The rate for synthesis of ammonia from nitrogen and hydrogen can be determined from the solution of differential equations
Differential equations provide the medium for the interaction between mathematics (especially calculus) and various branches of science and engineering. The physicist, chemist, biologist, engineer, or social scientist models a process with a differential equation and then turns the equation over to the mathematician who tries to provide a solution which results in a complete description of the original process. Differential equations come in all levels of complexity and even today there is active mathematical research in differential equations. In this section, we will look at a couple of the simplest, yet most important, differential equations.
Radioactive decay: Over a given time interval, each atom in a sample of a radioactive element has the same (usually fairly or extremely small) chance for disintegration as any other. Thus, if we have a quantity N of radioactive atoms of one type, then we can expect twice as much decay if the sample size is doubled to 2N and three times as much decay if the supply is tripled to 3N. That means that the rate of decay (i.e., the rate at which the number N of radioactive atoms is decreasing) is proportional to the supply of atoms at any time. Symbolically, we should write
N´ (t) = -a N(t)
Chemical reactions: Gaseous dinitrogen pentoxide decomposes into nitrogen dioxide, NO_2, and oxygen (the reaction is ). The decomposition is caused by collisions of two dinitrogen pentoxide molecules. So doubling the concentration of the gas doubles the rate of decomposition (since the molecules are then twice as likely to interact), tripling the concentration triples the rate of decomposition, etc.. Thus, if C(t) is the concentration of dinitrogen pentoxide at time t, then C´ (t) is proportional to C(t). In other words,
C´ (t) = -a C(t)
Newton's law of cooling: Newton determined empirically that if a hot or cold object is left in an environment of constant temperature, that the rate of change of the temperature of the object is proportional to the difference between the object's temperature and the ambient temperature. In other words, if T(t) is the temperature of the object at time t, then
T´ (t) = a(T(t) - A)
The differential equation y'(t)=ay(t): In general, a differential equation gives a relationship between (among) a function (or functions) and its (their) derivatives (of various orders). To solve a differential equation means to find a function (or functions) whose derivatives obey the specified relationship. For example, the differential equation
y´ (x) = ay(x)
First of all, if y(x_0) > 0 for some x_0, then y´ (x_0) = ay(x_0) > 0, so y(x) is increasing as it goes through x_0. As x pushes on, the fact that y(x) is going up and the fact that y´ (x) = ay(x) means that y(x) goes up faster and faster (i.e., the graph of y is concave up) as x advances to infinity.
On the other hand, if y(x_0) < 0 for some x_0 then y´ (x_0) = ay(x_0) < 0, so y(x) is going down as x goes through x_0. As x pushes on, the fact that y(x) is going down and the fact that y´ (x_0) = ay(x_0) means that y(x) goes down faster and faster (i.e., the graph of y is concave down) as x advances to infinity.
As a result of this analysis, we see that y(x) is either always positive, increasing and concave up or always negative, decreasing and concave down (unless y(x) is dead zero, of course).
The rest of your reading for this section is contained in the exercises 2-13 to 2-18. You should go through the exercises to be sure you have understood the material covered so far. The next section begins applications of differential equations to specific science topics.
Next: The Growth of Populations Up: Describing How Things Change Previous: Derivatives at Work
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