Silver is composed of two naturally occurring isotopes: Ag-107 (51.839%) and Ag-109. The ratio of the masses of the two isotopes is 1.0187. What is the mass of Ag-107?

Video transcript

Welcome back everyone to another video, silver is composed of two naturally occurring isotopes. Silver 107 51.839% and silver 109. The ratio of the masses of the two isotopes is a 1.0187. What is the mass of silver of 107? We are given four answer choices. A 106.69 B 106.91. D 107.32 and D 107.19. All of these are given in atomic mass units. So we can begin solving our problem by understanding the information given in this problem. We have two unknowns. We once identify one of them. Let's suppose that the mass of silver 107 is X. And let's suppose that the mass of silver 109 is Y according to the problem, the ratio of their masses is 1.0187. So we can set up our first equation and we can say that 1.0187 is equal to the ratio of the two masses. And because we have a number that is greater than one, the heavier one goes into the numerator, the heavier one is 109. So that would be Y divided by X. We have got our first equation. So let's label it as equation number two. And let's proceed and identify the second equation. The second equation will come from the weighted average formula. And that equation tells us that if we have an element that consists of two isotopes, we can find that its atomic mass by taking the weighted averages of the two isotopes, we are taking the mass of the first isotope multiplying by its abundance in a decimal form. And we are adding the mass of the second isotope multiplied by its abundance in a decimal form. So we're going to use this formula. And for the purposes of the equation, we need to look up the mass of silver from the periodic table. If we look at the periodic table, we notice that the mass or the atomic mass of silver is 107.87. So we substitute this in. Now, the mass of the first size of toe is X, right? And we need to multiply by the abundance to 0.51839 multiplied by a notice how we have converted our percentage into a decimal form. And now we are going to use the second abundance. We don't know what it is. But we know that we can essentially subtract one minus 0.51839. The reason why we're doing this is because we only have two isotopes. So the two decimal forms of the percentages, they must add up to one. And now we multiply by the mass, which is why. And that would be our second equation. It doesn't look too good yet. But we are happy to see two equations and two unknowns because whenever we have two equations and two unknowns, we can identify each unknown. Now, we have to understand that we are looking for X. So from the first equation, we can state that X is equal to Y divided by 1.0187. So we have our expression for X and now we can calculate the result or our subtraction and we can simplify the second equation as follows. 107.87 must be equal to 0.51839 X plus. Now, if we subtract, we get 0.48161 and now we plug in our Y, right? Essentially we want to plug in our Y in terms of X. So in reality, we actually need to rearrange our previous equation and state that Y is equal to 1.0187 X. So we can now plug in for Y, meaning we're now multiplying by 1.0187 X. And now we simply need to combine the like terms and solve the equation for our unknown. Now, what we end up with is 0.51839. Now we need to multiply the two numbers. Let's see what we got. The result that we get is 0.490616. Let's add an additional significant figure here. That's X and now let's add them together. 107.87 is equal to. Now, if we add two numbers together, the two numbers together that we had previously, where we have two axes, we get a number 1.009 01 X. And finally, if we want to solve for X, we take 107.87 and divide that by 1.00901. Now let's calculate the result that we get here. That would be 106 point 91 atomic mass units. So based on our answer, we can essentially conclude that the correct answer choice to this problem is B 106.91 atomic mass unit. Thank you for watching.