Posts Tagged ‘Temperature’

This prototype displays temperature, relative humidity, dew point temperature and absolute humidity

As shown in previous posts on the CarnotCycle blog, it is possible to compute dew point temperature and absolute humidity (defined as water vapor density in g/m^3) from ambient temperature and relative humidity. This adds value to the output of RH&T sensors like the DHT22 pictured above, and extends the range of useful parameters that can be displayed or toggled on temperature-humidity gauges employing these sensors.

Meteorological opinion* suggests that dew point temperature is a more dependable parameter than relative humidity for assessing climate comfort especially during summer, while absolute humidity quantifies water vapor in terms of mass per unit volume. In effect this added parameter turns an ordinary temperature-humidity gauge into a gas analyzer.


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I used an Arduino Uno microprocessor and a wired DHT22 sensor with data output to a 16×2 liquid crystal display. Circuit components are uncomplicated: a 10 kΩ potentiometer, 220 Ω resistor and a few jumper and breadboard wires are all that is needed, power supplied by a 9V battery* after programming via USB.


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I wired the LCD as per guidance on the Arduino website. The pot controls contrast on the LCD. The DHT22 was wired to take 5V from the breadboard power rail with sensor data routed to digital pin 7. The sensor version that I used (Adafruit AM2302) has a built-in 5.1 kΩ pull-up resistor.

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Note: the last line – consisting solely of a closing brace } – got chopped off this image.

The DHT22 has a sampling rate of 0.5 Hz which some regard as a weakness, but in the context of a temperature-humidity gauge the criticism is rather academic since it would serve no purpose to output data to the LCD at such a rapid rate. I set the display refresh to 30 seconds. Note the built-in option to display ambient temperature and dew point temperature in Celsius or Fahrenheit.

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I used the unit to investigate the change in temperature and humidity parameters in a bathroom (enclosed volume 11.6 m^3) before and after operating the shower at a temperature of 40°C for about 5 minutes. The sensor was placed 60 cm above floor level at the midpoint of the room.

Here is the data display before the shower

and after the shower

The displayed data shows that bathroom temperature stayed constant during the experiment while the relative humidity increased markedly. This result could have been obtained with an ordinary temperature-humidity gauge, but the smart gauge gives additional information.

In contrast to the steady ambient temperature, the dewpoint temperature shows a sharp rise from a comfortable 11.8°C (53°F) to a humid 18.3°C (65°F). The absolute humidity data shows an even greater increase – a 50% hike in water vapor concentration from 10 to 15 grams per m^3 in a matter of minutes.

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© P Mander, June 2018




The Arrhenius equation explains why chemical reactions generally go much faster when you heat them up. The equation was actually first given by the Dutch physical chemist JH van ‘t Hoff in 1884, but it was the Swedish physical chemist Svante Arrhenius (pictured above) who in 1889 interpreted the equation in terms of activation energy, thereby opening up an important new dimension to the study of reaction rates.

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Temperature and reaction rate

The systematic study of chemical kinetics can be said to have begun in 1850 with Ludwig Wilhelmy’s pioneering work on the kinetics of sucrose inversion. Right from the start, it was realized that reaction rates showed an appreciable dependence on temperature, but it took four decades before real progress was made towards quantitative understanding of the phenomenon.

In 1889, Arrhenius penned a classic paper in which he considered eight sets of published data on the effect of temperature on reaction rates. In each case he showed that the rate constant could be represented as an explicit function of the absolute temperature:


where both A and C are constants for the particular reaction taking place at temperature T. In his paper, Arrhenius listed the eight sets of published data together with the equations put forward by their respective authors to express the temperature dependence of the rate constant. In one case, the equation – stated in logarithmic form – was identical to that proposed by Arrhenius


where T is the absolute temperature and a and b are constants. This equation was published five years before Arrhenius’ paper in a book entitled Études de Dynamique Chimique. The author was J. H. van ‘t Hoff.

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Dynamic equilibrium

In the Études of 1884, van ‘t Hoff compiled a contemporary encyclopædia of chemical kinetics. It is an extraordinary work, containing all that was previously known as well as a great deal that was entirely new. At the start of the section on chemical equilibrium he states (without proof) the thermodynamic equation, sometimes called the van ‘t Hoff isochore, which quantifies the displacement of equilibrium with temperature. In modern notation it reads:


where Kc is the equilibrium constant expressed in terms of concentrations, ΔH is the heat of reaction and T is the absolute temperature. In a footnote to this famous and thermodynamically exact equation, van ‘t Hoff builds a bridge from thermodynamics to kinetics by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant Kc is in fact the quotient of the kinetic velocity constants for the forward (k1) and reverse (k-1) reactions


Substituting this quotient in the original equation leads immediately to


van ‘t Hoff then argues that the rate constants will be influenced by two different energy terms E1 and E-1, and splits the above into two equations


where the two energies are such that E1 – E-1 = ΔH

In the Études, van ‘t Hoff recognized that ΔH might or might not be temperature independent, and considered both possibilities. In the former case, he could integrate the equation to give the solution


From a starting point in thermodynamics, van ‘t Hoff engineered this kinetic equation through a characteristically self-assured thought process. And it was this equation that the equally self-assured Svante Arrhenius seized upon for his own purposes, expanding its application to explain the results of other researchers, and enriching it with his own idea for how the equation should be interpreted.

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Activation energy

It is a well-known result of the kinetic theory of gases that the average kinetic energy per mole of gas (EK) is given by


Since the only variable on the RHS is the absolute temperature T, we can conclude that doubling the temperature will double the average kinetic energy of the molecules. This set Arrhenius thinking, because the eight sets of published data in his 1889 paper showed that the effect of temperature on the rates of chemical processes was generally much too large to be explained on the basis of how temperature affects the average kinetic energy of the molecules.

The clue to solving this mystery was provided by James Clerk Maxwell, who in 1860 had worked out the distribution of molecular velocities from the laws of probability. Maxwell’s distribution law enables the fraction of molecules possessing a kinetic energy exceeding some arbitrary value E to be calculated.

It is convenient to consider the distribution of molecular velocities in two dimensions instead of three, since the distribution law so obtained gives very similar results and is much simpler to apply. At absolute temperature T, the proportion of molecules for which the kinetic energy exceeds E is given by


where n is the number of molecules with kinetic energy greater than E, and N is the total number of molecules. This is exactly the exponential expression which occurs in the velocity constant equation derived by van ‘t Hoff from thermodynamic principles, which Arrhenius showed could be fitted to temperature dependence data from several published sources.

Compared with the average kinetic energy calculation, this exponential expression yields very different results. At 1000K, the fraction of molecules having a greater energy than, say, 80 KJ is 0.0000662, while at 2000K the fraction is 0.00814. So the temperature change which doubles the number of molecules with the average energy will increase the number of molecules with E > 80 KJ by a factor of more than a hundred.

Here was the clue Arrhenius was seeking to explain why increased temperature had such a marked effect on reaction rate. He reasoned it was because molecules needed sufficiently more energy than the average – the activation energy E – to undergo reaction, and that the fraction of these molecules in the reaction mix was an exponential function of temperature.

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The meaning of A

But back to the Arrhenius equation


A clue to the proper meaning of A is to note that e^(–E/RT) is dimensionless. The units of A are therefore the same as the units of k. But what are the units of k?

The answer depends on whether one’s interest area is kinetics or thermodynamics. In kinetics, the concentration of chemical species present at equilibrium is generally expressed as molar concentration, giving rise to a range of possibilities for the units of the velocity constant k.

In thermodynamics however, the dimensions of k are uniform. This is because the chemical potential of reactants and products in any arbitrarily chosen state is expressed in terms of activity a, which is defined as a ratio in relation to a standard state and is therefore dimensionless.

When the arbitrarily chosen conditions represent those for equilibrium, the equilibrium constant K is expressed in terms of reactant (aA + bB + …) and product (mM + nN + …) activities


where the subscript e indicates that the activities are those for the system at equilibrium.

As students we often substitute molar concentrations for activities, since in many situations the activity of a chemical species is approximately proportional to its concentration. But if an equation is arrived at from consideration of the thermodynamic equilibrium constant K – as the Arrhenius equation was – it is important to remember that the associated concentration terms are strictly dimensionless and so the reaction rate, and therefore the velocity constant k, and therefore A, has the units of frequency (t^-1).

OK, so back again to the Arrhenius equation


We have determined the dimensions of A; now let us turn our attention to the role of the dimensionless exponential factor. The values this term may take range between 0 and 1, and specifically when E = 0, e^(–E/RT) = 1. This allows us to assign a physical meaning to A since when E = 0, A = k. We can think of A as the velocity constant when the activation energy is zero – in other words when each collision between reactant molecules results in a reaction taking place.

Since there are zillions of molecular collisions taking place every second just at room temperature, any reaction in these circumstances would be uber-explosive. So the exponential term can be seen as a modifier of A whose value reflects the range of reaction velocity from extremely slow at one end of the scale (high E/low T) to extremely fast at the other (low E/high T).

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P Mander September 2016

Relative humidity (RH) and temperature (T) data from an RH&T sensor like the DHT22 can be used to compute not only absolute humidity AH but also dew point temperature TD

There has been a fair amount of interest in my formula which computes AH from measured RH and T, since it adds value to the output of RH&T sensors. To further extend this value, I have developed another formula which computes dew point temperature TD from measured RH and T.

Formula for computing dew point temperature TD

In this formula (P Mander 2017) the measured temperature T and the computed dew point temperature TD are expressed in degrees Celsius, and the measured relative humidity RH is expressed in %

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Strategy for computing TD from RH and T

1. The dew point temperature TD is defined in the following relation where RH is expressed in %

2. To obtain values for Psat, we can use the Bolton formula[REF, eq.10] which generates saturation vapor pressure Psat (hectopascals) as a function of temperature T (Celsius)

These formulas are stated to be accurate to within 0.1% over the temperature range –30°C to +35°C

3. Substituting in the first equation yields

Taking logarithms


Separating TD terms on one side yields

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Spreadsheet formula for computing TD from RH and T

1) Set up data entry cells for RH in % and T in degrees Celsius.

2) Depending on whether your spreadsheet uses a full point (.) or comma (,) for the decimal separator, copy the appropriate formula below and paste it into the computation cell for TD.

Formula for TD (decimal separator = .)


Formula for TD (decimal separator = ,)


3) Replace T and RH in the formula with the respective cell references. (see comment)

Your spreadsheet is now complete. Enter values for RH and T, and the TD computation cell will return the dew point temperature. If an object whose temperature is at or below this temperature is present in the local space, the thermodynamic conditions are satisfied for water vapor to condense (or freeze if TD is below 0°C) on the surface of the object.

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P Mander August 2017


The above image from the northern latitudes of Mars, taken by the Viking 2 Lander in May  1979, revealed the presence of a white substance on and around the red rocks. We get this white stuff on Earth too. It’s called frost.

I can remember when I first saw this image confirming the presence of water frost on the Martian surface, I took an instant step closer to believing in the possibility of Little Green Men. During the Martian summer, the temperature can rise above freezing point, and lo! there would be liquid water, the swimming pool of life.

After revelling in this idea for a while, the thought struck me that I had directly transferred the phase change behavior of water on Earth to another planet. On reflection however, this appeared perfectly valid. There are no planet-dependent terms in the Clausius-Clapeyron equation, so the phase diagram of water must look exactly the same up there as it does down here.

Sound enough reasoning. The phase diagram is a PT plane, and according to what I was taught, those  thermodynamic variables, along with state functions such as enthalpy of fusion, were considered to apply universally. But then duh! the realization dawned on me – I had failed to consider differences in atmospheric pressure.

At the time, I knew nothing at all about the atmosphere on Mars, apart from the fact that it had one. But at least I could calculate the minimum surface pressure that the atmosphere would need to exert in order for water to undergo a solid-liquid phase change at 0°C. The Magnus-Tetens approximation, which generates saturated vapor pressure (hPa) as a function of temperature (°C)


made things simple since when T=0 the answer is immediately 6.1094 hectopascals, or if you prefer, 6.1094 millibars.

Back then, I never did find out what the atmospheric pressure on Mars was, and my inquiry stalled. Years later I came upon the information by happenstance. The atmospheric surface pressure on Mars is just below 6.1 hectopascals.


Now a glance at the phase diagram for water shows that the triple point temperature is a gnat’s whisker from 0°C, so for practical purposes the triple point pressure can be taken as 6.1 hPa. This is significant because at no point in the PT plane below the horizontal drawn through the triple point does the liquid phase exist. Since the atmospheric surface pressure on Mars lies below this horizontal, the only possible phase change is between solid and vapor. No liquid water can exist on the surface of Mars.

Except. Well, there are craters on Mars, so if at the bottom of a sufficiently deep one the atmospheric pressure were to nudge above 6.1 hPa, and if the temperature crept above 0°C, and if there was surface frost to start with, then liquid water would form. That’s a lot of ifs, but a possibility all the same. There might not be Little Green Men down there, but there might be something else that answers the description of ‘life form’.


Photo credits: (top) Mars / NASA, (below) Little Green Men / Wikimedia
Triple point diagram credit: National Metrology Institute of Japan
(I have added the horizontal line through the triple point)

Sadi Carnot (1796-1832) One of the great original thinkers.

Sadi Carnot (1796-1832) One of the great original thinkers.

The milestone memoir Réflexions sur la Puissance Motrice du Feu (Reflections on the Motive Power of Fire), published by French engineer Sadi Carnot in 1824, marks the starting point of thermodynamics as a theory-based science. In this work, Carnot developed his powerful ideas with the aid of the caloric theory, which viewed heat as an effect caused by an all-pervading, invisible fluid called caloric. An important tenet of the theory was that caloric was considered to be conserved in all thermal processes.

The fact that Carnot published no other work during his short life led later theoreticians, notably Rudolf Clausius and James Clerk Maxwell, into the error of assuming that Carnot never questioned the validity of the caloric theory. But after Carnot’s death in 1834, a bundle of his papers was found whose contents reveal that he had not only questioned the caloric hypothesis, but had reached the point where he felt compelled to abandon it in favour of its eventual successor, the dynamical theory.

This cannot have been an easy decision for Carnot, since the rejection of the caloric theory in favour of the dynamical theory robbed him of the very principle he had employed in Réflexions sur la Puissance Motrice du Feu to reach his groundbreaking conclusions regarding the motive power of heat.

In this post, I will examine the dilemma that Carnot faced in contemplating this conceptual shift.

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When Carnot began thinking about how steam engines turned heat into work, the theory of machines that produced work through mechanical force supplied by men or animals, the wind or a waterfall was already well understood.

In his memoir, Carnot begins by saying that “a similar theory is evidently needed for heat-engines”, and commences his analysis by noting that the production of motive force in steam engines is always accompanied by the transportation of caloric from a body at elevated temperature to another whose temperature is lower.

It should be noted that in the early 19th century when Carnot was writing his memoir, steam engines had such low thermal efficiency that any consumption of caloric in its passage between the hot and cold reservoirs would barely have been perceptible. So it was quite reasonable for Carnot to think that caloric was conserved by a steam engine in motion.

But how did a steam engine produce motive power? With the aim in mind of ‘a similar theory’ to the mechanical engine, Carnot likened the passage of caloric to the passage of water from a higher reservoir to a lower reservoir as it drives a water wheel – there is no loss of water, and motive power depends on the quantity of water transported and the height of the waterfall.

In an equivalent way, Carnot saw the motive power of a steam engine arising from the quantity of caloric transported, and the ‘height of its fall‘, by which he meant the difference in temperature between the hot and cold bodies. And as with the water wheel analogy, the process involved no loss of caloric.

With this model-based principle in place, Carnot was ready to start finding answers to important questions.

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The first question Carnot poses is one that had long been asked by engineers: Is the motive power of heat invariable or does it vary with the working substance employed to realize it?

To furnish an answer, Carnot conducts a thought experiment involving an imaginary steam engine undergoing a cyclical sequence of four entirely reversible isothermal and adiabatic operations, the hot and cold reservoirs being maintained at given temperatures – a stroke of conceptual genius which we know today as the Carnot cycle.

In this steam engine, Carnot imagines the transfer of a quantity of caloric from a body A to a colder body B, both maintained at a given temperature. Assuming no loss of motive power of caloric, he shows that the motive power produced in a single Carnot cycle can be used to return the same quantity of caloric from body B to body A by operating the cycle in the reverse direction. He then concludes: “An indefinite number of alternative operations of this sort could be carried on without in the end having either produced motive power or transferred caloric from one body to the other.”

He then puts the question “Can there exist a working substance that makes better use of the heat than the steam employed in the cycle just described?” and uses the following argumentation to supply the answer:

If it were possible by any method whatever to make the caloric produce a quantity of motive power greater than we have made it produce by our first series of operations, it would suffice to divert a portion of this power in order by the method just indicated to make the caloric of the body B return to the body A from the refrigerator to the furnace, and thus be ready to commence again an operation precisely similar to the former, and so on: this would be not only perpetual motion, but an unlimited creation of motive power without the consumption either of caloric or of any agent whatever. Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible.

To put it in modern parlance, in the first series of operations using steam as the working substance, a quantity C of caloric is transported from body A to body B, producing a quantity M of motive power. Then operating the cycle in reverse, the same quantity M of motive power is used to transport the same quantity C of caloric from body B back to body A, thus restoring the initial conditions. The net result is that no caloric is transferred and no motive power produced.

In the second series of operations using a working substance that is imagined to be more effective than steam, an identical quantity C of caloric is transported from body A to body B, but this time it produces a quantity M+m of motive power. Then operating the previously described cycle (using steam) in reverse, the quantity M of motive power is used, as before, to transport the quantity C of caloric from body B back to body A, thus restoring the initial conditions. The net result is that no caloric is transferred, but a quantity m of motive power is produced – an operation which could be endlessly repeated, producing an unlimited quantity of motive power from nothing. This violates the principle and disproves the initial assertion. Carnot thus concludes: “the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable by any means whatever“.

Carnot has proved that the  motive power of heat is independent of the working substance employed to realize it.

He then asks a second question – whether the motive power of heat is unbounded or subject to an assignable limit – and goes on to prove by further thought experiment that its quantity is fixed solely by the temperatures of the bodies between which the transfer of caloric is effected.

The important point to note is that Carnot arrives at these powerful results by applying the principle that  caloric is a conserved quantity in thermal processes.

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But even as his memoir was going to print (he published it at his own expense), it appears that doubts about the caloric theory were already forming in Carnot’s mind. A sentence in the manuscript which describes the theory as “beyond doubt” was changed at proof stage into less certain language: “the theory … does not appear to be of unquestionable solidity. New experiments alone can decide the question.

The kind of experiments Carnot had in mind are revealed in the bundle of papers found after his death. The 23 loose sheets, containing questions, speculations, fragments of essays, and proposed experiments almost identical to those Joule was later to conduct, chart Carnot’s increasing belief that heat and work are equivalent. He lists Rumford’s experiments on the boring of cannon and the friction of wheels on their spindles among the experimental facts undermining the caloric theory.

One of the sheets contains the following paragraph, in which Carnot identifies heat and work as interconvertible forms of a conserved quantity, and effectively states the first law of thermodynamics in relation to cyclical thermodynamic processes (ΔU = 0, Q – W = 0) :
Heat is simply motive power, or rather motion which has changed its form. It is a movement among the particles of bodies. Wherever there is destruction of motive power, there is at the same time production of heat in quantity exactly proportional to the quantity of motive power destroyed. Reciprocally, wherever there is destruction of heat, there is production of motive power.

On one of the last sheets, Carnot writes:
When a hypothesis no longer suffices to explain phenomena, it should be abandoned. This is the case with the hypothesis which regards caloric as matter, as a subtle fluid.

But as Carnot becomes increasingly convinced that heat and motive power are interconvertible, he is at the same time caught in a dilemma. Because whereas the caloric theory enabled him to prove an assertion by disproving the contrary assertion, the dynamical theory fails to do so.

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As Carnot contemplated the interconvertability of heat and work, he would no doubt have re-run the reversible steam engine thought experiment on the basis of the dynamical theory:

In the first series of operations using steam as the working substance, a quantity H of heat is taken from body A, a quantity M of motive power is produced, and a quantity H-M of heat is transferred to body B. Then operating the cycle in reverse, the quantity H-M of heat is taken from body B, a quantity M of motive power is consumed, and a quantity H of heat is transferred to body A, thus restoring the initial conditions. The net result is that no heat is transferred and no motive power produced.

In the second series of operations using a working substance that is imagined to be more effective than steam, an identical quantity H of heat is taken from body A, but this time a quantity M+m of motive power is produced. Taking h to be the quantity of heat equivalent to the quantity m of motive power, a quantity H-M-h of heat is transferred to body B. Then operating the previously described cycle (using steam) in reverse, the quantity H-M of heat is taken as before from body B, a quantity M of motive power is consumed, and a quantity H of heat is transferred to body A, thus restoring the initial conditions.

The net result is that a quantity h of heat has been taken from the colder body B and a quantity m of motive power has been produced, an operation which could be endlessly repeated. But since we have defined the equivalence relation h=m, there is no violation of principle and the dynamical theory therefore fails to disprove the absurd result of the limitless production of motive power solely by consuming the heat of a body.

This is the core of the dilemma Carnot faced.
He doubted the caloric theory, but it proved his assertions.
He favoured the dynamical theory, but it did not prove his assertions.

One of the loose sheets found after Carnot’s death reveals this difficulty in a passage where he convinces himself that heat can be converted into motive power and vice versa. But while he sees many advantages in this hypothesis, he notes “it would be difficult to explain why, in the development of motive power by heat, a cold body is necessary; why motion cannot be produced by consuming the heat of a body.

The explanation, as we now know, is provided by the second law of thermodynamics. Although Carnot had no time in his short life to grapple with this dilemma, it is not fanciful to suggest that he would have found the solution had he lived. More than twenty years before Clausius, Sadi Carnot had already effectively stated the first law, and as the last quote shows, he only needed to turn his question into an assertion to find the essential statement of the second law.

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