was equal to negative 6.63 times 10 to the seven joules per mole. And so, given that for positron emission, we have a decrease in mass accompanied by a release of energy. This is then most likely to occur compared to electron capture. Thanks for watching. Hope. This help.