# What are the units of Q10 (temperature sensitivity)?

$Q_{10}$ is the increase in a rate (e.g. activity of an enzyme) observed with a 10° temperature increase.

According to Wikipedia:

$$Q_{10}=\left(\frac{R_2}{R_1}\right)^{{10}/\left({T_2-T_1}\right)}$$

It is apparent that the units of $R$ (e.g. $mol/g/s$) cancel out, but what about the units of temperature? Does $Q_{10}$ have units? It seems the units would be "per 10 °C", but it is not clear how this comes from the referenced equation

Figure. A schematic diagram showing the effect of the temperature on the stability of an enzyme catalysed reaction. The curves show the percentage activity remaining as the incubation period increases. From the top they represent equal increases in the incubation temperature (50 °C, 55 °C, 60 °C, 65 °C and 70 °C).

The $Q_{10}$ is a unitless number, that summarizes the effect of raising temperature 10°C on the rate of a chemical reaction. A $Q_{10}$ of 2.0 suggests that raising the temperature of a system by 10 °C will effectively double the rate of the reaction. This value would be expected for most chemical reactions occurring within normal physiological temperatures.

Mathematically, $Q_{10}$ can be represented by the following expression:

$$Q_{10}=\left(\frac{k_2}{k_1}\right)^{\frac{10}{t_2-t_1}}$$

$t_2$ = higher temperature
$k_2$ = rate at $t_2$
$t_1$ = lower temperature
$k_1$ = rate at $t_1$

Usually the temperature difference is about 10 °C, then you can simplify the equation

$$Q_{10}=\left(\frac{k_1}{k_2}\right)^{\frac{10}{10}}=\frac{k_1}{k_2}$$

Edit: You can easily calculate $k$ form Arrhenius equation

$$k=Ae^{\frac{-\Delta G^*}{RT}}$$

where $k$ is the kinetic rate constant for the reaction, $A$ is the Arrhenius constant, also known as the frequency factor, $-\Delta G^*$ is the standard free energy of activation ($kJ/mol$) which depends on entropic and enthalpic factors, $R$ is the gas law constant and $T$ is the absolute temperature.

• how do you calculate k, are the curves in the figure something like e^{-kt}?
– Abe
Jun 12, 2012 at 17:12
• Here k is a rate constant like R in your wikipedia example, and for the figure the curves show the percentage activity remaining as the incubation period increases. Jun 12, 2012 at 17:44
• But how do you estimate the pre-exponent factor A? Jul 15, 2015 at 8:05
• My understanding of Q10 it that it is the ratio of the rate of a chemical reaction at a specified temp to the rate of a chemical reaction 10 degrees higher. For example if the rate of an enzyme-catalysed reaction at 37 C is twice that at 25 C, then can speak of a Q10 being equal to 2. Q10 makes no assumptions about the underlying mechanism which, for an enzyme catalysed reaction, is likely to be complex, and to involve temperature effects on multiple rate constants (both first-order and second-order). I find your diagram of (first-order) decay curves misleading. These may be used to ... May 19, 2023 at 10:39
• calculate the half-life of an enzyme at different temperatures, but surely have nothing to do with Q10? May 19, 2023 at 10:39

This is a nice example of mathematical sloppiness which is more often present in textbooks on biology than on chemistry or physics where perople are more careful about dimensional analysis*.

The correct form of equation (with your notation) is

$$Q_{10} = \left(\frac{R_2}{R_1}\right)^\frac{10\,^\circ\mathrm{C}}{T_2-T_1}.$$

From this form, it is clear that the units of temperature cancel out, and that $$Q_{10}$$ is therefore unitless.

* For example, exponential function – such as the one present in the equation for $$Q_{10}$$ – cannot take units as an input, only dimensionless numbers.