Ex: Exponential Model - Determine Age Using Carbon-14 Given Half LifeDifferential equations form a large subject by itself. It can be considered as an extension of "Integral Calculus" too. Here we study elementary differential equationsthe reason is that you learn integration better and also you find immeditate real-world applications which are fascinating. We shall not use partial differentialsso they are called 'Ordinary' DE's. Let us take a simple example of a DE: the rate of change of a variable or derivative depends on the value of the variable at that instant. This is the key to DE's.
This production compensates the decay. And therefore the ratio in living beings will be the same like the one of the earth atmosphere since our metabolism is taking in carbon of that particular ratio at any time.
Until we die. And here comes the decay handy: With the death of an organism no more is replaced. And if we know the decay-function the function that describes the time dependency of the decay we can determine how long ago this organism died — means we can determine its age!
What is left is to determine the value of k.
Since we know now the exact law for decay, we are able to determine the age of the fossil immediately. A nice feature of the method is that we do not need to know the absolute measurement of but only a relative one. Other radionuclides are used to date minerals or even water.
Although their methods take much more parameters into account than we will do here. Problem : A copper ball with temperature 0 C is dropped into a basin of water with constant temperature 30 0 C. After 3 minutes the temperature of the ball has decreased to 70 0 C. When will it reach a temperature of 31 0 C? This is a very simple case because copper is a very good conductor and we have a highly symmetrical shape.
T temperature, is the temperature of the surrounding medium, and k is a constant.
This law simply says that the speed of cooling is proportional to the temperature difference between the body and the surrounding medium. This makes sense: I am expecting to see something to chill off faster when I drop it into ice water than if I would drop it into hot water…. I call the integration constant because I want to use later on …. And we found a general solution for our problem. Note: there is a like one of many general solution but the particular one….
Get the value of k from 2 nd condition:. Now we have the solution function for our copper ball and we can go ahead and answer the question when it will reach a temperature of 31 centigrades:.
Archaeologists use the exponential, radioactive decay of carbon 14 to estimate the death dates of organic material. The stable form of carbon is carbon 12 and the radioactive isotope carbon 14 decays over time into nitrogen 14 and other particles. Carbon is naturally in all living organisms and is replenished in the tissues by eating other organisms or by breathing air that contains carbon. At any particular time all living organisms have approximately the same ratio of carbon 12 to carbon 14 in their tissues.
As a side note, Carbon is an important tool in the archeological research known as radiocarbon dating. Example: A radioactive isotope has a half-life of 16 [Differential Equations] [First Order D.E.] · [Geometry] [Algebra] [Trigonometry ]. Archaeologists use the exponential, radioactive decay of carbon 14 to Radiocarbon dating can be used on samples of bone, cloth, wood and plant If the fossil has 35% of its carbon 14 still, then we can substitute values into our equation.
When an organism dies it ceases to replenish carbon in its tissues and the decay of carbon 14 to nitrogen 14 changes the ratio of carbon 12 to carbon Experts can compare the ratio of carbon 12 to carbon 14 in dead material to the ratio when the organism was alive to estimate the date of its death.
Note that doubling time is another way of expressing the growth constant,'k'. You can convert dougling time into k and then proceed. For exponential decay problems,we use the term "Half Life" --the time it takes for the variable to dacay to half its original value. Example 4 The half life of decay of certain species of beetles was found to be 21 days. If a laboratory had beetles, find out when the popualtion of beetles will reduce to Check: We can check the result using the half-life data.
After 21 days, the population will decrease to beetles and after another 21 days,it will decrease to beetlesso, a total of 42 days! Example 5 Drosophila or fruit flies are used often in Genetics research laboratories. The number of beetles double every 1. Example 6 Madame Marie Curie extracted one gram of Radiuma radioactive element which decays into other elements,from a ton of Pitch Blende, a mineral of Lead and Uranium,[from Joachimstal mines] by a tedious chemical crystallisation process.
It had decayed about 0. Example 7 John uses calculus to track his sales record of new cell phone models. He finds that the sales have doubled in 42 days. If the original sales quantity is cell phones per day, find the likely sales at the end of the quarter, after 90 days. For 90 days, it is more thanThe half-life of this unstable isotope is years.
When a plant or a tree is living ,the C14 content is constant. When it dies, the C14 content slowly decays and decreases. By comparing the C14 of a fossil sample with that of a living plant today,we can find the date of the fossil. Find the age of the sample. Find an exponential model for both the cities and predict the population figures for the year If a medical clinic buys 10 grams of this isotope, how much of Cobalt 60 will be present after 2 years.
Using a exponential decay model for the demand ,predict the demand for In many cases, the growth variable reaches a steady value after a long time,not increasing exponentially for ever.
Then the fish popualtion reaches a nearly maximum value ,called 'saturation value'. Find the population equation and estimate the crab population for the third year.
The population increases from and slowly reaches nearly after a long time [asymptotically]. Draw the graph of P versus t in a graph paper on in a graphing calculator and check your answer for P 3.
Example 10 Newton's Law of Cooling Suppose you have a hot mug of coffee in your hand at temp of deg F. The surrounding temp of air is 60 deg F. Then the coffee mug cools slowly, the temperature decreases exponentially and reaches the surrounding temp of 60 deg F. Newton found that ,as a first apporximation, the rate of cooling is proportional to the temperature difference between the mug and the surrounding. This is called "Newton's Law of Cooling.Exponential Equations: Half-Life Applications
Try this with graph paper or graphing calculator. Parachute Fall: A paratrooper drops from an aircraft and while falling,experiences air resistance [due to his parachute and his body] a drag force which slows his speed.
The drag or air-frictional term is 'kv', where k is a constant. We assume that this drag force is proportional to velocity v--a good assumption for low speeds. The answers will be very approximate. This method serves to illustrate the DE approach to such problems. The average energy burned by a person in the body is about This equation is exactly similar to the equation we had for Newton's law of coolingExample It loses its mass in burning at the rate of 1 kg per second.
20.6: The Kinetics of Radioactive Decay and Radiometric Dating
The solution is : The solution is of the general form : logistic-growth function You will come across this type of 'growth function' in many situations;medical research, business growth,chemical processes and so on.
This curve is also called. The spread of the virus among the students was given by the sigmoidal curve: where y is the number of infected students at time t.
Find the number of students infected after 5 days. Plot the graph of y versus t. After 5 days :. Example 2 The sale of mobile phones in a city increase slowly for the first six months and then rapidly rises for thenext two years,tapering off to a total of 0.
Find the growth rate after 1 year and the max growth rate. Draw the curve of y vs t.
Separation of Variables Method We have used this method in previous examples without giving it a name because it was 'obvious' to do. Now we shall see more difficult examples.
of an ordinary differential equations of the second order, if the hypothesis there is organic material present, radiocarbon dating is a universal and most popular. The Kinetics of Radioactive Decay and Radiometric Dating .. either the differential rate law (Equation ) or the integrated rate law. Knowing that Carbon half-live is ? = years, date the remains. Solution: Set t To construct a model means to find the differential equation that takes into .
In this method, we try to separate the y terms and bring them to the left side and all the x terms to the right. Then proceed to integrate. Step 2: Multiply both sides by the integrating factor: F Then the solution:.
Radiocarbon dating differential equation
The current flowing in the circuit is given by the DE:. IR is the voltage drop across the resistance.
We continue with some practical examples: Modeling: Separable Differential Equations. The first example deals with radiocarbon dating. This sounds highly. First, we can solve the differential equation. Since m has This fact is used in radiocarbon dating to determine the age of fossils up to 60 years old. Roughly. A brief tutorial on basic elementary differential equations with several Example 8 Carbon Dating in Archaeology:[This technique is due to Willard Libby.
Mathematical modellers are fond of writing a system of first order equationstwo or more interconnected differential equations. These are called "Dynamic systems. Here ,for illustration,let us consider Epidemic modelsa subject of ' Epidemiology. We can write three first order diff equations for this situation: Since susceptibles decrease by becoming infected,we have a negative factor.
I 1 The change in infectives is due to people getting infected from S and the loss of infectives due to death or isolation:. How to solve this set of diff equation? Firstly, we will consider only equations 1 and 2.
We can use the difference equation or Euler's method to solve them. Take the initial value of S and I. We increment the previous valus of S and I by the changes: It is easy to do the caluclations with a tabular format:. Day 0. This is typical of many epidemics