Question

The growth of Pseudomonas bacteria is modeled as a first-order process with \(k=0.035 \mathrm{~min}^{-1}\) at \(37^{\circ} \mathrm{C}\). The initial \(P\) seudomonas population density is \(1.0 \times 10^{3}\) cells/L. (a) What is the population density after \(2 \mathrm{~h}\) ? (b) What is the time required for the population to go from \(1.0 \times 10^{3}\) to \(2.0 \times 10^{3}\) cells/L?

Short answer

The population density after 2 hours is 66690 cells/L. The time required for the population to double is approximately 19.8 minutes.

Step-by-step solution

- Understand the given parameters

Firstly, note down the given parameters: the growth rate constant is \(k=0.035 \mathrm{~min}^{-1}\), the initial population density is \(P(0)=1.0 \times 10^{3} \) cells/L.

- Convert the time from hours to minutes

For part (a), convert the given time of 2 hours into minutes: \(t = 2 \mathrm{~h} \times 60 \mathrm{~min/h} = 120 \mathrm{~min}\).

- Use the first-order kinetic equation

The first-order kinetic equation for population growth is: \[ P(t) = P(0) e^{kt} \] Substitute the known values into the equation: \[ P(120) = 1.0 \times 10^3 \times e^{(0.035 \times 120)} \]

- Calculate the population density after 2 hours

Calculate the exponent first: \(0.035 \times 120 = 4.2\). Then calculate \(e^{4.2}\): \(e^{4.2} \approx 66.69\). Therefore, \(P(120) = 1.0 \times 10^3 \times 66.69 = 66690 \) cells/L.

- Determine the time required for the population to double

For part (b), set the population at time \(t\) to be \(2.0 \times 10^3\) cells/L and use the first-order kinetic equation: \[ 2.0 \times 10^3 = 1.0 \times 10^3 \times e^{0.035t} \]

- Solve for time (t)

Divide both sides by \(1.0 \times 10^3\): \(2 = e^{0.035t}\). Take the natural logarithm of both sides: \[ \ln(2) = 0.035t \] Recall that \(\ln(2) \approx 0.693\), So, solve for \(t\): \[ t = \frac{0.693}{0.035} \approx 19.8 \text{ min} \]

Key concepts

Population Growth

Population growth refers to the change in the number of individuals in a species over a certain period. Many types of growth patterns exist, such as linear, logistic, and exponential. In this context, we focus on exponential growth, which appears when the population size increases at a rate proportional to its current size. Understanding these patterns is crucial in many fields like biology, ecology, and environmental science.
In our example, we're dealing with the growth of Pseudomonas bacteria, where the population grows exponentially. This means that as the number of bacteria increases, the growth rate also increases.

Pseudomonas Bacteria

Pseudomonas bacteria are a type of Gram-negative bacteria, famous for their diverse metabolic capabilities and resistance to antibiotics. They can be found in various environments, including soil, water, and plant surfaces. Because of their ability to adapt and thrive in different conditions, studying their growth helps scientists understand microbial ecology and disease pathways.
In the exercise, we model the population growth of Pseudomonas bacteria using first-order kinetics, which is common in microbial growth studies.

• The initial population density given is 1.0 × 10³ cells/L.
• These bacteria grow best at around 37°C, which is the temperature used in the study.

Exponential Growth

Exponential growth occurs when the growth rate of a population is proportional to its current size. This leads to a rapid increase in population over time. Mathematically, this can be expressed using the formula:
P(t) = P_0 e^{kt} ,
where:
- P(t) is the population at time t.
- P_0 is the initial population size.
- k is the growth rate constant.
This formula shows that as time passes, the population grows exponentially instead of linearly.
In our exercise, the bacteria population grows exponentially, and we need to calculate the population size after a certain time and the time taken for the population to double. This widespread growth pattern is seen in various biological systems, from bacteria to human populations.

Reaction Rate Constant

The reaction rate constant, often represented as k, measures how rapidly a given reaction occurs. In the context of population growth, it signifies how quickly the population increases. The unit of k depends on the order of the reaction; for first-order kinetics, it is time^-1.
The value given in our example is k=0.035 min^-1. This constant helps determine how the population size changes over time. By using this rate constant in our exponential growth formula, we can predict future population sizes accurately.
Understanding the reaction rate constant is essential for professionals involved in modeling biological processes, environmental engineering, and even pharmacology, where the growth or decay of substances needs precise calculation.