Question

Given the "z-value" of a microorganism as 4°C. What will be the reaction quotient (Q\(_{10}\)) ?

• A. 613.23
• B. 31.32
• C. 316.23
• D. 61.23

Hint:

To compute powers of 10 with fractional exponents:
For \( 10^{2.5} \), rewrite it as \( 10^2 \times \sqrt{10} \).
Since \( \sqrt{10} \approx 3.16 \), the result is roughly \( 100 \times 3.16 = 316 \).
This estimation helps identify the correct option immediately.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The \( z \)-value represents the temperature increase required to reduce the decimal reduction time (\( D \)-value) of a microorganism by \( 90\% \) (or by one log cycle).
The reaction quotient, \( Q_{10} \), is a parameter used in biochemical and thermal death kinetics that describes the rate of a reaction's acceleration when the temperature is increased by \( 10^\circ\text{C} \).
Key Formula or Approach:
The mathematical relationship between the \( z \)-value and the temperature coefficient \( Q_{10} \) is expressed as:
\[ Q_{10} = 10^{\frac{10}{z}} \]

Step 2: Detailed Explanation:
We are given that the \( z \)-value of the microorganism is \( 4^\circ\text{C} \).
Let us substitute this value into our relationship formula:
\[ Q_{10} = 10^{\frac{10}{4}} \]
Simplify the fraction in the exponent:
\[ \frac{10}{4} = 2.5 \]
Now, calculate the value of \( 10^{2.5} \):
\[ 10^{2.5} = 10^2 \times 10^{0.5} \]
Since \( 10^2 = 100 \) and \( 10^{0.5} = \sqrt{10} \):
We know that the square root of 10 is approximately \( 3.162277 \).
Therefore:
\[ Q_{10} = 100 \times 3.162277 = 316.2277 \]
Rounding this to two decimal places yields \( 316.23 \).
Thus, the reaction quotient \( Q_{10} \) is \( 316.23 \).

Step 3: Final Answer:
The reaction quotient \( Q_{10} \) for a \( z \)-value of \( 4^\circ\text{C} \) is \( 316.23 \).