Each statement that follows refers to a comparison between two radioisotopes, A and X. Indicate whether each of the following statements is true or false. (a) If the half-life for A is shorter than the half-life for X, A has a larger decay rate constant.

Video transcript

welcome back everyone. We need to consider to radioactive isotopes M and N. And determine whether the following statement is true or false. So if M has a greater decay rate constant, it will have a longer half life than N. What we should recall is that our half life for first order kinetic reactions is equal to 0.6 93 divided by our rate constant K. So this is our first order kinetics, which is what we're assuming here Half life. And we want to isolate to solve for our rate constant K. So we can determine whether are given statement is true or false. So we would isolate K by having K equal to .693 divided by our half life. And sorry, this is T 1/2 here. And we can see that we have an inverse relationship between our rate constant and our half life since it's in the denominator. And because we have an inverse relationship as our rate constant increases, we would notice a decrease in our half life or as our rate constant decreases, we would notice an increase in our half life. And so going back to our statement from the prompt, if isotope M has a greater decay rate constant K. Then it would therefore have a longer half life than N. And because we have associated that our rate constant has an inverse relationship with half life meaning that as the rate constant increases, half life should decrease. We would correct this to say that it would have a shorter half life than isotope N. Meaning that the given statement is going to be false and so false would be our final answer. To complete this example. I hope everything I reviewed was clear. If you have any questions, please leave them down below and I will see everyone in the next practice video.