Exponential growth. Laws of power in business

When Albert Einstein was asked to name the most powerful force in the world, he answered without hesitation: “Compound interest.”

To truly understand the nature and consequences of a long period of growth requires a genius. Experiments have shown that even educated people who are good at math tend to significantly underestimate the effects of growth. For example, in one study* subjects were asked to estimate the required productivity of a tractor factory that began operating in 1976 with production of 1,000 tractors per year, after which demand increased by 6 percent each year. How many tractors, they were asked, would the plant need to produce in 1990, 2020, 2050 and 2080? Typical answers were based on gradual linear increases, and therefore demand estimates before 1990 were fairly close to the correct answer. But subsequent numbers of correct answers jumped “exponentially,” while responders’ scores continued to be based on a steady increase. Most respondents answered that in 2080 the demand will be about 30,000 tractors, while the correct answer is about 350,000, which is more than 10 times more!

Now guess the riddle. In a pond with an area of ​​13 thousand square meters. feet, one water lily leaf floats, occupying an area of ​​1 square. foot. A week later there are already two leaves. In two weeks four. Calculate how long it will take for the water lilies to cover the entire pond.

In 16 weeks they will cover half the pond. Now tell me, how long will it take until the entire pond is covered with water lilies? It took 16 weeks for the water lilies to cover half the pond. But to cover the second half, one week will be enough, since the leaf area doubles every week. The final answer is 17 weeks.

* Cm.: ^ Dietrich Dörner. The logic of failure: why things go wrong and what we can do to improve them (Dietrich Dorner. The Logic of Failure: Why Things Go Wrong and What We Can Do to Make Them Right. 1996, Metropolitan Books, New York). The original was published in Germany in 1989 under the title "Die Logik des Misslingcns" by Rowohlt Verlag.

Do you remember the fable about the Indian king who wanted to reward the inventor of chess? The inventor asked for just a few grains of rice: put one on one cell, two on the second, four on the third, and so on on all the other cells. The king thought that the sage was being modest - until it turned out that just one last cell would have to put 9,223,372,036,000,000,000 grains, or about 153 billion tons, or more than two and a half million huge (60,000 tons each) dry cargo ships , filled to the very sides with rice. And it’s all due to “exponential” growth, in this case the doubling of rice grains on each cell.

^ What is the essence of exponential growth?

An exponent is a number that shows how many times a quantity must be multiplied by itself. For example, if the exponent is 3 and the magnitude is 4, then the expression 4 3 means 4 x 4x4, which is 64. Mathematical expression at 2 means at X at, A the number 2 is the exponent.

How does exponential growth differ from linear growth? With linear growth, the value increases at each stage by same thing ok and not on multiple number. If my starting capital is $1,000 and increases by $100 every year, then in 10 years I will double it and have $2,000. This is linear growth, by the same amount every year. But if my starting capital of $1,000 increases by 10 percent every year, then after ten years I will have $2,594. This is an example of exponential growth with a constant annual increase multiple of 1.1. If I continue my business for another 10 years, then linear growth will give me a total of $3,000, while exponential growth will give me a total of $6,727.

Any market or business that maintains a growth rate of 10 percent or more over an extended period of time will experience far greater value creation than we intuitively estimate. Some companies - such as IBM or McDonald's for the period from 1950 to

1985 or Microsoft in the 1990s - managed to achieve growth rates exceeding 15 percent per year and increased their capital many times over. If you start with $100 and grow your capital at 15 percent per year for 15 years, you'll end up with $3,292, nearly 33 times what you started with. A small increase in growth percentage makes a big difference in results.

For example, American stockbroker William O'Neill created a fund for his classmates and managed it from 1961 to 1986. During this time, the initial $850 turned into $51,653 after paying all taxes *. Over 25 years, the average increase was 17. 85 percent per year, which translated into a 61-fold increase in the original amount. Thus, we see that if over 25 years, 15 percent growth increases capital by 33 times, then adding less than 3 percentage points to the annual growth rate increases the result by 33 times. 61 times.

Exponential growth changes things not only quantitatively, but also qualitatively. For example, with the rapid growth of the industry - Peter Drucker puts the figure at 40 percent in 10 years - its very structure changes, and new market leaders come to the fore. Rapid growth of markets is driven by innovation, lack of patterns, new products, technologies or consumers. Innovators, by definition, do things differently than everyone else. New ways rarely coexist with the habits, ideas, procedures and structures of existing firms. Innovators often have the opportunity to skim off for several years until traditional leaders decide to launch a counterattack, but then it may be too late.

^ Fibonacci Rabbits

I would like to offer you an interesting riddle on the topic of exponential growth. In 1220, Leonardo of Pisa, who received the nickname “Fibonacci” 600 years later, came up with the following:

* ^ William J. O'Neil. How to make money on exchanges ( William J. About "Neil. How to Make Money in Stocks. 1991, McGraw-Hill, New York. P. 132).

real scenario. Let's start with a couple of rabbits. Then imagine that each couple gives birth to another pair a year later, and another one a year later. After this, the rabbits become too old to breed. How will the number of pairs increase, and is there anything great about this model?

If you want, you can sequence the annual number of pairs yourself, but you can look at the answer right away:

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...

Notice anything unusual?

Strictly speaking, there are two interesting points here. One is that starting from the third, each subsequent digit is the sum of the previous two. The second is that the ratio of the number of each year (after the third) to the number of the previous one is an almost constant coefficient, which soon approaches 1.618. In other words, there is a constant growth rate of just over 60 percent.

Over time a mystery ^ Rabbits Fibonacci received a comprehensive mathematical explanation, but, fortunately, there is no place for it here*. However, these rabbits are a great illustration of exponential growth, as well as the fact that even such apparently limited growth cannot continue for very long. In 144 years, the volume of Fibonacci rabbits will exceed the volume of the Universe, and all people will die, suffocating under the fluffy mass. This is really far-fetched!

^ Big Bang

Another, more extreme form of exponential growth may underlie the origin of the universe. Nowadays, almost all astronomers and physicists agree with The Big Bang Theory, according to which the universe began

* Mathematics enthusiasts may want to look through Peter M. Higgins's book "Math for the Curious" (Peter M. Higgins. Mathematics for the Curious. 1998, Oxford University Press, Oxford).

from an unimaginably small volume, and then in a split second doubled its size 100 times, making it look like a small grapefruit. This period of “bulging,” or exponential growth, then ended, giving way to linear growth, during which an expanding fireball created the Universe today.

Exponential growth is an integral part of creativity of any kind. The interesting lesson is that with exponential growth, you don't have to start with something big. In fact, you can start with the smallest things. If the Universe could begin with something so small that we cannot imagine it, and expand to its present unimaginably infinite size, then the factor of the initial size of the new business should be considered completely irrelevant. The key indicator is a period of exponential growth followed by a longer period of linear growth.

^ Conclusions from the concept of growth

The best opportunities for creativity and growth occur during periods of disequilibrium, or in other words, when a tipping point is reached and immediately after.

Disequilibrium and tipping points do not occur suddenly. There is always a period, sometimes quite long, of preliminary warm-up, when the existing system shows signs of instability, and the new one quietly gains strength. In everything that concerns new technologies or types of products, the tipping point is reached only after the innovation receives a “registration” on the mass market. This means that its sale must be based on traditional criteria of profit, and the revolutionary nature of the change (if there is one) must be camouflaged.

Periods of rapid change and high exponential growth usually do not last long. It will not be long before a new equilibrium is established with a new dominant technology and/or a new competitive situation. Hence the feeling of fascination and unusual uncertainty associated with periods of imbalance. Hence the exceptional benefits that people who managed to seize dominant positions in this short period derive. This dominance is more a result of clever marketing and positioning than the superiority of the technology itself.

Most innovators fail. To achieve success, they must “cross the chasm”—or pass the tipping point—and penetrate the mass market. The key factor here is acceleration. Until a new product or technology begins to multiply rapidly, it has little chance of survival.

^ Say's Law of Economic Arbitration

In 1803, the French economist Jean-Baptiste Say (1767-1832) published a remarkable work, Treatise on Political Economy. Thomas Jefferson said this about her:

"An excellent work...brilliantly laid out, clear in ideas, clear in style, and the whole work twice as subtle as [Adam] Smith's book."*

The treatise contained many startling innovations, including the term "entrepreneur" and the first theory of economic arbitrage, formulated in the same sentence.

An entrepreneur moves economic resources from an area of ​​lower productivity to an area of ​​higher productivity and benefits from it.

Long before the concept of return on capital was popularized, Say identified it as one of the most important engines of economic creativity and progress. Resources are by definition limited, so growth depends less on the exploration and exploitation of natural resources than on the ability to more fully

* Thomas Jefferson in a letter to Joseph Milligan, April 6, 1816. This is an excellent article and I used it in my report.

efficient use of each unit of resource. This is partly a function of better technology and techniques, but the entrepreneur's ability to get those resources to where they will be most productive cannot be discounted.

^ Freud's reality principle

In 1900, Sigmund Freud (1856-1939) published The Interpretation of Dreams and founded the new science of psychoanalysis. One of his key concepts was The reality principle asserts that the only thing that keeps us from using other people for selfish purposes is that they seek to do the same to us. When faced with reality (reality), we are forced to adapt to the needs of other people and the demands of the outside world in order to be able to satisfy our own instincts.

Freud's concept certainly has great value, but a rather unexpected twist on the same idea was given by his contemporary, playwright George Bernard Shaw:

“The rational man adapts himself to the world [according to Freud’s reality principle]: the unreasonable man persistently tries to adapt the world to himself. Consequently, any progress depends on the unreasonable person."

Creativity and entrepreneurship need to be fueled by new ideas, new methods and unwise approaches. Was Henry Ford being reasonable when he insisted that automobiles should be accessible to the working man? It clearly did not follow demand, since the demand for cars existed only among the rich. Ford refused to accept the world that existed around him; he continued to try to adjust the world to his vision. Using an assembly line and maximum standardization, Ford reduced the cost of the Model T from $850 in 1908 to $300 in 1922 and succeeded in its mission to "democratize the automobile."

^ Successful Entrepreneur

The book of Genesis and the Big Bang theory agree on one thing: there was only one original creation of the world. Therefore, progress is just a rearrangement of terms. There is nothing new under the sun.

This point of view is by no means gloomy, and it is encouraging. All that human well-being needs is to take a certain set of resources and move them from areas of low productivity to areas of high productivity.

All economic progress is based on this type of economic arbitrage. It's a good news. It is easier to engage in arbitrage than creativity. Everyone should be able to come up with something that can benefit from economic arbitrage, from identifying resources that can be used more efficiently.

True entrepreneurs don't wait for market researchers to tell them what to do. They have their own vision of how to do something better and differently. They develop ways to achieve more with less effort. They exchange less profitable uses of resources for more profitable ones and continue to be persistent and unreasonable until the world accepts their point of view.

^ Law of Diminishing Returns

One of the most influential and popular concepts of how markets and enterprise work is Law of Diminishing Returns, which was formulated around 1767 by the French economist Robert Jacques Turgot.

The law states that after a certain point, the return on additional effort or investment decreases, that is, the increase in value decreases. For a hungry person, a loaf of bread is very valuable. The value of the second loaf is less. The tenth will no longer have almost any value. If you hire several additional farmers to cultivate one piece of land, then after a certain point the law of diminishing returns will come into play.

A hundred years later, British classical economists, led by Alfred Marshall, extended this idea to markets and firms. Market-leading products or companies become trapped in diminishing returns. The price of large size in business - large market share, large factory, wide variety - peaks and then declines. Well, that sounds quite reasonable.

But classical economists went further. They stated that sooner or later a predictable equilibrium of prices and market share would be reached and that fair competition in cooperation with the law of diminishing returns would ultimately lead to the impossibility of making excess profits. This theory justified government regulation of markets - if profits are very high, this means only one thing: monopolists are artificially inflating prices and preventing fair competition.

The expression “exponential growth” has entered our lexicon to mean rapid, usually uncontrollable, increase. It is often used, for example, to describe the rapid growth of cities or an increase in population. However, in mathematics this term has a precise meaning and denotes a certain type of growth.

Exponential growth occurs in those populations in which the increase in population (the number of births minus the number of deaths) is proportional to the number of individuals in the population. For a human population, for example, the birth rate is approximately proportional to the number of reproductive pairs, and the death rate is approximately proportional to the number of people in the population (we denote it N). Then, to a reasonable approximation,

population growth = number of births - number of deaths

(Here r- so-called proportionality factor, which allows us to write the proportionality expression as an equation.)

Let d N— number of individuals added to the population during time d t, then if in the population in total N individuals, then the conditions for exponential growth will be satisfied if

d N = rN d t

Since Isaac Newton invented differential calculus in the 17th century, we know how to solve this equation for N— population size at any given time. (For reference: this equation is called differential.) Here is his solution:

N=N0 e rt

Where N 0 is the number of individuals in the population at the beginning of the countdown, and t- the time that has passed since this moment. The symbol e denotes such a special number, it is called base of natural logarithm(and is approximately equal to 2.7), and the entire right side of the equation is called exponential function.

To better understand what exponential growth is, imagine a population consisting initially of one bacterium. After a certain time (a few hours or minutes), the bacterium divides in two, thereby doubling the population size. After the next period of time, each of these two bacteria will again split in two, and the population size will double again - there will now be four bacteria. After ten such doublings there will be more than a thousand bacteria, after twenty - more than a million, and so on. If the population doubles with each division, its growth will continue indefinitely.

There is a legend (most likely not true) that the man who invented chess gave his Sultan such pleasure that he promised to fulfill any of his requests. The man asked the Sultan to place one grain of wheat on the first square of the chessboard, two on the second, four on the third, and so on. The Sultan, considering this demand insignificant compared to the service he had provided, asked his subject to come up with another request, but he refused. Naturally, by the 64th doubling, the number of grains became such that there would not be enough wheat in the whole world to satisfy this request. In the version of the legend that is known to me, the Sultan at that moment ordered the inventor’s head to be cut off. The moral, as I tell my students, is: sometimes you shouldn't be too smart!

The chessboard example (as well as the imaginary bacteria) shows us that no population can grow forever. Sooner or later, it will simply run out of resources - space, energy, water, whatever. Therefore, populations can only grow exponentially for a while, and sooner or later their growth must slow down. To do this, you need to change the equation so that when the population size approaches the maximum possible (which can be supported by the external environment), the growth rate slows down. Let's call this maximum population size K. Then the modified equation will look like this:

d N = rN(1 — (N/K)) d t

When N much less K, member N/K can be neglected, and we return to the original equation of ordinary exponential growth. However, when N approaches its maximum value K, value 1 - ( N/K) tends to zero, and accordingly the population growth tends to zero. The total population size in this case stabilizes and remains at the level K. The curve described by this equation, as well as the equation itself, have several names - S-curve, logistic equation, Volterra's equation, Lotka–Volterra equation. (Vito Volt e RRA, 1860-1940 - outstanding Italian mathematician and teacher; Alfred Lotka, 1880-1949 - American mathematician and insurance analyst.) Whatever it is called, it is a fairly simple expression of the size of a population growing sharply exponentially, and then slowing down as it approaches some limit. And it reflects the growth of real populations much better than the usual exponential function.

EXPONENTIAL DEPENDENCE IN NATURAL PROCESSES

Stoykov Dmitry

10 “A” class MBOU secondary school No. 177, g. Kazan

Khabibullina Alfiya Yakubovna

scientific supervisor, mathematics teacher of the highest category, MBOU Secondary School No. 177, Kazan

Introduction

In nature and human life, there are a large number of processes in which some quantities change in such a way that the ratio of a given quantity at regular intervals does not depend on time. Among these are the radioactive decay of substances, the increase in the amount in a bank account, etc. All these processes are described by an exponential function. I was interested in the question of why the occurrence of these processes does not depend on time. After all, logically, any changing processes must be correlated with an independent quantity - time. In reality, this rule does not always work.

Purpose of the research work : experimentally confirm the occurrence of certain chemical processes in accordance with the exponential dependence described by the Arrhenius equation.

Tasks :

·Study the exponential function;

·Study exponential dependence as a special case of an exponential function;

·Study the Arrhenius equation describing exponential dependence;

·Study examples of chemical processes occurring in accordance with exponential dependence;

· Conduct a series of experiments and confirm in practice the occurrence of certain chemical processes in accordance with the exponential dependence described by the Arrhenius equation.

Research hypothesis : Using the Arrhenius equation, you can describe some chemical processes.

Object of study : exponential function as an element of applied mathematics.

Research methods :

1. Study of literature and electronic resources on the research topic.

2. Analysis of the application of exponential dependence

3. Chemical experiments to confirm the Arrhenius equation.

Exponential function

Let X R, a ≠ 0, (r n ) is a sequence of rational numbers converging to x. Let's define the number a x as a limit. An exponential function with a base a > 0 and a ≠ 1 is a function of the form y=a x, X R

This limit does not depend on the choice of the sequence r n leading to the number x. The domain of definition of the exponential function is the entire number line. This function is continuous and increases monotonically for a > 1 and decreases monotonically at 0< a < 1 . Функция никогда не обращается в ноль, но имеет горизонтальную асимптоту y = 0.

Graph of exponential function y=0.5 x

Exponential dependence

Of particular importance in applications is the exponential function, the base of which is the number e, defined as

Numerically it is equal e= 2.71828182845904523536 and is called Euler's constant.

A function defined in this way is called exponential or simply exponential and is denoted at= e x ≡ exp x.

Consider the graph of the exponential function y = e x. Since 2< e < 3, то функция at= e x monotonically increasing throughout the entire domain of definition. At point (0;1) the tangent is inclined to the abscissa axis at an angle of 45 o (π/4). The derivative of this function at zero is equal to 1. This is the only function whose derivative and antiderivative coincide with itself.

Arrhenius equation

Swedish physicist and chemist Svante Arrhenius received the Nobel Prize in Chemistry in 1903 for his theory of electrolytic dissociation. In his doctoral dissertation (Uppsala University), Arrhenius suggested that “molecules” such as sodium chloride spontaneously disintegrate in solution, forming ions that act as reactants in electrolysis. However, Arrhenius is best known for his equation that determines the temperature dependence of the reaction rate constant.

Arrhenius first established the exact relationship between reaction rates and temperature in 1889. This relationship, called the Arrhenius equation, has the form

,

Where: To— reaction rate constant;

A— a constant characterizing each specific reaction (Arrhenius constant);

e— exponent;

E a- another constant, characteristic of each reaction and called activation energy;

R— gas constant;

T— absolute temperature in degrees Kelvin.

Note that this equation relates temperature not to the rate of reaction, but to the rate constant.

The relationship between reaction rate and temperature was derived from the results of the first kinetic studies in 1880–1884. and got the name van't Hoff's rules: the rate of many reactions when heated by 10 o C increases by 2–4 times. This rule applies to relatively slow reactions in solutions and is therefore not universal. When solving some problems, you can use the Van't Hoff formula:

Where: γ — van’t Hoff coefficient (= 2–4),

T- temperature in degrees on the Celsius or Kelvin scale (since the difference is used, the scale does not matter).

Arrhenius equation b more accurately and more universally expresses the dependence of the reaction rate constant on temperature. Factor A in this equation is related to the frequency of particle collisions and their orientation during collisions.

Examples of natural processes occurring in accordance with the Arrhenius equation

Example 1. The speed (frequency) of crickets' beeping obeys, although not quite strictly, the Arrhenius equation, gradually increasing in the temperature range from 14.2°C to 27°C, with an effective activation energy E a = 51 kJ/mol. Based on the frequency of chirps, you can determine the temperature quite accurately: you need to count their number in 15 seconds and add 40, you get the temperature in degrees Fahrenheit (F) (Americans still use this temperature scale). So, at 55 F (12.8°C) the chirping frequency is 1 chirp/s, and at 100 F (37.8°C) it is 4 chirps/s.

Example 2. In the temperature range from 18°C ​​to 34°C, the sea turtle's heart rate is consistent with the Arrhenius equation, which gives the activation energy E a = 76.6 kJ/mol, but at lower temperatures the activation energy increases sharply. This may be due to the fact that at low temperatures the turtle does not feel very well and its heart rate begins to be controlled by other biochemical reactions.

Example 3. Particularly interesting are attempts to “put Arrhenius dependence” on human psychological processes. Thus, people with different body temperatures (from 36.4°C to 39°C) were asked to count the seconds. It turned out that the higher the temperature, the faster the counting ( E a = 100.4 kJ/mol). Thus, our subjective sense of time obeys the Arrhenius equation. The author of the sociological study, G. Hoagland, suggested that this is due to certain biochemical processes in the human brain.

The German researcher H. von Foerstler measured the rate of forgetting in people with different temperatures. He gave people a sequence of different signs and measured the time during which people remembered this sequence. The result was the same as Hoagland's: Arrhenius dependence with E a = 100.4 kJ/mol.

Rich folk experience suggests many conclusions that are scientifically confirmed. There has long been a saying in Rus': “Keep your feet warm and your head cold.” The Arrhenius equation substantiates this statement.

Dependence of the rate of chemical reactions on temperature

A change in temperature has a dramatic effect on the rate constant, and therefore on the rate of a chemical reaction. In the vast majority of cases, the rate of a chemical reaction increases with heating.

In accordance with Van't Hoff's rule, with every 10 degree increase in temperature, the rate of a chemical reaction increases, on average, 2-4 times:

v 2 = v 1 ×γ (T 2- T 1)/10,

where: γ is the temperature coefficient, which can be calculated using the formula:

γ = k T +10 /k T ,

Where: k T T;

k T+10- reaction rate constant at temperature (T+10).

Experiments

Experiment No. 1: Reaction of zinc with dilute sulfuric acid.

They took several pieces of zinc with a precisely known mass and placed them in equal volumes of dilute sulfuric acid solutions at different temperatures. We measured the time for complete dissolution of zinc, which takes place in accordance with the reaction:

Zn + H 2 SO 4 = ZnSO 4 + H 2

The reaction rate was calculated in µmol/s (the amount of substance per mole was calculated through the mass divided by the atomic mass of zinc). The results are presented in the table and in the form of a graph of the reaction rate versus temperature.

Table 1.

Dependence of the rate of interaction of zinc with dilute sulfuric acid on the temperature of the sulfuric acid:

Experiment No. 2: INinfluence of temperature on the rate of enzymatic reaction.

As a model enzymatic reaction, we took the reaction of butyrylcholine hydrolysis, catalyzed by the enzyme butyrylcholinesterase:

(CH 3) 3 N + -CH 2 -CH 2 -O-C(O)-C 3 H 7 + H 2 O → (CH 3) 3 N-CH 2 -CH 2 -OH + HO-C(O)- C3H7

Three-dimensional model of the butyrylcholinesterase molecule.

To carry out the reaction, a plate for immunochemical studies was used (see Figure 1 of Appendix 1). Solutions of the substrate, butyrylcholine, and the enzyme, butyrylcholinesterase, were prepared by dissolving an accurately weighed portion of the substance in a 0.002 mol/L phosphate buffer solution containing the acid-base indicator bromothymol blue, with pH = 8. The indicator is blue in an alkaline medium (pH>7) and yellow coloring in an acidic environment. Since acid is formed as a result of the enzymatic hydrolysis reaction, the pH of the solution decreases and the color of the indicator changes from blue through green to yellow. Thus, the rate of a chemical reaction can be assessed by the rate of color change of the indicator.

Carrying out the reaction. The substrate and enzyme solutions were cooled or heated to the desired temperature (5°C, 15°C, 25°C, 35°C) using snow and a water bath. The maximum temperature chosen was 35°C, since the enzyme cholinesterase has a temperature optimum of 37°C (the temperature at which the enzyme activity is maximum). An enzyme solution of a certain temperature was added into the plate cell using a 100 μl dispenser, then 100 μl of a substrate solution, and the time was recorded using a stopwatch. The time from the start of the reaction (the moment the substrate was added to the enzyme) until the color of the indicator changed to yellow was measured. At each temperature, the experiment was carried out in triplicate, then the average time of color change was calculated.

Dependence of the time of change in the color of the indicator on the temperature of the solutions:

This method for assessing the reaction rate is a variant of the catalytic method of analysis - the fixed concentration method. This is a method in which the reaction is carried out to a strictly defined (fixed) concentration of the indicator substance and the time to reach this concentration is measured. In this reaction, the indicator substance is butyric acid, the concentration of which determines the color of the indicator. The time to reach a certain concentration is a measure of the rate of reaction. The graph is plotted in coordinates: the reciprocal of the time to reach a fixed concentration is the parameter being studied (temperature).

Dependence of the rate of change of color of the indicator on temperature:

Experiment No. 3(calculation problem): Ethyl alcohol dehydration rate.

Problem: How many times will the rate of dehydration of ethyl alcohol increase when the temperature increases from 180 o C to 200 o C, if the temperature coefficient of the reaction is three?

Solution: In accordance with Van't Hoff's rule, v 2 = v 1 ×γ (T 2- T 1)/10, hence

V 2 /v 1 = γ (T 2- T 1)/10, where γ = 3, T1= 180, T2= 200. Thus, v 2 /v 1 = 3 (200-180)/10 = 9, i.e. the speed will increase 9 times when the temperature increases by 20 degrees.

Based on the data obtained, it is possible to construct a graphical dependence of the rate of dehydration of ethyl alcohol on temperature (with an increase in temperature for every 10 degrees, the reaction rate increases 3 times).

Dependence of reaction rate on temperature:

Conclusions. Conclusion

In the process of working on the research topic, the exponential function, exponential dependence, as a special case of the exponential function, as well as the Arrhenius equation describing the exponential dependence were studied.

After considering examples of natural processes occurring in accordance with an exponential dependence, a number of chemical experiments were carried out and the occurrence of some chemical processes in accordance with the exponential dependence described by the Arrhenius equation was confirmed in practice.

We believe that the research hypothesis “using the Arrhenius equation can describe some chemical processes” was confirmed. Thus, when Zn is dissolved in sulfuric acid at different temperatures, the rate of the chemical reaction changes exponentially. In addition, the rate of enzymatic reaction in neurons of the human brain also changes with temperature according to the Arrhenius equation.

Thus, the occurrence of some chemical processes in accordance with the Arrhenius equation, described by an exponential dependence, was experimentally confirmed.

Bibliography:

1. Lavrentiev M.A., Shabat B.V. Methods of the theory of functions of a complex variable. M.: Nauka, 1987. - 688 p.

2.Leenson I.A. Why is the rule outdated? Encyclopedia for children. T. 17. Chemistry. - M.: Avanta+, 2000. - 640 p.

3.Leenson I.A. Why and how chemical reactions occur. - M.: MIROS, 1994. - 176 p.

4. Khaplanov M.G. Theory of functions of a complex variable (short course). M.: Education, 1965. - 209 p.

Exponential growth

The expression “exponential growth” has entered our lexicon to mean rapid, usually uncontrollable, increase. It is often used, for example, to describe the rapid growth of cities or an increase in population. However, in mathematics this term has a precise meaning and denotes a certain type of growth.

Exponential growth occurs in those populations in which the increase in population (the number of births minus the number of deaths) is proportional to the number of individuals in the population. For a human population, for example, the birth rate is approximately proportional to the number of reproductive pairs, and the death rate is approximately proportional to the number of people in the population (we denote it ) . Then, to a reasonable approximation,

population growth = number of births - number of deaths

Let be the number of individuals added to the population during time , then if there are a total of individuals in the population, then the conditions for exponential growth will be satisfied if

To better understand what exponential growth is, imagine a population consisting initially of one bacterium. After a certain time (a few hours or minutes), the bacterium divides in two, thereby doubling the population size. After the next period of time, each of these two bacteria will again split in two, and the population size will double again - there will now be four bacteria. After ten such doublings there will be more than a thousand bacteria, after twenty - more than a million, and so on. If the population doubles with each division, its growth will continue indefinitely.

There is a legend (most likely not true) that the man who invented chess gave his Sultan such pleasure that he promised to fulfill any of his requests. The man asked the Sultan to place one grain of wheat on the first square of the chessboard, two on the second, four on the third, and so on. The Sultan, considering this demand insignificant compared to the service he had provided, asked his subject to come up with another request, but he refused. Naturally, by the 64th doubling, the number of grains became such that there would not be enough wheat in the whole world to satisfy this request. In the version of the legend that is known to me, the Sultan at that moment ordered the inventor’s head to be cut off. The moral, as I tell my students, is: sometimes you shouldn't be too smart!

The chessboard example (as well as the imaginary bacteria) shows us that no population can grow forever. Sooner or later it will simply run out of resources - space, energy, water, whatever. Therefore, populations can only grow exponentially for a while, and sooner or later their growth must slow down. To do this, you need to change the equation so that when the population size approaches the maximum possible (which can be supported by the external environment), the growth rate slows down. Let's call this maximum population size . Then the modified equation will look like this:

Predator-prey relationship

Sometimes a simple mathematical model describes a complex biological system well. An example of this is the long-term relationship between predator and prey species in an ecosystem. Mathematical calculations of population growth for a single species (see above) show that the limits of population density can be described by simple equations that produce a characteristic S-shaped curve. It is a population curve that grows exponentially while it is small and then levels off as it reaches the limits of the ecosystem's ability to support it. A simple extension of this concept allows us to understand an ecosystem in which two species, predator and prey, interact.

So, if the number of herbivorous prey is , and the number of carnivorous predators is , then the probability that a predator will meet a herbivore is proportional to the product . In other words, the higher the abundance of one of the species, the higher the likelihood of such encounters. In the absence of predators, the prey population will grow exponentially (at least initially), and in the absence of prey, the predator population will decline to zero - either due to starvation or migration. Now, if is the change in the population of herbivores over time, and the change in the population of carnivores over the same time interval, then the two populations are described by the equations:

Solving these equations shows that both populations develop cyclically. If the herbivore population increases, the probability of predator-prey encounters increases, and, accordingly (after some time delay), the predator population increases. But an increase in the population of predators leads to a decrease in the population of herbivores (also after some delay), which leads to a decrease in the number of offspring of predators, and this increases the number of herbivores, and so on. These two populations seem to be dancing a waltz in time - when one of them changes, the other changes after it.

Encyclopedia by James Trefil “The Nature of Science. 200 laws of the universe."
James Trefil is a professor of physics at George Mason University (USA), one of the most famous Western authors of popular science books.

Exponential growth

Exponential growth- an increase in a quantity, when the growth rate is proportional to the value of the quantity itself. They say that such growth obeys exponential law. Exponential growth is contrasted with slower (over a sufficiently long period of time) linear, power or geometric dependencies.

Properties

For any exponentially growing quantity, the larger the value it takes, the faster it grows. This also means that the magnitude of the dependent variable and the rate of its growth are directly proportional. But at the same time, unlike a hyperbolic curve, an exponential curve never goes to infinity in a finite period of time.

Exponential growth ultimately turns out to be faster than any geometric progression, than any power progression, and even more so than any linear growth.

Mathematical notation

Exponential growth is described by the differential equation:

The solution to this differential equation is exponential:

Examples

An example of exponential growth would be the increase in the number of bacteria in a colony before resource limitation occurs. Another example of exponential growth is compound interest.

see also

Links

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See what “Exponential growth” is in other dictionaries:

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Books

• Great Lakes of the World, V.A. Rumyantsev, V. G. Drabkova, A. V. Izmailova. Exponential population growth and subsequent growth of industry and agriculture leads not only to a catastrophic shortage of fresh water reserves, but also to their deterioration...