## Interplay of Theory Development with Physical Reality

What’s the relation of theories of physics to the actual physical reality they are describing? A naive answer would be something like “Theories are formed by making enough experiments and deducing a common law from the results. As more experiments are made, the theory with its laws becomes a more correct description of reality”. This, however is a simplistic way to describe the situation, as no amount of supporting special cases counts as an actual logical proof for a general statement, and usually there’s a need to already have at least some degree of a “pre-theory” before one can even make any experiments.

Suppose someone wants to study the behavior of matter at different temperatures. The things measured could involve thermal expansion of a metal rod, pressure changes upon heating a gas in a rigid container, or something similar. Actually, to even be able to measure a temperature, one has to make several assumptions about the physical laws related to it. When we immerse one end of a thermometer in a kettleful of hot water, we assume that only the temperature of the thermometer changes significantly, and the hot water stays at a very close to constant temperature as the thermometer and water reach thermal equilibrium. So there we already need to have some kind of a common-sense notion of heat capacity, whatever we decide to call that thing at that point of constructing a theory of thermodynamics. In particular, we assume that if an object has either a very small mass or small volume, it takes only a small amount of “thermal energy” to change its temperature.

Also, the concept of thermal equilibrium itself requires some assumptions that are not obvious in principle. The Zeroth Law of Thermodynamics says that if Object 1 and Object 2 are in thermal equilibrium with each other, and Object 2 is in equilibrium with Object 3, then the Objects 1 and 3 are also in mutual thermal equilibrium. A kind of an equivalence relation, but this time in physics instead of pure mathematics (another example of an equivalence relation in physics is found when forming a set of all scalar-vector potential combinations resulting in the same electric and magnetic fields). We need to do this assumption before we can measure temperatures based on the thermal expansion of some material, as the equilibration needs to be able to be transmitted forward through the material that we are trying to expand by heating it. So, to measure temperature we need to have some kind of an intuitive notion of thermal equilibrium, and also assume that the length of a metal rod or column of mercury or ethanol is a single-valued function of temperature in the range of conditions we are doing the experiments in. This may seem like a chicken-and-egg problem where we need to define temperature to study thermal expansion, and also know a law of thermal expansion to be able to define temperature. The same situation is apparent in the SI definition of the unit of length as an exact fraction of the distance travelled by light in a second, despite the fact that distance measurements were done by people for quite a long time before someone even got the idea of the constancy of speed of light. This circularity problem doesn’t cause as much trouble as one may think, as you can try different kind of theories combining assumptions about temperature and thermal expansion and make a lot of tests on every theory, trying to find a contradiction between the theory and experimental fact, and trust that in the case of wrong assumptions it will become apparent that something fishy is going on. Or of course if there’s a logical contradiction in the theory with itself then it must be wrong no matter what the experiments tell.

Some things that are usually assumed about thermal equilibration include that it proceeds monotonously towards its conclusion, and doesn’t do strange things like the thermometer reading oscillating around some value before setting to equilibrium (in other words, the thermal impact given to the thermometer by the hot liquid doesn’t cause an effect similar to how a mechanical impact affects a physical pendulum in absence of supercritical damping). This assumption is not always true in thermodynamics, as oscillating chemical reactions like the BZ reaction show (first academic papers about oscillating reactions were actually initially rejected from publication because they were so much against the common sense notions about chemical kinetics and thermodynamics at the time).

The temperature/thermal equilibrium example was a rather simple one, but a lot more difficult problem are the differences of the theory of relativity and quantum mechanics to the classical Newtonian mechanics, where one finds out a necessity to accept physical laws that are very difficult to form a mental image about and that are against conventional common sense (however, it must be admitted that even the classical Newton’s first law is not intuitively obvious on the surface of the Earth where there are frictional forces damping almost every form of motion).

Philosophical theories about the nature of successive scientific theories include the Karl Popper’s theory where the most important feature of a scientific theory is thought to be its falsifiability, which means that the theory must be “vulnerable” to experimental evidence in the sense that the theory can’t be somehow cleverly bent to fit any possible experimental results (in the same way how illogical features in a religious belief system can be justified by saying “Inscrutable are the Lord’s ways”). Other philosophers who have investigated the subject include Thomas Kuhn and Paul Feyerabend. In Kuhn’s theory in particular, the very concept of “objective physical reality” is questioned, and it’s said that to be able to talk about physical reality one always has to do that in reference to some particular theoretical framework.

Actually I recently got (for free) a book related to this subject, C. Dilworth’s “Scientific Progress – A Study Concerning the Nature of the Relation Between Successive Scientific Theories”, and it seems to contain a lot of interesting material but I haven’t managed to read very far through it. I hope the links in this blog post help someone who’s trying to find info about this subject.

## What are the most energetic combustible materials?

Most people have seen several examples of combustion reactions that produce different flame temperatures. The temperature of a campfire can be somewhat over 800 deg C, while the flame of an oxyacetylene torch in welding can reach a temperature of over 3000 deg C. The energeticity of combustion reactions can be quantified in several ways:

1. Maximum temperature of the flame that is produced
2. Amount of heat released in combustion (per unit mass or volume of fuel)
3. In the cases of gaseous fuels, aerosols or vapors, the magnitude of sudden pressure increase in the explosion of fuel-oxygen mixture

Note that the amount of heat released per mole of fuel is not a good measure of the energeticity, as you can make the molar heat of combustion of an organic compound practically as large as you want by building longer and longer hydrocarbon chains.

The heat release of combustion reactions, more formally called enthalpy of combustion $\Delta H_c$, is rather easy to measure in the laboratory using a calorimeter. A typical example of a combustion reaction is the burning of methane, described by the reaction equation below:

$CH_4 + 2O_2 \rightarrow CO_2 + 2H_2 O$ .

The enthalpy change in this reaction can be calculated from the enthalpies of formation ($\Delta H_f$) of the reactants and products, which can be found from tables of thermodynamic quantities (in books or online). The enthalpy change in the combustion of a mole of methane is

$\Delta_c (CH_4) = \Delta H_f(CO_2 ) + 2\Delta H_f(H_2 O) - \Delta H_f (CH_4 )$,

where the enthalpy of formation of elemental oxygen has not been included as it is zero (as is the $\Delta H_f$ of any element in its most stable form).

The thermodynamic quantities in this are

$\Delta H_f(CO_2 ) = -393 kJ/mol$
$\Delta H_f(H_2 O) = -242 kJ/mol$
$\Delta H_f(CH_4 ) = -74.9 kJ/mol$

which give a value of about -802 kJ/mol or -50 kJ/g for the quantity $\Delta H_c (CH_4 )$. An enthalpy change that is negative means that heat is released to the surroundings by the reaction. The formal definition of enthalpy is $H=E+PV$, or internal energy plus pressure times volume, and in constant-pressure processes of simple systems its change is $\Delta H = \Delta E + P\Delta V$. A system that is “simple” in the sense meant here can do work on the surroundings only by expanding: $W = -P\Delta V$. Some systems can do other kinds of work, too, like elastic deformation (which can require an application of force over some distance even without involving a change of volume) or electrical work (charging a battery).

As the heat of combustion increases linearly with increasing $\Delta H_f$:s of the products and decreases linearly with increasing $\Delta H_f$:s of the reactants, the most energetic fuels are those that have a positive heat of formation, which means that energy is consumed when they are formed from their elemental constituents. Straight-chain alkane hydrocarbons like ethane, propane and butane all have negative heats of formation. To make a hydrocarbon that has a positive $\Delta H_f$, we need to have triple bonds between carbon atoms, or highly strained carbon-carbon bonds (such as in small alicyclic rings). An example of the former case is acetylene C2H2, for which $\Delta H_f = 227.4 kJ/mol = 8.7 kJ/g$ and an example of the latter is cyclopropane $(C_3 H_6)$, for which $\Delta H_f = 53.2 kJ/mol = 1.26 kJ/g$. Cyclopropane was used in the past as an anesthetic gas in surgical operations, but this use was discontinued because of the fire/explosion hazard related to it. In addition to molecules with small ring structures, there are also other molecules with large “steric hindrance” like tetra-tert-butylmethane or cubane that have or are predicted to have a positive heat of formation.

Inorganic combustible substances with positive $\Delta H_f$ include hydrazine ($N_2 H_4$) and cyanogen ($C_2 N_2$). Both are nasty toxic compounds, and hydrazine can also explode when heated, even with no oxygen present, as it decomposes violently to elemental hydrogen and nitrogen if it’s given enough activation energy (this can happen with acetylene, too). Cyanogen is a gaseous compound that is formed from two cyano groups (-CN). The cyano group is called a pseudohalogen, as it is often found in organic molecules in positions where there could also be a halogen (fluorine, chlorine, bromine or iodine) atom (see chlorobenzene and cyanobenzene). A stoichiometric mixture of cyanogen and oxygen can reach a flame temperature of about 4500 degrees Celsius.

Figure 1. Molecular structures of cyanobenzene (left) and chlorobenzene (right).

Reactive metals like magnesium or aluminum often have large heats of combustion, for example the $\Delta H_c (Al)$ is about -838 kJ/mol or -31 kJ/g which is less per gram than the typical values of hydrocarbons, but more per unit volume because Al metal has a significantly higher density than liquefied hydrocarbons. Aluminum is not something that a layman would think of as a combustible fuel, but it is actually very flammable when it’s in the form of very fine powder, and it is used in thermite mixtures and flash powders (pyrotechnic mixtures of Al powder with oxidizers such as potassium perchlorate, which produce a very loud bang and a temperature of over 3000 deg C when ignited in a confined space).

Figure 2. Magnesium metal burns with a really high-temperature flame, which makes it useful in firestarters (source: https://en.wikipedia.org/wiki/Magnesium#/media/File:Magnesium_Sparks.jpg )

Figure 3. The thermite mixture, made from finely powdered aluminum and iron oxide, burns with a high temperature flame and has been used in the welding of railroad tracks. (source: https://commons.wikimedia.org/wiki/File:ThermiteReaction.jpg )

When estimating the maximum temperature that can be reached in the combustion of some substance, there is a need to consider not only the enthalpy changes of the reactions, but also the heat capacities of the products that are formed. A quantity denoted $T_f$, and called adiabatic flame temperature, is the temperature that would theoretically be reached when the fuel reacts with oxygen in a system that is thermally insulated (to prevent heat loss to the surroundings) but isobaric (can do work on surroundings by expanding). A basic estimate of $T_f$ is

where $\Delta H_c$ is the heat produced in the combustion of 1 mole of the fuel and

$C_p$ is the constant-pressure heat capacity

of the product mixture at a temperature of about 1000 deg C (partial derivative of enthalpy with respect to temperature at constant pressure). This formula is not accurate for the most energetic combustions, as most combustion reactions don’t proceed all the way to the stoichiometric end products in high temperatures. This is because at high temperatures, the molar entropy change $\Delta S$ of the reaction starts to be a significant factor in determining the molar Gibbs energy change (and equilibrium constant) of the process, and smaller molecules (like $H_2$ and $O_2$ instead of $H_2 O$) usually have a larger molar entropy than large ones. At high temperatures the heat capacity also increases dramatically, because the Boltzmann factor $k_B T$, which gives the energy scale corresponding to absolute temperature T, starts to approach the energy scale of molecular vibrational (and later also electronic) transitions, and energy is therefore distributed to the vibrational and electronic degrees of freedom of the reaction products. At really high temperatures, like inside the Sun, matter is in the form of ionized plasma, but that kind of extreme conditions can’t be produced chemically.

When considering the combustion of reactive metals, another thing that limits the maximum reaction temperature is the boiling point of the reaction products such as $MgO$ or $Al_2 O_3$. The flame temperature in metal combustion can’t usually exceed the boiling point of the most volatile product, as the heat of combustion is not as large as the heat of vaporization of the reaction products. The combustion of zirconium metal in pure oxygen can raise the temperature up to 4000 degrees Celsius, because of the very high boiling point of zirconium oxide.

The factors mentioned above limit the maximum temperature attainable in a chemical combustion reaction to about 5000 degrees Celsius, which is approximately the adiabatic flame temperature of dicyanoacetylene, a derivative of cyanogen. The amount of pressure rise in the combustion reaction, which was mentioned as one measure of the energeticity of the reaction, depends on both the flame temperature and the difference in the number of moles of gas is the reactants and products. Nuclear reactions such as the fission of uranium-235 are able to produce much higher temperatures, up to millions of degrees Celsius, due to the very large energy release in a relatively small volume.