Problem 100 - Chemical Principles

Question

Models of population growth are analogous to chemical reaction rate equations. In the model developed by Malthus in 1798 , the rate of change of the population \(N\) of Earth is \(\mathrm{d} N / \mathrm{d} t=\) births - deaths. The numbers of births and deaths are proportional to the population, with proportionality constants \(b\) and \(d\). Derive the integrated rate law for population change. How well does it fit the approximate data for the population of Earth over time given below? $$ \begin{array}{lccccccc} \text { Year } & 1750 & 1825 & 1922 & 1960 & 1974 & 1987 & 2000 \\ N / 10^{9} & 0.5 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} $$

Step-by-Step Solution

Answer

The integrated rate law for population change according to the Malthusian model is \(N = N_0e^{rt}\), where \(N\) is the population at time \(t\), \(N_0\) is the initial population, and \(r\) is the net growth rate. To evaluate the model fit, the actual population data should be compared against the model's prediction graphically or numerically, which cannot be done here.

Step 1: Express rate of change differential equation for population growth

We begin by framing the relation given by Malthus. The rate of change of the population, represented by the differential equation is \[\begin{equation}\frac{dN}{dt} = bN - dN,\end{equation}\]where \(b\) and \(d\) are the birth and death rates respectively, and \(N\) is the population. Since both births and deaths are proportional to the current population, the equation simplifies by combining constants into one.

Step 2: Combine proportionality constants and simplify differential equation

We combine the proportionality constants \(b\) and \(d\) into a single constant, \(r = b - d\), which represents the net growth rate. By doing this, we simplify the differential equation to \[\begin{equation}\frac{dN}{dt}= rN.\end{equation}\]This equation states that the rate of population growth is directly proportional to the current population.

Step 3: Separate variables and integrate the equation

To solve the differential equation, we separate the variables \(N\) and \(t\). This yields\[\begin{equation}\frac{1}{N} dN = r dt.\end{equation}\]We then integrate both sides of the equation. The left side is integrated with respect to \(N\), and the right side with respect to \(t\), resulting in:\[\begin{equation}\ln(N) = rt + C,\end{equation}\]where \(C\) is the integration constant.

Step 4: Solve for the population \(N\)

To solve for \(N\), we exponentiate both sides of the equation to remove the natural logarithm:\[\begin{equation}N = e^{rt + C}.\end{equation}\]We can rewrite the constant \(e^C\) as a new constant \(N_0\), which represents the initial population at \(t = 0\):\[\begin{equation}N = N_0e^{rt}.\end{equation}\]This is the integrated rate law for population change according to the Malthusian model.

Step 5: Evaluate the fit of the Malthusian model

To evaluate the fit of the Malthusian model to the provided data, we would plot the population data points against time and compare it to the curve represented by \(N = N_0e^{rt}\). If the points closely follow the curve, it suggests the model is a reasonable estimate for the population growth over the time period covered by the data. However, this step requires numerical evaluation or graphical analysis which is not possible within the scope of this explanation.

Key Concepts

Malthusian Growth Model

The Malthusian growth model is a cornerstone concept in understanding population dynamics. This model, developed by Thomas Malthus in the late 18th century, posits that populations grow exponentially under ideal conditions with abundant resources. When applying this to real-world scenarios, the model assumes that the rate of population growth at any given moment is directly proportional to the current population size.

According to this simple yet powerful framework, if we denote the population at any time as N, the rate of change of the population (that is, how quickly it grows or shrinks over time) can be represented by the birth rate minus the death rate, both of which are in turn proportional to N. Thus, the Malthusian model boils down to a basic differential equation that describes population dynamics over time.

Looking at historical data, such as the population numbers from the 18th to the 20th century, one can assess the model's applicability. For example, during periods of relative stability without major events that dramatically affect birth or death rates, the Malthusian model can provide a good approximation of population growth.

Differential Equations

Differential equations are mathematical tools that describe relationships involving rates of change. At their core, they are equations that involve derivatives, which are mathematical expressions representing how a quantity changes in response to changes in another quantity, usually time.

An understanding of differential equations is crucial when it comes to population growth models as they enable us to express the rate of population change, denoted by \( \frac{dN}{dt} \), where N refers to the population and t stands for time. The rate of change depends on the current population, which leads to an equation that can differ in complexity depending on the model used.

In the Malthusian model, the differential equation is relatively simple, and solving it allows us to predict future populations based on current trends. These powerful equations are not just limited to populations but are used in a myriad of fields, including physics, engineering, and economics, to describe dynamic systems.

Integrated Rate Law

The integrated rate law is a mathematical expression obtained from rearranging and integrating a differential equation. It provides the means to calculate the amount of a substance or, in our context, the size of a population, as a function of time.

Starting with the separated variables from the differential equation, integrating both sides gives us the logarithmic relationship between population and time. Exponentiating this relationship then eliminates the logarithm, providing a direct equation for the population N in terms of the net growth rate r and initial population N_0. Specifically, the equation \(N = N_0e^{rt}\) allows us to calculate the expected size of a population at any future time point, given its current size and growth rate.

This process of integration is not unique to populations: in chemistry, physics, and other sciences, integrated rate laws are fundamental for understanding how systems evolve over time. While it originated in the realms of math and physics, the integrated rate law is a key piece of the puzzle in predicting biological and ecological changes as well.

Exponential Population Growth

Exponential population growth is a pattern of growth where the increase becomes more rapid in proportion to the growing total number or size. It is commonly illustrated by the J-shaped curve when the population is graphed over time.

In the context of the Malthusian model, if resources are unlimited and the birth rate exceeds death rate, population growth can be considered exponential, given by the equation \(N = N_0e^{rt}\), which reflects a constant doubling time. Here, N represents the future population size, N_0 the initial population size, r the net growth rate, and t the time elapsed.

However, real-life scenarios often challenge this model. Factors such as competition, limited resources, and environmental constraints introduce complexities into the growth patterns. Understanding exponential growth and its limitations is essential not only in demography but also in areas such as epidemiology, finance, and even in understanding the spread of information and ideas.