![]() ![]() During life, any organism exchanges carbon with its environment. One important method of dating fossil remains is to determine what portion of the carbon content of a fossil is the radioactive isotope carbon-14. SOLUTION: If P dollars are invested for t yr at 5%, the amount will grow to A = Pe 0.05 t in t yr. (See Figure N9–7b.)Īt a yearly rate of 5% compounded continuously, how long does it take (to the nearest year) for an investment to triple? (8) As a beam of light passes through murky water or air, its intensity at any depth (or distance) decreases at a rate proportional to the intensity at that depth.Įach of the above four quantities (5 through 8) is a function of the form ce −kt ( k > 0). (7) It is common for the concentration of a drug in the bloodstream to drop at a rate proportional to the existing concentration. (6) If P is the present value of a fixed sum of money A due t years from now, where the interest is compounded continuously, then P decreases at a rate proportional to the value of the investment. (5) Radioactive isotopes, such as uranium-235, strontium-90, iodine-131, and carbon-14, decay at a rate proportional to the amount still present. Where k > 0 if y is increasing and k 0 if it decays (or diminishes, or decomposes), then k 0). We list several applications of each case, and present relevant problems involving some of the applications.Īn interesting special differential equation with wide applications is defined by the following statement: “A positive quantity y increases (or decreases) at a rate that at any time t is proportional to the amount present.” It follows that the quantity y satisfies the d.e. In each of the three cases, we describe the rate of change of a quantity, write the differential equation that follows from the description, then solve-or, in some cases, just give the solution of-the d.e. ![]() We now apply the method of separation of variables to three classes of functions associated with different rates of change. "Radiogenic Isotope Geology." United Kingdom: University Press, Cambridge.Calculus AB and Calculus BC CHAPTER 9 Differential Equations KAERI Chart of the Nuclides has lots of half-lives you can look up.The Radioactive Decay page has a much more detailed and thorough description of radioactive decay, including situations with more complex decay chains. ![]() There are many steps along the way, but they all have very short half-lives compared to that of U-238 so they can be ignored on these long (geologic) time scales. The decay of Uranium-238 eventually leads to Lead-206. Notice that the sum of these two curves is always 1.0.įigure 2. $$-\frac N\\Īn example of Uranium-238 (with a 4.47 billion year half-life) decaying into Lead-206 is shown in Figure 2, where we assume \(D_0 = 0\). If you had 10 jumping beansĪnd saw one jump every second, you’d expect to see about 10 In a sample is proportional to the number of radioactive atoms in the We start by noting that the speed of radioactive decays occurring Try out our Interactive half-life calculator! Number of atoms at any given time WARNING: there is a little bit of calculus involved. That deal with chains of nuclides, but here we only worry about theīasics. There are many general forms of the equation ThisĮquation allows us to figure out how many radioactive atoms are leftĪfter any amount of time. This page derives the basic equation of radioactive decay.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |