Potassium ion, K+, is present in most foods and is an essen-tial nutrient in the human body. Potassium-40, however, which has a natural abundance of 0.0117%, is radioactive with t1/2 = 1.25 x 10^9 years. What is the decay constant of 40K? How many 40K+ ions are present in 1.00 g of KCl? How many disintegration/s does 1.00 g of KCl undergo?

Video transcript

Hello everyone today. We have the following problem. Calcium is an essential mineral required by the body. One isotope of calcium, calcium 48 with an abundance of 480.187% is radioactive with a half life of 6.4 times 10 to the 19th years, calculate the rate constant for calcium 48 the number of disintegration per day in 1.5 metric ton calcium chloride. So first we need to get this half life from years two days. So we're gonna take our half life which was 6.4 times 10 to the 19th years. We're gonna multiply by the conversion factor that in one year we have 365 days When our units of years cancel out. We'll be left with 2.336 times 10 to the 22nd days or day. We'll save that number for later. Next we're going to solve for our constant. Our constant is going to be K equals to the natural log of two divided by our halftime or a half life. That we just calculated four in days. So that's going to be the natural log of two divided by our 2.336 times 10 to the 22nd days. And that is going to give us 2.967 times 10 to the negative 23rd days raised to the -1. And so we'll circle that number there and keep it for later. Next we're going to calculate the number of calcium Adams which can be represented by N. And so to do that first, we need to find the abundance that we have, Which was said to be .187%. We're gonna divide that by 100 so that we can get the approximate number. That's going to be 1.87 times 10 to the negative third. And now we can solve for it. So we have in or a number is going to be equal to the abundance or the amount that we have, which is 1.87 times 10 to the negative three. We're gonna multiply by that 1.5 metric ton. And we're gonna use the conversion factor that in one metric ton We have 10 to the third kg. We're then going to convert those kilograms into regular grams by using the conversion factor that one kg is equal to 10 to the third grams. We're then gonna multiply by the molar mass of calcium chloride, which if you take the molar masses of one calcium and two chlorides chlorine from the periodic table, you will get 110.98 g. What? And then lastly, we're gonna multiply by the multiple ratio that in one mole of calcium chloride we have only one calcium or one mole of calcium. Two plus. And so ultimately all of our units will cancel out except for the number of ions that we have. And lastly After. So after we calculate for our multiple ratio, we're actually going to take this equation down here a little bit and to get rid of our moles of calcium, we're going to say that in one mold of calcium we're going to multiply this by a big number or 6.2 times 10 to the 23rd Adams. And now we can officially say that we have won five times 10 to the 25th calcium two plus ions. Lastly, what we're gonna do is we're gonna multiply by our rate. We're gonna find the rate and that's gonna be found by taking a constant and multiplying the number of atoms we have, R. K. Constant, which we saw earlier was 2.9796, 7 times 10 to the negative 23rd days. Multiplied by the number. We just found a number of ions here which was 1.5 times 10 to the 25th calcium two plus ions. And in solving for this, we're going to get 450. This integration per day and this is going to be our final answer. Overall, I do hope this helped until next time