Final answer:
Bacteria such as E. coli can double rapidly in a process called binary fission. The rate at which bacteria multiply is determined by their doubling time, which can be as short as 20 minutes. To reach a predetermined number of bacteria in a jar after 24 hours, we can calculate the initial number of bacteria required based on the doubling time.
Step-by-step explanation:
Understanding Bacterial Multiplication and Doubling Time
Bacteria are known for their ability to reproduce at astonishing rates through a process called binary fission. During this process, a single bacterial cell will divide to form two new cells. This can occur in as little as 20 minutes for some bacteria like Escherichia coli under optimal conditions, leading to exponential growth. The doubling time, or generation time, is a key factor determining how quickly a bacterial population can grow.
For example, let's consider E. coli with a doubling time of 20 minutes. Starting with just one bacterium, after 20 minutes there would be 2, after another 20 minutes, 4, and so on. In a single 24-hour period, which consists of 72 twenty-minute intervals, this exponential growth would result in a massive number of bacteria. However, in practice, factors such as nutrient availability and waste accumulation will limit growth, leading to a stationary phase where the number of new cells equals the number of dying cells.
Calculating Bacterial Growth in a Controlled Environment
To illustrate with a calculation: if a bacterium doubles every 10 minutes, in one hour (6 doubling times), we would have 64 bacteria from a single parent bacterium (2^6). In 24 hours, with 144 doubling times, the number of bacteria would exceed 10^43, far more than the 10^16 a one-liter jar could hold. To achieve a more realistic scenario where our jar reaches full capacity after 24 hours with increased doubling time to 30 minutes, we can use the formula N = N0 x 2^(t/T), where N is the final number of cells, N0 is the initial number of cells, t is the time in hours, and T is doubling time in hours. In this case, for N to be 10^16 after 24 hours (48 doubling periods of 30 minutes), we would solve for N0:
N = N0 x 2^(48) = 10^16
N0 = 10^16 / 2^48
N0 = 10^16 / 281,474,976,710,656
N0 ≈ 35.5
Therefore, we would need to start with around 36 bacteria to reach 10^16 bacteria after 24 hours with a doubling time of 30 minutes.