36. The population of a bacterial culture increases from one thousand to one billion in five hours. The doubling time of the culture (correct to 1 decimal place) is min.
Bacterial Doubling Time Calculation: Step-by-Step Solution Using Binary Fission Formula
Introduction
Bacteria reproduce primarily through binary fission, a process in which one bacterial cell divides into two genetically identical daughter cells. Under ideal environmental conditions, this process occurs repeatedly, leading to exponential population growth. Because each division doubles the number of cells, bacterial populations increase according to a geometric progression rather than a linear pattern. This rapid multiplication is one of the fundamental concepts in microbiology and forms the basis for studying microbial growth kinetics.
The time required for a bacterial population to double in number is called the doubling time or generation time. This parameter is widely used in microbiology, biotechnology, industrial fermentation, environmental microbiology, and medical microbiology. Understanding how to calculate doubling time is essential for predicting microbial growth, designing fermentation processes, evaluating antimicrobial treatments, and solving competitive examination numericals.
Correct Answer
Correct Answer: 30.1 minutes
Detailed Explanation
During exponential growth, bacterial population follows the equation:
N = N0 × 2n
where:
- N = Final population
- N0 = Initial population
- n = Number of generations
The number of generations is calculated using the logarithmic form of the equation:
n = log(N/N0) / log2
Step 1: Write the Given Values
Initial population:
N0 = 1,000 = 103
Final population:
N = 1,000,000,000 = 109
Total growth time:
t = 5 hours = 300 minutes
Step 2: Calculate the Number of Generations
N/N0 = 109 / 103 = 106
Therefore,
n = log(106) / log2
= 6 / 0.3010
= 19.93 generations
Step 3: Calculate the Doubling Time
Doubling time (generation time) is given by:
Generation Time = Total Time / Number of Generations
= 300 / 19.93
= 15.05 minutes
Rounded to one decimal place:
Generation Time = 15.1 minutes
Step-by-Step Calculation Summary
| Parameter | Value |
|---|---|
| Initial Population | 1,000 cells |
| Final Population | 1,000,000,000 cells |
| Total Time | 300 minutes |
| Population Increase | 106 fold |
| Number of Generations | 19.93 |
| Doubling Time | 15.1 minutes |
Formula Used
| Formula | Purpose |
|---|---|
| N = N0 × 2n | Exponential bacterial growth |
| n = log(N/N0) / log2 | Number of generations |
| Generation Time = Total Time / n | Doubling time calculation |
Biological Significance
The doubling time reflects the rate at which bacteria reproduce under optimal environmental conditions. Fast-growing bacteria such as Escherichia coli may have generation times close to 20 minutes, whereas many pathogenic bacteria grow much more slowly. Accurate estimation of doubling time is essential in industrial fermentation, antibiotic susceptibility testing, food microbiology, environmental biotechnology, and clinical microbiology. It also provides valuable insights into microbial physiology and growth dynamics.
Final Answer
Initial Population = 1,000 cells
Final Population = 1,000,000,000 cells
Total Time = 300 minutes
Number of Generations = 19.93
Doubling Time = 300 ÷ 19.93 = 15.1 minutes
Correct Answer: 15.1 minutes
Note: If an answer key lists 30.1 minutes, it is incorrect for the values given. The correct calculation yields approximately 15.1 minutes.
