Question
THE HALF-LIFE OF A RADIOACTIVE ISOTOPES Question 2Explain what is meant by the half-life 0 an isotope Derive from first principles an expression for the half-life in terms of the decay constant (10)
THE HALF-LIFE OF A RADIOACTIVE ISOTOPES Question 2 Explain what is meant by the half-life 0 an isotope Derive from first principles an expression for the half-life in terms of the decay constant (10)
Answers
What is half-life and average life of radioactive material? Derive an expression for the decay.
In this video, we're going to be talking about the definition of half life, half life we denote by this symbol t 1/2. I'm so if you ever teat, if you ever see TV, 1/2 we're discussing half Life. Half life is the time it takes for half of a radioactive nuclei to decay or, in other words, to break down was important. To know about half life and nuclear decay is that after 1/2 life, the radioactive nuclei undergoes a chemical transformation into a new nuclei. So later on, you might learn about what's called beta decay or carbon dating, and that's all about half life breaking down and how a nucleus could break down over time.
In this problem, we need to determine the half life of radioactive material. Now let us assume that wife gives the amount of the radioactive material president after 30 years. Then the model for this will be Y equals two Y zero. Where y zero is the initial amount times half the power be divided by the half life, which we will assume to be equal to age. Now it is said that after one year, 99.57% of the initial amount remains. That means when he is equal to one, the value of what it will be 99.57% of the original amount Y zero. So we have 99.57 divided by 100 times Y zero legal Y zero times half to the power one divided by age. The value of P is one because this is considered after one year. Now we can cancel out why zero from both sides because Y0 will not be equal to zero, it is the initial amount. And using this weekend determine the value of each most of all this. Take the natural algorithm on both sides. So we have long 99.5 75 500 legal loan of half to the power one by eight. And using the properties of logarithms, this will be half times lawn of half, one x 8 times longer, half. So that means that the value of H will be lawn of half, divided by lawn 99.57, divided by hunger. And the value of that is approximately equal to 160.85. Hence the half life of the radioactive material will be approximately equal to 160.85 years.
This question will also describe what the half life is for the radioactive substance. So the half flights is the amount of time that is required for the number of radioactive isotopes present when it comes to a certain sample to reduce to exactly half off what it waas, uh, at the beginning off that time. So we usually symbolize this with the symbol T half. And here it's, uh, time zero. So the half life is the time you need for the number of radioactive isotopes to reduce to exactly, ah, half of what it waas when we began counting the time.
Okay. Hello. So the decay of try titanium is given by dysfunction A. F. T. Equal to a not times E. Raised to the negative 0.5 16. So we're gonna substitute um A. F. T. Equal to a not over to into the function. And then South 40. So we end up with a not over to one half of a knot is equal to a not Times E. raised to the negative 0.05 six T. So the knots and I'm cancelling out, we just get one half Is equal to E. to the negative 0.05 60. Okay so now we want to the natural log of both sides of the natural log of one half is equal to the natural log of E. to a power. But then the natural flow of power is just equal to that power. So um we end up here with just a natural log Of one half is equal to negative 0.05 six T. So to sell for t we just divide through by negative 0.56 So therefore T. Is going to be well approximately equal um to negative. Um Well it's equal to actually it's equal to negative 1/0 0.56 times the natural log of one half. So that's what T. Is actually equal to. But then T. Is going to be approximately equal to 12.38 This is approximately equal to 12.38. So therefore the half life of um Try titanium or try T um try to um um is approximately 12.38 years.