The Height of Temperatures. An Dinner Poser

How high can a temperature get?

By lim juboo



I was having a quiet and uneventful dinner last evening, and was least expecting anything to happen. So I thought?



A Medical Emergency?


All of a sudden my dinner was rudely interrupted. A Malay neighbour’s wife a few doors away rushed into my kitchen and called me to help in an emergency. I left everything, food still in my mouth, rushed over expecting some kind of life-death tragedy.



There I saw was her daughter barely 4 years old crying badly. I was told she knocked a pot of hot water in the kitchen over her. It was an accident. I dutifully examined her, expecting a bad scald. Fortunately it was not.



The very fact she was crying in pain immediately assured me the scald was not serious enough. Had it been much hotter burns from a flame, the burns would have destroyed the deeper dermis of her skin, and damage the nerves. In that case there would no longer be any pain. That would be a very serious burn. Even a boiling water scald can never exceed 100 deg C, and boiling water scalds do cause very severe pains, but no nerve damage.



In most cases of 30 burns, the patient may need antibiotics, sedatives, ATS / ATT (anti-tetanus serum / adsorbed tetanus toxoid) injections, and at a later stage, skin grafting. The burns in such cases are severe and would be a real medical emergency.



But in her case, there was just redness over her arms, legs, and neck, but fortunately her face and eyes were spared. Her thick non-absorbent clothes protected her body to a great extent. I assessed her as a first degree burn, and it also did not involve a very large area of her body.



That how we do a primary survey on a casualty to triage cases in a mass disaster, giving low priority for cases where a victim shouts a lot in pain.



But patients never understand why we need to leave them alone first, and very urgently attend to those who are unconscious and unable to utter a word.



Priority is given (triaged as red) for those who lose consciousness, and need instant attention to manage ABC (Airways, Breathing and Circulation) as a Trainer for St John Ambulance Malaysia.



But hers was just a minor event with a lot of shouting and crying. In fact I have not seen any real extensive 3 degree burns that require extensive care in my clinical attachment at the Emergency Department of Kuala Lumpur Hospital as a Trainer with St John Ambulance Malaysia.



Of course her mother viewed this differently, and I do not blame her for her worries and anxiety. In her case, cooling the affected area with running water will do, and some moderate analgesic like paracetamol, or ibuprofen may be given. Other non-steroidal anti-inflammatory drugs (NSAIDs), although may be gastric irritating, may also be given to relieve pain. Her mother asked me if she could give her daughter aspirin for the pain, and I advised her not because of the risk of Reye's syndrome with aspirin for children.



I then told her to observe for post-burns infections, pus formation or fever. If blisters developed, I advised not to break them. They are there to protect against post burns infections. I also told them not to apply oil, butter, lotions, ointments, or even put any dressing on. Just leave the affected part dry. The pain and redness will soon go away, and the skin will quickly heal. If any signs of infection set in, she needs to be hospitalized immediately.



Had it been a serve thermal or electrical burn, I would immediately advise them to go to a hospital that has a special burn unit for intensive management. She may possibly need to sleep on an air-fluidized bed to prevent bed-skin contacts. But all these were unnecessary after I examined and assessed her.



It was not much of an event after all, but it was sufficient to spoil my quiet and peaceful dinner anyway!



How cold is coldest, how hot is the hottest?


I went back to my kitchen to finish my half-eaten dinner, and observed the beautiful hot blue propane-butane flame heating up the kettle of water for my mug of tea. I then thought to myself, just how hot can a flame get?



We know that there is a lower limit for temperature. It is the absolute zero, defined as a temperature of precisely 0 Kelvin, which is equal to −273.15 °C or −459.68 °F.


But I asked myself, if there is a lower scale of temperature, would there be an upper limit for temperature? I pondered over this as I continued my dinner.



This thought triggered me to write this article. But before that…



The School Experiments that Fascinated Me:


I remember enjoying seeing and hearing the hissing hot greenish-blue flame jetting out from an air-acetylene Bunsen burner in my school science laboratory when I first studied science in school. I used to be very fascinated by the colour and hissing sound of this flame. In fact I was fascinated by most of the experiments our school science teacher demonstrated to us.



I remember how a glowing wooden splinter burst into flame when trusted into a jar of oxygen, and how magnesium burns even more violently and brilliantly in oxygen. I remember how potassium and sodium burst into flames when put into a trough of water, and how carbon dioxide turns lime water (calcium hydroxide solution) milky as calcium carbonate is formed. It was the test for carbon dioxide.



I remember how blue litmus paper turned red in acid, and red litmus turned blue in alkaline. I remember how we can make ‘ammonia water fountain’ inside a round bottom flask because of the great solubility of ammonia in water that generated a ‘vacuum’ inside the flask for the water to be sucked up.



I remember a101 things about all those experiments and observations even as a small boy. I suppose these were the things that triggered my interest in science and mathematics (taught in English) as a young lad.



The Hottest Flame as a Lad:


By the time I reached Form 5, we were taught that the hottest chemical flame possible on Earth is the oxy-acetylene flame that reaches a maximum temperature of some 3,100 degrees Celsius (or Centigrade as we used in school those days).


When mixed with air, the air-acetylene flame is about 2,400 0 C. This was the prettiest and fascinating blue-green flames I used to enjoy watching hissing out warningly from a Bunsen burner.



But the oxy-hydrogen flame, attains a maximum of temperature of about 2,700 0 C, which is sufficiently hot for wielding, but when mixed with air, it is about 2,050 0 C.


The butane-propane gas mixture from our kitchen gas cylinder will burn in air with a flame that can reach about 1900 0 C. All these are fascinating figures to me.

Heat and Velocity:

By the time I got into upper Form 6 (A Levels Science) in Singapore, I learnt in classical mechanics the ideal gas law directly provides a relationship between kinetic energy of the particles and the temperature (T). It is given by:
1/2 mv2 rms = 3/2 kT
(where m is particle mass, and rms, the root-mean-square particle for the speed v)

In other words, the higher the temperature, the more energetic the gas particles, and the faster they move. So far, so good!


My career:


After completion of my university education in various courses, I embarked on a 25-year long professional career as a senior medical researcher in nutrition, rural health, community medicine, behavioural medicine, biostatistics, epidemiology, clinical research, traditional and complementary medicine at the Institute for Medical Research, Ministry of Health Malaysia and in food and medical toxicological research at MIT.


The Sun and Stars that Fascinate:


However, what fascinate me most are the stars and galaxies. I then began to study astronomy as a hobby instead. It was my childhood hobby, but set it aside for years because I wanted to concentrate on my higher education and professional career.



I began to recall what I studied in Science in school. I realize in chemistry class in school, the temperature from chemical combustion such as the oxy-acetylene flame actually pales into nothing in comparison to the temperature of nuclear reactions in the Sun. The temperature on the surface of the Sun for instance, is about 5,778 Kelvin (5505 °C). The Sun converts about 600,000,000 tons of hydrogen nuclei each second into helium nuclei.



These fusion reactions of hydrogen nuclei convert part of their atomic mass of some 4 million tons into energy, and release an enormous amount of energy into the Solar System. In short, the Sun loses some 4 million tons of mass each second. The Sun will run out of fuel in about 5 billion (5,000,000,000) years at this rate of nuclear consumption.



Surprisingly, 2,000 km above the Sun’s surface and temperatures soar to 100,000 Kelvin. The outer layer of the Sun, called the chromospheres, the temperatures arrive at 1 million Kelvin? The outermost most layer, the corona, the temperatures get into several million (degrees) Kelvin?



The temperature in the interior of the sun is even higher, to the tune of 15 million Kelvin?



The hottest stars found have a surface temperature of some 220,000 Kelvin or at least 40 times as hot as the surface of our sun. These belong to the O Class of bluish white stars which on an average have surface temperatures of 25,000-50,000 K or higher. That is searing hot. We have no idea what their interior temperatures are?



The Lower Scale:


So far so good! That is as far as we can get in practice. But what I have in mind as I write this thought here is, whether or not there a limit to the maximum height of temperatures? We know there is a limit of ‘coldness’ which is precisely absolute zero (zero Kelvin) or minus 273.15 ° Celsius. (minus 459.68 °F).



How Hot Up?


I reckon, if there a limit how cold we can go down the temperature scale, there must be a limit how hot we can get up the scale? But is there a upper theoretical limit to temperature? This is what I attempt to pen as my after dinner thoughts.



One source claimed that the maximum possible temperature is the Planck temperature, or 1.41679 x 1032 Kelvins. But how valid is this? I shall get back to this later.



The sun is pretty hot as I mentioned. It is easy for us, particularly me as a non physicist, to calculate out the amount of radiation the Sun emits by measuring the solar constant on Earth, namely the energy the sun radiated out within a sphere enclosed by a radius between the Sun and Earth whose mean distance is 1.496x108 km. We can measure the heat received per square metre here on the Earth’s surface.



We can then calculate back how much heat the Sun emitted over that Earth-Sun distance. I have done that a few times, and it tallied very precisely with the figures solar scientists have done using very advanced measurements and sophisticated calculations. But we will not go into that as it is not the subject of my thought for the moment.



Kinetics of Hot Gases:


Let us take a look at the kinetic energy of molecules instead. I remember well my ‘O Level’ physics even till this day. It tells us that the kinetic energy (E) of a moving particle is equal to ½ mv2



Where, ‘m’ symbolizes the mass of a particle and ‘v’ its velocity. If we solve ‘v’ in the equation E = ½ mv2, we get v = √ (2E /m) = 1.414 √ (E/m) …Equation 1



But since the kinetic energy is also a measure of its temperature (T) as I said earlier, we can then substitute ‘T’ for ‘E’ in Equation 1. The numerical constant will be changed for the equation to be expressed in SI Unit. We can then say that v = 0.158 √ (T/m) …Equation 2.



Now then, if in Equation 2 the temperature (T) is given in Kelvin, and the mass (m) of an atomic particle is given in atomic units, then the average velocity (v) of the particles will come out in kilometers per second.



Since 1 km = 0.621 miles, if we want to change the formula so that the results will come out in miles per second, all we need to do is to replace the constant 0.158 in the equation with 0.158 x 0.621 = 0.098 to give the answer in miles per second.



Molecular velocities:



For instance, the atomic mass unit of hydrogen atom, the commonest and most abundant atom in the universe is 1. But helium gas which the Sun converts from hydrogen, is composed of individual helium atoms, each with an atomic mass unit (amu) of 4, or more precisely, 4.002602 amu. Suppose we decide to freeze a cylinder of helium gas down to the freezing point of water (0 deg C) or 273 Kelvin. We now substitute 273 for ‘T’ and 4 in ‘m’ as in Equation 2.



We find that even if we decide to cool helium to 0 deg C, the helium atoms with an atomic mass of 4, will still be able to move about with an average velocity of 1.31 kilometers per second or 0.81 miles per second. The muzzle velocity of the fastest rifle bullet is 1500 m/s. This means a helium atom even if you decide to ‘freeze’ it to zero degree C, will still be able to fly out at the speed of the fastest rifle bullet.



Similarly we can work out for other values of ‘T’ and ‘m’. The velocity of oxygen molecules with an atomic or molar mass of 32 in a tropical temperature like Malaysia of 30 deg C (303 Kelvin) for instance, works out to be 0.158 √ (303 / 32) = 0.49 km / sec. or 0.30 miles per second.



The velocity of carbon dioxide molecules (molar mass of 44) at boiling point of water (373 K) turns out to be 0.46 km /sec. Argon atoms with an atomic mass unit of 39.948 when heated in an electric arc to 3000 0 C (3273 Kelvin) will move at a velocity of 1.4 km per second. We can work on this on and on for various atoms and molecules so long we know their atomic mass numbers and the temperatures they are being heated.



Equation 2 tells us that at any given temperature, the less massive the particle, the faster it moves. It also tells us that at absolute zero (where T = 0) the velocity of any atom or molecule, irrespective its mass, is always zero. This is another view point about the absoluteness of absolute zero. It is the temperature of almost absolute atomic or molecular rest where there is no heat or temperature. It is when temperature is at its lowest, and absolutely nothing could move.



But if a velocity of zero marks the lower limit, we may then ask isn’t there an upper limit to the velocity too? Isn’t this upper limit constrained by the velocity of light? In fact ‘yes’ this is true, at least theoretically.


So Have We Arrive?



When the temperature goes so high such that ‘v’ reaches the speed of light and can no longer climb higher, we may ask if we have reached the maximum and absolute upper limit of temperatures? It is tempting to say ‘yes’ and that could be the ultimate height of temperature or hotness? Let’s presumed it be so for the moment. But let us have another look at the equation


If we solve ‘T’ in Equation 2, it turns out:


T = 40 mv2

(where ‘T’ is in Kelvin, and ‘v’ in kilometers per second)



Let’s us now substitute the value ‘v’ (the molecular velocity) as equivalent to the maximum possible velocity which is the velocity of light at 299 792.458 kilometers per second. When we do that, it turns out that the maximum possible temperature
(T max) must be:



T Max = 3 600 000 000 000 Kelvin (Equation 3)



Now we may ask what about the value of ‘m’ (the mass of the particles in question)…Equation 4



We can see, the more massive the particle, the higher the maximum temperature! This puts the value of T max into question because of the varying values of atomic mass. We are right to question this validity.



Nuclear Fusion & Temperatures:


At temperatures to the tune of millions of Kelvin, all molecules and atoms will be broken down into nuclei. At temperatures of hundreds of millions and into the lower billions as in the Sun and stars, nuclear (fusion) reactions begin between simple nuclei to build up into complicated nuclei



At still upper temperatures, the reversal takes place where all nuclei will break apart into simple sub-atomic particles of protons and neutrons.



Let us make a theoretical assumption that in the region of the maximum possible temperature of over a trillion degree, only protons and neutrons can exist. These have a mass of 1 on the atomic scale.


Mathematically from Equation 3, we must are forced to wrap up that the maximum possible temperature is:


3 600 000 000 000 Kelvin (degrees) Kelvin


Or shall we rest at this point? Let us have another look.



Not Yet…?


For one thing, if we look at Equation 1 again we are only assuming that the value of ‘m’ always remain constant whatever its kinetic / dynamic status. But this is an assumption. We only assume that if a helium atom has an atomic mass of 4, it will always assume a mass of 4 under whatever conditions? But is this true? Well as a matter of fact the physics I studied in school makes no other assumption.



This would be true if we view the universe in the eyes of Sir Isaac Newton. But in a Newtonian universe there is no mathematical prescription to limit velocity and neither is there an upper limit to temperature.


Special Theory of Relativity:



But Albert Einstein saw this differently. The Einsteinian view of the universe he prescribed an upper limit of velocity, namely the velocity of light, and this offers us some doubt on an upper limit of heat if we do not consider mass a constant under any circumstances.


But the mass of any object no matter how infinitesimal small under ordinary conditions, as long as it is greater than zero, will increase in mass as its velocity increases. At least this is Einstein’s view in his equation on Special Relativity:


M = m / √ (1 – v 2 / c2)



Where M is new mass, m = rest mass at 0 velocity, v = velocity of mass, c = velocity of light



We can easily see that as the initial mass increases in velocity (v), and becomes closer and closer to light (c) the denominator (1 – v 2 / c2) tends towards zero. Hence when the rest mass (m) is divided by a zero denominator, the new mass (M) tends towards infinity. This means it could even be more massive than the Universe. In short, it tells us that any mass no matter how small, becomes indefinitely large as it gets closer and closer to the velocity of light.



At ordinary practical velocities say of no more than a few thousand kilometers per second, the increase in mass is quite insignificant, and there is nothing to worry about for Newtonian physicists except for hair-splitting and neurotic calculations by some scientists with obsessive compulsive disorder.



But when we are working at velocities close to the velocity of light, ‘m’ becomes very large and tends towards infinity irrespective its initial particle mass. If we do that, we find consequently there will be no upper limit for ‘T max’



No Answer?


We have now no choice but to conclude there may after all no ‘maximum’ practical or theoretical limits to temperature, whether in an Newtonian or an Einsteinian universe. So is that all, and there is really no maximum to the height of temperature?



However, I have earlier promised I shall get back to what some physicists claim that the maximum possible temperature is the Planck temperature of 1.41679 x 1032 Kelvins?



Disturbed Thought:


Now, this is very disturbing to me.


If the Planck temperature of 1.41679 x 1032 Kelvin is truely the maximum temperature theoretically possible according to scientists specializing in quantum physics, it would be interesting to work out what the velocity would be, say of a hydrogen atom at that temperature?



We shall use this same equation in classical mechanics, a principle first developed by Gottfried Leibniz and Johann Bernoulli to determine the velocity of atomic particle



I shall show you how at very high temperatures this equation in classical mechanics is longer valid and need to be replaced by relativistic mechanics. Let us give it a try.


Since T = 40 mv2


Rearrange,

V2 = T/40m

∴ V = √ (T/40m)

Since 1.41679 x 1032 =40 mv2

Let the atomic mass unit (m) of hydrogen is 1


Then V = √ (1.41679 x 1032 /40) = 1.8820 x 1015 km s-1



As we can see, at Planck temperature of 1.41679 x 1032 Kelvins, the velocity of a hydrogen atom, or any particle would be 1.8820 x 1015 km per sec?



We know that the velocity of light is 299 792 458 metres (299 792.458 km) per sec. But this is a lot faster than light. Naturally this is not possible. It only shows that classical mechanics of kinetic energy can no longer be applied for such extremely ultra high temperature. It is only valid for speeds are lower than that of light.


Relativistic Mechanics


We need to switch into relativistic mechanics to deal with motions where energy levels subject particles near the speed of light or even faster than light.


Let us assume Planck temperature is possible, and that sub atomic particles travel faster than light. If that is so, let us get back to the equation:


M = m / √ (1 – v 2 / c2)


If physicists think that ‘v’ (velocity of sub-atomic particles at Planck temperature) > ‘c’ (velocity of light), then (v 2 / c2) would be greater than 1 (where v > c)


In that case (1 - >1) = a negative number. But in mathematics we know that we cannot get the square root of a negative number. Such a number is not real. It is an imaginary number (i)


An Imaginary Universe?


If such temperature 1.41679 x 1032 is possible, than it must belong to another universe, an imaginary universe, or a negative universe opposite to ours?


I really do not know the answer myself from the mathematical logic I have just work out for you. I leave this to you to strain your own imagination.



But if you think this temperature is possible (like some physicist think), then I have to admit this type of quantum physics beats me. It is completely beyond me. Perhaps the brainer physicists can explain how they arrive at Planck temperature. I am not capable of doing that. As far as I have worked out in this simple article of mine, they may not be an upper limit to temperature after all, whether from the classical Newtonian universe point of view, or in an Einsteinian universe.



The Real Universe:


So far spectral measurements of some of the hottest stars are in the region of only 50 000 -60 000 Kelvin. They are the blue stars in the ‘O’ Class. But none has been found to exceed a temperature of 250,000 Kelvin, and it may well be that is probably the hottest in a practical sense we can get to in the universe. In such case, even in temperature of 250 000 Kelvin, subatomic particles would have a velocity of just 79 kilometers per second.


My Simple Dinner Thoughts:


I am after all a retired government medical scientist and now a free lance physician who would shy from arguing with the brainer quantum physicists.



My thought on the physics of temperatures is only a simple one. I mulled over this question during my interrupted dinner after attending a medical emergency call from a neighbor.



It was not much of a medical event for me after all, but a deeper thought for me and others to argue about.


ljb


December 12, 2011