# Definition of heat and temperature

This chapter is theoretically relevant to Section R1(i) of the 2023 CICM Primary Syllabus, which expects the trainees to "define heat and temperature". It is not something that has appeared in previous papers, nor does one genuinely expect a solid SAQ with this subject as a sole focus, but to expect a definition in the opening statement of an answer to some other temperature-related question would be a common trope of the CICM First Part exam.  For this reason, the trainees should be prepared to confidently define "heat" and "temperature".  To extrapolate further, historical SAQs where basic scientific definitions were asked had also often implied that the candidates had to give a range of other, unasked definitions, as if this was an essential part of a complete answer, which means the discussion of such esoterica as "latent heat" and "specific heat" would not be a completely pointless digression. Still, the reader is reminded that a workmanlike understanding of these concepts is likely all that is required for exam purposes, which means one could realistically restrict oneself to memorising the contents of this grey box:

• Internal energy of a thermodynamic system (U) is the total energy contained within the system (i.e. excluding kinetic energy or potential energy in relation to other systems), and can be conceptualised as the combination of:
• microscopic kinetic energy, or thermal energy, which is the sum of the motions of all the particles in the system, and
• microscopic potential energy, which is the sum of all  chemical and nuclear particle bond forces, internal electric or magnetic dipole moments, as well as the energy of deformation of solids (stress-strain)
• as well as many other forms of energy
• Temperature is the average kinetic energy of the molecules or atoms of a system (i.e. the average microscopic kinetic energy).
• Thermal energy is the sum kinetic energy of the molecules or atoms of a system.
• Heat is a difference in thermal energy, the net energy transferred from one object to another because of a temperature difference.
• Heat is measured in joules (J), or in calories (cal), where 1 calorie is the energy required to raise the temperature of 1ml of water by 1 °C
• Heat capacity is the change in thermal energy due to a change in temperature.
• Specific heat capacity is the thermal energy required to raise the temperature per unit of mass, measured in joules per kelvin per kilogram.
• T​​​​​​he specific heat capacity of a substance is mostly determined by the complexity of its molecules and the molar mass (low molecular mass substances have higher specific heat capacities)
• The specific heat capacity of the human body is about 2.98 kJ·kg−1·°C−1
• Molar heat capacity is the thermal energy required to raise the temperature per molar amount of substance, measured in joules per kelvin per mole
• Latent heat is thermal energy required to produce a phase change, and does not change the temperature, whereas sensible heat is thermal energy which produces a temperature change but no phase change
• Specific latent heat is the thermal energy required to convert 1 kilogram of a substance from one phase to another at a given temperature (in joules per kg).
• Latent heat of vapourisation is the heat required to convert a substance from liquid to vapour at a given temperature. Latent heat of vapourisation decreases as ambient temperature increases, and is reduced to zero at the critical temperature of that substance.

Of the peer-reviewed resources available to explain this subject  Sullivan & Spencer (2022) and Kenny & Jay (2013) were the most lucid, or at least they appeared that way after looking at all the others. Wilson (2024) is a worthy contender but would be lacking in detail for the level expected from the crosstable viva. For a raw béton brut treatment which consists mostly of definitions, the reader can grind through the 3rd edition of the Glossary of Terms for Thermal Physiology by the IUPS (2001).  No CICM exam candidate should ever have to handle a textbook on thermodynamics, and they are redirected from any such thinking by stern warnings about time management and mental health. Nor would one be essential to grasping these basic concepts; nor did any specific physics textbook stand out in its clarity among the others, though this is perhaps a comment on the authors's own ability to comprehend their content.

## Definitions of heat and temperature

In the process of pursuing a well-digested CICM-exam-ready definition of heat and temperature, one's attention cannot help but become distracted by the enormous body of literature which deals with the difficulty of defining and teaching these terms. Probably the most interesting was a series of reflections by a Turkish professor in the Boğaziçi University Journal of Education (Sözbilir, 2003), but other works were also illuminating (Mac & Young, 1987Cartlon, 2000; Doige & Day, 2012), in the sense that it was helpful to understand just how random and erratic the definitions of these terms are across textbooks, even within the same discipline. In the context of having seen this, the incoherence of the cryptic engravings on the official CICM physiology monoliths is much less baffling. It is not that the textbook authors are intentionally obscuring the facts,  or were themselves confused - no, in fact the concepts are just genuinely difficult to communicate. Still, these are the official textbooks, and the trainees of CICM are expected to learn their definitions for heat and temperature. Here they are, arranged in order of least to most verbose.

Brandis, year unknown but purchased by the author in the early 2000s, p.276:

"Heat is a form of energy."

Kam (2015, p.382):

"Temperature is a measure of the average kinetic energy of a substance per degree of freedom of its constituent molecules."

Middleton (2021, ch.4):

"Heat: the quantity of thermal energy contained in a substance."

"Temperature: when a body is at any temperature the individual molecules are at a range of energy levels; the temperature is related to the mean energy of the molecules."

Davis and Kenny (5th ed, p. 114):

"Heat is a form of energy that can be transferred from a hotter substance to a colder substance, the energy being in the form of the kinetic energy of the molecules of the substance."

"Temperature is the thermal state of a substance which determines whether it will give heat to another substance or receive heat from it, heat being transferred from the substance at the higher temperature to the substance at the lower temperature."

Is there an even more official answer that supercedes the college textbooks? Perhaps the Commission for Thermal Physiology of the International Union of Physiological Sciences, who published the last changes to their  Glossary of Terms in 2001:

Temperature: A measure of the mean kinetic energy of the molecules in a volume.

Heat: see Energy.

Energy: Energy may occur either as chemical, electromagnetic, or mechanical energy. [Ws = J]. Synonymous to mechanical energy (force times length, height or distance) is work, synonymous to thermal energy is heat. Work or heat per unit time (→ work rate or → heat flow) constitute physically → power, and are used as components in the body heat balance equation.

It is perhaps fortunate that the rest of the official textbooks do not make any attempt at this, because the trainee is already at risk of drowning in definitions. Guyton & Hall, 13th ed, have an entire chapter on heat and body temperature regulation (p.911-922), but define neither. Nor do Ganong (23rd ed., p.285) or Boron & Boulpaep (2017, p.1193). Similarly, Stoelting (2015, p. 77) prefer to just focus on the pragmatic bread and butter of making sure the patient does not freeze on the table during your anaesthetic. On reflection, this is probably the right approach, as the life of a patient in the hands of a CICM fellow is never going to balance precariously on the results of their debate with a physicist.

Still, it would feel wrong to leave things there. A hostile physicist, should one ever be encountered in the ICU, would probably be informed by some classical introductory text such as Pearson (2020) or Fermi (2012), and their definitions of heat and temperature would probably be grounded in the theory of thermodynamics, which treats heat and temperature as basically mechanical phenomena, except referring to the mechanical interactions of the incomprehensibly vast writhing mass of molecules and atoms which makes up a physically tangible solid. When contemplating the collective movement of this many tiny objects, all individual interactions lose their meaning and importance, and only the average properties of the whole system need to be considered. The simplification leading from this is that the temperature of a system can be defined as a measure of the average kinetic energy of the atoms and molecules in the system, and this is the approach taken by the branch of statistical mechanics, to which Gibbs, Boltzmann and Maxwell had contributed, and which has led to the development of chemical thermodynamics used to describe the energy interactions in the theatre of cellular biochemistry which the intensivist will be more familiar with. That's where all that talk of "energy" and "calories" comes from. To put an attractive box around it:

"Heat: the quantity of energy that flows across the boundary between the system and surroundings because of a temperature difference between the system and the surroundings."

To elaborate further, heat is something that only appears during an exchange of energy, i.e. when the "internal energy" or "thermodynamic energy" of a system (all that collective kinetic energy of its vibrating particles) is being added to or subtracted from, which is what a rise or drop in temperature is. The definition of internal energy by IUPAC actually incorporates this concept:

"Internal energy, U: Quantity the change in which is equal to the sum of heat, q, brought to the system and work, w, performed on it, ΔU=q+w. Also called thermodynamic energy"

This "internal energy" is integrated with the pressure and volume of a system to give us the concept of enthalpy, which is defined by IUPAC as:

"Enthalpy: Internal energy of a system plus the product of pressure and volume. Its change in a system is equal to the heat brought to the system at constant pressure."

These concepts all form a part of the equation of state, to elaborate on which would be an unforgivable digression even in the extremely distorted view of this writer. It will suffice to leave a Wikipedia link, and mention offhandedly that this is a relationship that describes the conditions of any matter at any given temperature pressure and volume, and that the ideal gas law is an example of such a relationship.

The reader satisfied with easily memorised definitions would surely not have read as far as this, and would therefore be expecting greater depth, asking perhaps of the particles, the atoms and molecules., why do they vibrate so? And how is this different to the mechanical energy of, for example, the vibration of an extremely anxious fidgety human? Well. The macroscopic movement of those sweaty fingers is the combined change in coordinates for a whole mass of atoms which was directionally coherent enough to survive the effects of averaging all the tiny microscopic oscillations of those atoms. Sure, each atom may be oscillating chaotically, but as a group they all oscillated vaguely more in the same direction on average, and this has produced the macroscopically observable effect of fiddling with a clicky pen. This is the kind of physics you would have to call mechanics, and the energy imparted to the pen lid is kinetic energy. Thermodynamics, on the other hand, is the physics that refers to those microscopic movements of atoms that do not survive this statistical averaging. To borrow an excellent turn of phrase from Herbert B. Callen (1985),

"Thermodynamics... is concerned with the macroscopic consequences of the myriads of atomic coordinates, that, by virtue of the coarseness of macroscopic observations, do not appear explicitly in a macroscopic description of a system"

In other words, those atoms and molecules can be vibrating chaotically, and energy can be transferred to this chaos, increasing the frequency and amplitude of those vibrations, but because of their disorganised anarchy, no macroscopic change in the position of the observed material would take place.

This disorganised kinetic energy of randomly bouncing particles is the result of interactions between charged particles of that matter, more specifically, the electrons. An electron is a wave function and vibrates at a frequency which is proportional to its energy (well, this is a huge oversimplification, but the soft brain of the author clings to anything that might simplify these concepts). These frequencies are quantal, as there is a range of discrete orbital energy states, and electrons cannot occupy an "in-between" state. For hydrogen, the frequency for the first orbital is something like 1.3×1017 Hz, with an energy of 13.6 eV.  Electrons can gain energy, ascending to a higher energy state (an "excited" state), in which their frequency increases. This can happen due to the addition of more electromagnetic energy, for example with the absorption of electromagnetic radiation, or it can occur due to the collision of that charged particle with another charged particle, which happens more frequently when the substance is compressed (which is Charles' law). The electrons can also lose enthusiasm and drop to a lower energy state, releasing energy in the form of electromagnetic radiation, which at normal room temperature conditions usually ends up being in the infrared spectrum. In fact they constantly do so, unless the substance is at absolute zero (in which case the atoms still vibrate because of zero point energy, but would emit nothing).

Molecules also vibrate as the result of these electron frequencies interacting with one another, albeit with a total frequency which is lower, something like 1013-1014 Hz. This vibration describes the internal movement of the atoms within the molecule, in relation to a motionless centre of the molecule mass, and can occur in several degrees of freedom (consider that the atoms can move apart in three dimensions and also rotate in relation to their own axis). The vibration of a molecule, interacting elastically with neighbouring molecules, imparts a kinetic energy to it, and the average kinetic energy of a mass of such molecules is where we get the definition of temperature from.

For an ideal gas, which has nothing much going on except the random kinetic bouncing of perfectly elastic molecules, temperature can actually be described in terms of average kinetic energy, as follows:

where

•  is obviously temperature,
• N is the number of molecules of the gas,
• n is the number of moles,
• mv2 is the kinetic energy,
• R is the universal gas constant which relates the energy scale to the temperature scale, in terms of  work (joules) per degree per mole,
• k is the Boltzmann constant, which (at a risk of linking to a Feynmann lecture and completely losing all readership to a better science communicator) is the equivalent of the universal gas constant, except expressing its energy per temperature increment per each individual particle instead of per mole.

The temperature of two systems can be different, and if those systems can interact, they will exchange their internal energy (the higher energy system supplying energy to the lower) until they achieve an equilibrium, where the average kinetic energy of their microscopic constituents ends up being the same. This transfer of energy is the definition of heat, and the energy transferred in this fashion is referred to as thermal energy or heat energy, even though in reality it is really still just kinetic energy on a microscopic scale.

Heat, therefore, is a form of energy, we are forced to conclude, somberly reflecting that we spent two thousand words only to come back to the same one-line definition by Kerry Brandis. Chemical energy, which is a potential energy stored in chemical bonds, can be released or absorbed in the form of heat energy (when bonds break or form), which means that a mass of molecules at a constant non-zero temperature and pressure has the potential to do some chemistry. Specifically, the amount of work that can be done by the system is described by the Gibbs free energy of the system, and this determines whether or not a chemical reaction will run at the specified temperature and pressure, or what direction it will run in.

But this is already too much.

## Difference between heat and temperature

Temperature is not the total internal energy of a system, but rather the measure of the average internal energy, and cannot describe heat, because heat is the flow of energy. Usually, this is a concept that requires an example, and one which is commonly offered to students usually relies on describing two objects with markedly different masses and temperatures, to demonstrate how one may be much cooler than the other, but still possess more internal ("thermal") energy. One may have a grain of sand heated to a temperature of 500 degrees Celsius, and it has a higher temperature than a one-billion-ton iceberg, but the iceberg has much more internal energy than the superheated sand grain. Heat is the transfer of energy from the sand grain to the iceberg, the change in the thermal energy of each object which will continue until both are at an equilibrium.

## Units of measurement used to describe heat

It is all fine and good to discuss the agitated animation of the combined or averaged kinetic energy of 12.046 × 1023 atoms in the mole of salt, but to use these definition of internal energy or temperature, we need to operationalise them in some way before we descend into madness. It would be inconvenient to discuss this sort of stuff with a colleague or use it to describe the course of an experiment. At this stage one needs to put down the Fermi and pick up a Cengel

As mentioned already, the transfer of internal "thermal" energy from one system to another resembles mechanical work, which is the transfer of mechanical energy, and is therefore susceptible to being described in the same terms. They are of course not the same thing: work can be transformed into heat, and some heat (but not all!) can be transformed into work, but the two concepts are sufficiently distinct that most physics textbooks make a reasonable effort to dismiss any idea of their equivalence from the mind of the reader before moving on to the subject of joules.

The joule is the SI unit of energy, and is therefore the correct measure for heat as well as for the energy deposited in a gold foil by an Xray beam or the transfer of kinetic energy between a beer can and the head of a fratboy. This unit is equal to 1kg × m2 × s2, conventionally defined as the amount of work done when a force of one newton displaces a mass through a distance of one metre. It is also the heat produced when an electric current of one ampere passes through a resistance of one ohm for one second, or the work required to move an electric charge of one coulomb through an electrical potential difference of one volt, or the work required to produce one watt of power for one second, among many examples. These discussions put one in a position where the thermal energy of an object can be described using conventional macroscopic parameters, for example the change in thermal energy that results from a change in temperature.

## Specific heat capacity, latent heat and sensible heat

Heat capacity, the ability of a substance to store thermal energy, or rather its thermal energy response to the change in temperature, is defined by IUPAC as

"Heat capacity, C: Heat brought to a system to increase its temperature divided by that temperature increase"

Heat capacity and specific heat capacity are slightly different, where the former does not require a mass for the definition, whereas the latter is defined as:

"Specific heat capacity: the heat energy required to raise the temperature per unit of mass, measured in joules per kelvin per kilogram"

Each substance has its own specific heat capacity, as the degree to which the injection of energy is going to disturb one's molecules depends on what kind of molecules they are. In general the specific heat capacity of "simple" materials is somewhat lower than the specific heat capacity of more complex ones, as large molecules can absorb a lot of kinetic energy into various interesting interactions between internal structural elements instead of just bouncing around idiotically. Conversely, simple materials can still just absorb more energy because they often contain more atoms. Hydrogen, which does bounce around idiotically, has a specific heat capacity of 14.3 J/g.K because each gram contains roughly one mole of hydrogen, whereas mercury makes it markedly easier (0.14 J/g.K, where one mole of mercury weighs 200.59 g). This brings us to the concept of molar heat capacity:

"Molar heat capacity: the heat energy required to raise the temperature per molar amount of substance, measured in joules per kelvin per mole"

The specific heat capacity of water is 4.184 J per ml per °C, and this us the definition of the calorie, a non-SI unit of heat measurement slowly being moved towards obsolescence by the international community.

These definitions also give rise to allied terms used occasionally in medicine, or at least in the context of primary exams. One such term is the latent heat:

Latent heat is the heat released or absorbed by a system during a process that occurs without a change in temperature.

That definition is not an official quote from any specific textbook and merely represents an attempt by the author to produce a one-liner to explain this concept.  "Latent", a word often used to mean "dormant" or "inactive", in this context is an archaic use of the original latin root latere, which means "to lay hidden". The word was originally coined in the 18th century by Joseph Black, who observed that phase changes, for example the melting of ice, all seemed to require more energy than merely heating the same amount of water would. The definition specifically requires that the thermal energy change needs to occur without change of temperature, which means that it is usually going to be associated with a phase change, as there are few other ways to add thermal energy to a system without changing its temperature. For this reason most textbooks (eg. Cengal) seem to define latent heat as

"Latent heat: the amount of energy absorbed or released during a phase-change process"

This will obviously be different for every substance, as each substance will have molecular and atomic peculiarities which will make it easier, or more difficult, to compel into a new phase state. This is reflected through the concept of specific latent heat, which is a material characteristic. To borrow a definition from an ancient copy of Davis & Kenny's Basic Physics and measurement in Anaesthesia (1992),

Specific latent heat: the heat required to convert 1 kilogram of a substance from one phase to another at a given temperature (in joules per kg).

For the anaesthetist, this has a special interest, as their tendency to evaporate gases for profit directly connects this concept to their income. The most important anaesthetic latent heat definition is therefore the specific latent heat of vapourisation:

Latent heat of vapourisation is the heat required to convert a substance from liquid to vapour at a given temperature. Latent heat of vapourisation decreases as ambient temperrature increases, and is reduced to zero at the critical temperature of that substance.

(where the  critical temperature for gases is the temperature beyond which they cannot help but remain gases, i.e nothing will compel nitrous oxide to remain a liquid at any pressure if the temperature is higher than 36.5 °C.)

This is to be contrasted with the latent heat of condensation, which is the energy released during something returning into a liquid state after being a vapour, and which is the same amount of energy. Similarly the latent heat of fusion, which is the energy absorbed during something melting, is the same as the latent heat of solidification. The main point is that during these phase changes the temperature of the substance remains the same, even as it evaporates or consolidates. The student of physics, well accustomed to having their mind blown, will often be taken far aback by the realisation that boiling water will remain at 100 °C while it boils, no matter how high one turns up the burner of the stove.

Latent heat is itself the opposite of sensible heat, which is another archaic term used to describe an exchange in thermal energy that does change the temperature. Again, without a specific source, a one-line definition could be:

Sensible heat: a change in thermal energy that results in a change of temperature

Its terminology reflects that it can be sensed, i.e. it creates a detectable change (whereas latent heat is energy that appears to have gone missing). The original characterisation of sensible heat by James Joule was literally as "the heat that is indicated by the thermometer", to set it as distinct from Black's  latent heat, which did not change the measured temperature. When bringing a bucket of ice up to room temperature from 0°C, the latent heat is the heat required to melt the ice, and the sensible heat is the heat required for the rest of the 21°C. The reason the term "sensible heat" is encountered in human physiology is mostly related to the fact that a fair amount of human heat loss occurs in the form of latent heat, where thermal energy is used to evaporate sweat.

But this is again a digression. The bottom line is that it is possible to calculate the total thermal energy contained in a mass, provided one chooses a starting and finishing state.   If the body of a 80kg patient was considered to consist of mostly water (and let's face it, often in the ICU they actually are), to bring their body temperature all the way up from 0°K would require 310.2 K × 4184 J × 80 kg, or 103, 830,144 joules. In fact the specific heat capacity of all tissues is known, and is lower than water on average, approximated as 2.98 kJ · kg−1 · °C−1 by Xu et al (2023). The authors wen on to compile an excellent table of different specific heat properties of many tissues:

This means that the gain of one degree of fever, for an 80kg patient, would represent a change of 238,400 joules; and that it might even be possible to calculate exactly how much of this thermal energy was absorbed by their eyeballs or their skeleton.  That these calculations are possible does not diminish the fact that it would be patently insane to discuss the thermal energy content of a patient in those terms, or to try to represent their fever in terms of calculated thermal energy gain. Which brings us to:

## Thermometry and calorimetry

Thermometry is the measurement of temperature, whereas calorimetry is the measurement of energy, and the latter term is usually applied to measurements of metabolic energy expenditure by indirect calorimetry, which makes sense because that is the main source of thermal energy in the human organism. Direct calorimetric measurements of heat produced by the human body can be also obtained, but these require terrifying devices such as the Snellen flow calorimeter which can quantify body heat content from measurements of the temperature of known masses of air or water which exchange heat with the subject, as well as their oxygen consumption and CO2 production.  One would not expect a new mother to do this at the moonlit cot of a toddler, and so something more convenient must be made available, so that thermal energy changes in patients can be observed and reported.  Fortunately, the change in body thermal energy, ΔQ, can be represented as:

ΔQ = ΔT × M × Cp

where

• ΔT  is the change in temperature
• is the mass of the body
• Cp is the the average specific heat of the tissues of the body

Though it is true that the ICU patient is often observed to interact with their environment by ejecting meaningful amounts of their mass into it, we can generally expect their mass to remain mostly stable within a certain sensible range of values. Similarly the average specific heat of the tissues is not a characteristic we can expect to fluctuate wildly. Temperature is therefore an excellent surrogate for thermal energy measurement.

## Units of measuring temperature

The preferred SI unit of measurement for temperature is the kelvin (and not "degrees Kelvin"). The system was devised to count up from absolute zero, remains the system preferred for calculations in the basic sciences, and was defined specifically so that Boltzmann’s constant would be exactly 1.380649×10−23J/K, to remain consistent with Celsius (i.e. one degree Celsius and one degree Kelvin would be the same).

An important fact about the Celsius temperature scale is that it is not an absolute scale. That is to say, unlike the calorimetric measurement of thermal energy content, one cannot say that an object contains twice as much thermal energy if it has doubled in temperature. On the other hand the Kelvin scale is coupled to the definition of kinetic energy, which means a doubling of the Kelvin temperature does represent a doubling of the thermal energy.

An unimportant fact is that it seems medicine has settled on Celsius mostly arbitrarily, from among an entire range of possible competing temperature scales (except in America where a baffling attachment to colonial units of measurement has preserved the Fahrenheit system). Just like Fahrenheit, the Celsius scale was an arbitrary division of gradations between two arbitrarily chosen points. In the case of Celsius, the points were the freezing and boiling temperatures of water, and in the case of Fahrenheit, they were several, including the freezing temperature of a proprietary solution of ammonium chloride, boiling water, and the temperature of the human body.

To settle on Celsius seems to have been a consensus decision to use the base 10 metric system, though in all honesty the Fahrenheit scale is not exactly unparsable binary code, and could have served just as easily. Another element of the convenience of Celsius was probably the use of water, rather than some weird brine, to define the scale.  To settle on the term "degree" seems to have come from the gradations used on thermometers, with "degree" itself deriving etymologically from the old French degré , meaning step. Nobody seems to have a clear idea of how the superscripted circle (°) ended up the symbol for temperature notation, except that it started to appear in the literature in the 16th century, and was presumably a pragmatic recycling of the "zero" glyph, where typography struggled to catch up with the increasingly bizarre demands of printed mathematics.

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