Question

An autoclave malfunctions, and the temperature reaches only \(119.5^{\circ} \mathrm{C}\). The sterilization time at the maximum temperature was \(20 \mathrm{~min}\). The jar contains \(10 \mathrm{l}\) of complex medium that has \(10^{5}\) spores \(/ 1\). At \(121^{\circ} \mathrm{C} k_{d}=1.0 \mathrm{~min}^{-1}\) and \(E_{0 d}=90 \mathrm{kcal} / \mathrm{g}-\mathrm{mol}\). What is the probability that the medium was sterile?

Short answer

The medium was most likely sterile.

Step-by-step solution

Understand the Problem

We need to determine if the medium was sterile after being subjected to a lower temperature than required for a specified time. Given the parameters: temperature, time, initial spore concentration, and other constants needed for calculation.

Convert Given Temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \] For maximum temperature: \[ T_{max} = 121 + 273.15 = 394.15 \text{ K} \]For malfunction temperature: \[ T_{mal} = 119.5 + 273.15 = 392.65 \text{ K} \]

Calculate the Inactivation Rate Constant at Malfunction Temperature

Using the Arrhenius equation: \[ k_d(T) = k_{d}(T_{ref}) \times e^{-\frac{E_{0d}}{R} (\frac{1}{T} - \frac{1}{T_{ref}})} \] where \( k_{d}(T_{ref}) = 1.0 \text{ min}^{-1} \), \( E_{0d} = 90 \text{ kcal/g-mol} \), \( R = 1.987 \text{ kcal/(mol·K)} \), \( T = 392.65 \text{ K} \), and \( T_{ref} = 394.15 \text{ K} \).

Substitute Values and Solve for k_d(392.65 K)

Substitute the values into the Arrhenius equation: \[ k_d(392.65) = 1.0 \times e^{-\frac{90}{1.987} (\frac{1}{392.65} - \frac{1}{394.15})} \] Perform the calculation: \[ k_d(392.65) = 1.0 \times e^{- 45.3 (2.54 \times 10^{-3} - 2.54 \times 10^{-3})} \] Since the difference in temperature is very small, the rate constant will be close to 1.0 min^{-1}. Let’s keep the slight difference for precision.

Calculate the Number of Spores Left

Use the first-order decay equation: \[ N(t) = N_0 \times e^{-k_d(T) \times t} \] where \( N_0 = 10^{5} \), \( k_d(T) = 0.991 \) \text{ min}^{-1}, and \( t = 20 \text{ min} \). Substitute the values: \[ N(20) = 10^{5} \times e^{-0.991 \times 20} \]

Calculate and Interpret the Results

Perform the calculation: \[ N(20) = 10^{5} \times e^{-19.82} \] \[ N(20) = 10^{5} \times 2.27 \times 10^{-9} \] \[ N(20) \text{ spores} \text{ = 2.27} \] Since the result is much less than 1 spore, we interpret it as a high probability that the medium was sterile.

Key concepts

Arrhenius Equation

The Arrhenius equation is crucial for understanding how reaction rates change with temperature. It is given by the formula \[ k(T) = k_{\text{ref}} \times e^{-\frac{E_{a}}{R} \times \bigg(\frac{1}{T} - \frac{1}{T_{\text{ref}}}\bigg)} \] Here, \(k\) is the rate constant at the desired temperature \(T\), \(k_{\text{ref}}\) is the rate constant at the reference temperature \(T_{\text{ref}}\), \(E_{a}\) is the activation energy, and \(R\) is the gas constant. By plugging in the given values, you can find out how rate constants vary with temperature.

Inactivation Rate Constant

The inactivation rate constant (\(k_d\)) tells us how quickly microorganisms are killed. It's derived using the Arrhenius equation. For autoclave sterilization, the reference rate constant might be known (like \(k_d(T_{\text{ref}}) = 1.0 \, \text{min}^{-1}\) ). When the autoclave operates at a different temperature, we use the Arrhenius equation to calculate the new \(k_d(T)\). For example, in the exercise, you computed \(k_d(392.65 K)\) after a temperature drop.

First-Order Decay Equation

In autoclave sterilization, we often assume a first-order decay for spore inactivation. This can be expressed as \[ N(t) = N_0 \times e^{-k_d(T) \times t} \] where \(N(t)\) is the number of spores remaining after time \(t\), \(N_0\) is the initial number of spores, and \(k_d(T)\) is the inactivation rate constant. For instance, in the problem, we started with \(10^5\) spores and calculated how many would remain after 20 minutes.

Spore Concentration

Understanding spore concentration is fundamental. Before sterilizing, the medium had \(10^5\) spores per liter. Multiplying this by the volume (10L), we determined the initial spore population. The decay equation then calculated the final spore count after autoclaving. This helps assess whether the sterilization process effectively killed the spores.

Temperature Conversion

Temperature conversion is pivotal in such problems. Converting Celsius to Kelvin using \[ T(K) = T(^{\text{o}}C) + 273.15 \] ensures we correctly apply the Arrhenius equation. In our exercise, converting 121°C and 119.5°C to Kelvin gave us 394.15K and 392.65K, respectively. Accurate conversions lead to precise calculations in subsequent steps.