The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. How long will it take for 20% of the U-238 atoms in a sample of U-238 to decay?

Video transcript

Welcome back everyone to another video. The half life for the radioactive decay of uranium 238 is 4.5 billion years and is independent of initial concentration. How long will it take for 18% of the uranium 238 atoms in a sample of uranium 238 to DK? And we are given for answer choices. A 29 billion years. B 58 billion years, C 1.3 billion years and D 3.2 billion years. So first of all, let's understand something relatively simple in this problem, we're given a radioactive decay. And the problem also gives us a second hand, which essentially stays the same thing. It's the half life is independent of the initial concentration, which means that we are dealing with the first order kinetics. So even if we don't know that a radioactive DK follows the first order kinetics, we know that according to the problem, half life is independent of the initial concentration, meaning it's the first order of kinetics. And we can define that as the natural logarithm of the mass of the sample at time T divided by the mass of the sample initially is equal to negative rate constant multiplied by time. And we also know that the rate constant is equal to natural logarithm of two divided by half life T one half. So now what we're going to do here, we're going to understand one important thing. The mass at time T can be expressed as some fraction omega multiplied by the initial mass. In other words, we can always express mass at some T as the fraction of the initial mass. So our formula reduces to Ln of omega multiplied by the initial mass divided by the initial mass is equal to negative K. What is K? Well, it's Ln of two divided by half life. And then we multiply that by time. Now we clearly see that the initial mass cancels out. And now the question is, what is the omega? Well, first of all, it says that 18% decay, right? So what is the remaining mass? Well, essentially 100% minus 18% this gives us 82% remaining, right? So essentially, we can say that mt mass at the time of interest is equal to 0.82 of the initial mass because our omega is given in a decimal form. So we have our omega, we have the fraction remaining, we still have 0.82 of the sample remaining. We have our half life, we want to identify time. So now if we rearrange the formula, our time would be equal to negative half life multiplied by the Ln of Omega divided by a line of two. We have our final formula, we are ready to substitute the givens and we'll be done. So we have a negative 4.5 billion years, right? We can say negative 4.5 multiplied by since the ninth years. Now we multiply by the Ln of Omega. Omega is 0.82. That's the decimal expression or based on the fraction of the sample remaining. And now we have a lot of two in our denominator. If we do the math, we can say that the final answer is 1.3 multiplied by 10th, the ninth years. So basically we have 1.3 billion years which corresponds to the answer choice. C Thank you for watching.