Summary
This article discusses two methods for calculating the Dvalue of a microorganism, the Bigelow (D/z model) and Arrhenius models, and provides access to a free calculator that enables the magnitude of the differences between the methods to be determined along with the Arrhenius thermal constants activation energy (E_{a}) and k_{0}/s.
Using the reference decimal reduction and Zvalues to determine the heat resistance of a microorganism at an arbitrary temperature
The decimal reduction value, D, is an indicator of heat resistance, or how easy it is to kill organisms using heat. This is the time, in seconds or minutes, taken to decrease the number of microorganisms by a factor of 10 (or to reduce their number by 90%). With the Bigelow (1921) approach D is regarded as a constant at a constant temperature for a particular strain of microorganism and is usually referenced against the reference temperature, T_{ref }as follows e.g., D _{93.3°C} where the reference temperature was 93.3°C and D was determined at this temperature.
Knowing the Dvalue say for a Listeria monocyotegenes strain at 72°C as 2 s and the total equivalent processing time at this temperature, F, as 16 s we can determine the lethality of the process by dividing F by D.
No of log reductions = F = 16 = 8
D 2
A pasteurisationtype process giving 8 log reductions for this pathogen would be considered safe.
Food technologists and microbiologists generally have limited information on the effect of temperature on the Dvalues of pathogenic and spoilage organisms. It is often necessary to model the Dvalue at a higher or lower temperature than the reference value.
This can be done in two ways, using the Bigelow (1921) method, also known as the D/z model, or the Arrhenius equation. The most commonly used method is the Bigelow method.
Both methods discussed here require knowledge of the Zvalue. The Zvalue is measured in °C, and is the reciprocal of the slope of the thermal death curve for the target microorganism or spore; 10 °C is the value frequently used in Fo calculations performed on lowacid foods. Z is a thermal constant that expresses the increase in temperature necessary to obtain the same lethal effect in 1/10 of the time. The Zvalue has a significant effect on the F value of a process. Note it is the decimal reduction value, D, that is a measure of the resistance of a microorganism to heat. The zvalue provides information on how this heat resistance changes with temperature.
Given that the D value at 72°C for an organism is 14 seconds and the Z value is 7°C how can the Dvalue at 78°C be calculated using the Bigelow method?
Since 78°C is 7°C more than 71°C and the Zvalue is also 7°C the Dvalue is reduced by 1log cycle i.e. 14 seconds (the Dvalue at 72°C) is divided by 10 to give a predicted Dvalue at 78°C of 1.4 seconds. Given a Dvalue at a particular temperature, D values at other temperatures can also be calculated using equation 1.
Equation 1. D_{ref } = 10^(T_{ref}T1/Z) (Stumbo, 1973)
D1
Where:
D_{ref } = Reference Dvalue in minutes or seconds.
D1 = Arbitrary Dvalue in minutes or seconds.
Z = Thermal constant for the organism in °C
T_{ref} = Reference temperature in °C
T1 = Arbitrary temperature in °C.
This equation gives acceptably accurate results providing that the Z value is linear over the temperature range. Regrettably, this is often not the case and consequently, this approach should only be used when the temperature difference between D_{ref }and D1 is small.
The alternative, a more mathematically complex approach and potentially more accurate, is to use the Arrhenius equation (equation 2). This equation has been extensively studied and it can be argued e.g., Skoglund (2022, 2023) amongst others that this is a more accurate way of calculating the lethal effects of heat on microorganisms particularly when high temperatures are involved e.g., >121 °C.
Equation 2. D_{ref } = EXP Ea x (TrefT1)
D1 R T1xTref
Where:
D_{ref } = Reference Dvalue in minutes or seconds.
D1 = Arbitrary Dvalue in minutes or seconds.
EXP = The exponential constant
T_{ref} = Reference temperature in K
T1 = Arbitary temperature in K
Ea = Activation energy (J/mol)
Ea can be calculated using Equation 3.
Equation 3. Ea = LN(10) x R x (Tref^2)/Z
Where:
R = Gas constant (J/K⋅mol)
T_{ref} = Reference temperature in K
Z = Thermal constant for the organism in °C
How do the D values calculated using the Bigelow and Arrhenius methods differ?
Comparisons of the calculated values can be shown in Tables 2 and 3. At low temperatures and providing the temperature range is small the disparity between the Bigelow and Arrhenius methods is relatively small. However, as predicted by Skoglund (2022, 2023) the disparity at higher temperatures is significant.
However, commercial UHT and sterilisation processes add additional time to allow a margin of process safety and this combined with normally low concentrations of bacteria especially spore formers probably accounts for the safety of hightemperature processes modelled using the Bigelow method (Skoglund, 2022).
Table 1. Modelling the difference in Dvalue for Mycobacterium avium paratuberculosis when using a constant Z and Arrenhius adjusted Z over 63 to 80 °C . 

Dvalue, s 

Temperature, °C 
63 
72 
75 
80 
ConstantZ 
179.2

12.45 
5.12 
1.16 
Arrhenius adjusted Z 
13.34 
5.78 
1.48 

% difference between values (rounded) 

7 
13 
28 
Notes: Pooled strains of M. paratuberculosis. Z=7.77 °C
Table 2. Modelling the difference in Dvalue for Geobacillus stearothermophilus strain when using a constant Z and Arrenhius adjusted Z over 102 to 150 °C . 

Dvalue, s 

Temperature, °C 
102 
125 
130 
150 
D ConstantZ 
1500

4.17 
1.16 
0.007 
D Arrhenius 
5.86 
1.91 
0.028 

% difference between values (rounded) 

41 
65 
300 
Notes. Geobacillus stearothermophilus had a Z value of 9.
Users can also download an Excel spreadsheet below to do these calculations. The cells are all unlocked and you are free to use it without copyright declarations.
You can download this calculator by donating £10 to help towards hosting charges.
Acknowledgements
The generous help of Dr Tomas Skoglund in helping me to understand and appreciate the use of the Arrhenius equation and its applications in thermal processing is gratefully acknowledged.
Literature cited
Bigelow, W.G. (1921). The logarithmic nature of thermal death curves. The Journal of Infectious Diseases. 29, 528–538.
Skoglund, T. (2022) On the common misuse of a constant zvalue for calculations of thermal inactivation of microorganisms Journal of Food Engineering 314, 110766
Skoglund, T. (2023) Standard microbiological approach to calculating z values, and consequences of approximations. Paper presented at the Society of Dairy Technology conference at Penrith, Cumbria, UK on the 29^{th} March 2023.
Stumbo, C. R. (1973). Thermobacteriology in food processing, 2nd ed. Academic Press, New York.
How to cite this article
Mullan, W.M.A. (2024).
[Online]. Available from: https://www.dairyscience.info/thermalprocessing/451arrhenius.html . Accessed: 23 February, 2024.