Calculate the specific growth rate and generation time of a bacterial…

Question 3 of 10intermediate🔧 ApplyNumerical3 marks

Calculate the specific growth rate and generation time of a bacterial population in which the number of bacteria increases from $10^{4}$ cells/mL to $10^{7}$ cells/mL, during 4 hours of exponential growth.

Correct Answer

Given: Initial cell concentration ( $X_0$ ) = $10^4$ cells/mL Final cell concentration ( $X_t$ ) = $10^7$ cells/mL Time ( $t$ ) = 4 hours

Calculation of Specific Growth Rate ( $\mu$ ): The formula for specific growth rate is $\mu = 2.303 \frac{(\log X_t - \log X_0)}{t}$ . Substituting the values: $\mu = 2.303 \frac{(\log 10^7 - \log 10^4)}{4}$ $\mu = 2.303 \frac{(7 - 4)}{4}$ $\mu = 2.303 \times \frac{3}{4}$ $\mu = 1.727$ hr $^{-1}$

Calculation of Generation Time ( $t_d$ ): The formula for generation time (doubling time) is $t_d = \frac{0.693}{\mu}$ . Substituting the value of $\mu$ : $t_d = \frac{0.693}{1.727}$ $t_d = 0.40$ h

Answer: Specific growth rate ( $\mu$ ) = $1.727$ hr $^{-1}$ Generation time ( $t_d$ ) = $0.40$ h

Exercise: EXERCISES | Q: 12 | (Chapter: 25)

For More Understanding

Explanation

The solution applies the standard growth equations provided in the textbook context. The specific growth rate ( $\mu$ ) is calculated using the logarithmic formula relating cell concentration change over time. Once $\mu$ is determined, the generation time ( $t_d$ ) is derived using the relationship $t_d = 0.693 / \mu$ . The log values are calculated based on the exponents of the scientific notation (e.g., $\log 10^7 = 7$ ).

Solution Steps

Step 1: Identify the given values: $X_0 = 10^4$ cells/mL, $X_t = 10^7$ cells/mL, and $t = 4$ h.
Step 2: Use the specific growth rate formula $\mu = 2.303 \frac{(\log X_t - \log X_0)}{t}$ .
Step 3: Substitute values: $\mu = 2.303 \times \frac{(7 - 4)}{4} = 1.727$ hr $^{-1}$ .
Step 4: Use the generation time formula $t_d = \frac{0.693}{\mu}$ .
Step 5: Calculate generation time: $t_d = \frac{0.693}{1.727} \approx 0.40$ h.

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