Twice as Cold as 0°C

It probably says more about me than anything else, but I've been asked the following question at least half-a-dozen times. In the interests of having a place to point future questioners I offer the following answer to the (apparently common) question:

if the temperature today is 0°C, and it will be twice as cold tomorrow, what will the temperature be tomorrow?

And the instant answer is:


Which is, in human terms, rather cold. The lowest temperature recorded and confirmed on Earth is -89.4°C, recorded on 21st July 1983, at Vostok, a Russian research station in Antarctica.

It's also a rather dramatic drop in temperature. A lot colder than most people guess when they ask this or similar questions.

So, how did I arrive at this apparently drastic figure, and how do you divide zero by two and get a number other than zero in the first place?

The answers to both questions derive from the same, usually overlooked, point: the Celsius temperature scale, like the Fahrenheit scale (and many other, now obsolete, temperature scales such as the Newton, Romer, Delisle, Leyden, Dalton, Wedgewood, Hales, Ducrest, Edinburgh and Florentine scales) is a relative scale.

0°C isn't the same as 0mm wide or 0V of electrical potential. Both these latter are absolute measures. You can't get narrower than 0mm and you can't get less electrical potential than 0V.

You can get colder than 0°C, however, since 0°C is just the freezing point of water.

So, to arrive at the frigid forecast above I simply converted the first figure to an absolute temperature scale (the Kelvin scale), halved it and then converted it back to Celsius.

Makes sense now? I didn't think so.

Let's step back a bit and take a look at the relative temperature scales, starting with the oldest temperature scale still in regular use.

The Fahrenheit scale, developed in 1724 by Gabriel Fahrenheit, used mercury to measure changes in temperature, since mercury exhibits consistent changes when it undergoes thermal change. Mercury expands and contracts as the temperature changes and this volume change is both uniform across a wide range and large enough to measure accurately.

In addition, mercury is cohesive rather than adhesive, so it doesn't stick to the only transparent substance Fahrenheit had access to: glass. Finally, mercury is bright silver, making it easy to visually distinguish changes in liquid volume in a narrow tube.

Fahrenheit began by placing his mercury thermometer in a mixture of salt, ice and water. The point the mercury settled to on his thermometer was considered zero.

He then placed the thermometer in a mixture of ice and water. The point the mercury settled to this time was set as 30. Finally, 'if the thermometer is placed in the mouth so as to acquire the heat of a healthy man' the point the mercury reaches is set to 96.

Using this scale, water boils at 212 and it freezes at 32. This latter number is an adjusted figure on Fahrenheit's part: it made the difference between boiling and freezing a relatively clean 180.

[NB, the above chronology isn't the only possible process Fahrenheit undertook. The Wikipedia article on the Fahrenheit temperature scale notes several other mooted explanations. Cecil Adams's The Straight Dope site also covers the origins of the Fahrenheit scale, focusing on the more amusing (or bemusing) possibilities.]

Less than twenty years after Fahrenheit's scale was developed, the Celsius scale was created by Swedish astronomer Anders Celsius. His scale used the freezing and boiling points of water as the two key markers and put 100 degrees between the two temperatures.

Unlike today, however, in Celsius's original scale, water's boiling point was 0 and the freezing point 100.

In the years after his death in 1744, the numbering scheme was reversed. This change is routinely credited to another great Swede, Carl Linnaeus (also known as Carolus Linnaeus) but the evidence for this is circumstantial and not particularly convincing.

Numbering scheme aside, the modern Celsius scale (used pretty much everywhere on earth except the United States) is different from the one Celsius developed.

It doesn't make much difference in day-to-day use, but the basis of the modern Celsius scale is the triple-point of water. The triple-point of a substance is the temperature and pressure at which the solid, liquid and gaseous states of said substance can all co-exist in equilibrium. And the triple-point of water is defined as 0.01°C.

As well, each degree Celsius is now defined abstractly. In Celsius's original scale a one degree change in temperature was defined as a 1% change in relative temperature between two externally referenced circumstances (ie the boiling and freezing points of water).

Today, a degree Celsius is defined as the temperature change equivalent to a single degree change on the ideal gas scale.

The ideal gas scale brings us almost to the point (finally, I hear you cry). As noted at the beginning of all this, the temperature scales above are relative scales: they give you a useful number to describe the thermodynamic energy of a system but they do so by creating a scale which is relative to some physical standard (whether that be the triple-point of water or the 'heat of a healthy man').

Back in 1787, however, Jacques Charles was able to prove that, for any given increase in temperature, T, all gases undergo an equivalent increase in volume, V.

Rather handily, this allows us to predict gaseous behaviour without reference to the particular gas being examined. It's as if gases were fulfilling some Platonic conceit, all acting in a fashion essentially identical to an imagined ideal gas. Hence the 'ideal gas scale' which describes the behaviour of gases under changing pressure without reference to any particular gas.

The Platonic ideal falls apart at very high pressures because of simple physical and chemical interactions. For the sort of pressures needed to use a gas as a thermometric medium (ie measurer of temperature) on earth, however, all gases exhibit the same, very simple behaviour described by the following equation:

pV = [constant]T

or in words:

pressure multiplied by Volume = [a derived constant] multiplied by Temperature

Which means if you keep the pressure constant, as the temperature changes so does the volume. Or, if you change the temperature and keep the volume constant, the pressure goes up or down in direct relation to the temperature's rise or fall.

One very nifty thing about this is the way it makes it possible to create a temperature scale which is independent of the medium used to delineate the scale.

Back in 1887, P Chappuis conducted studies using gas thermometers which used hydrogen, nitrogen, and carbon dioxide as the thermometric media at the International Bureau of Weights and Measures (BIPM). Regardless of the gas he used, he found very little difference in the temperature scale generated. If the temperature of the gases changed by a value T, and the pressure, P, was held still, the increase in volume, V, was the same regardless of the gas being used to set the scale.

This change in thermodynamic activity has been recognised and accepted as the fundamental measure of temperature, since it is derived from measures of pressure and volume that aren't dependent on the substance being measured.

One of the most important consequences of this discovery is the recognition that there is a naturally defined absolute zero temperature value. When the pressure exerted by a gas reaches zero, the temperature is also zero. It is impossible to get 'colder' than this, since at this temperature all atomic and sub-atomic activity has ceased. (And, before anyone asks, yes I know what negative temperature is, and it isn't a temperature 'below absolute zero.' Systems with negative temperature are actually hotter than they are when they have positive temperature.)

In 1933 the International Committee of Weights and Measures adopted a scale system based on absolute temperature. It is called the Kelvin scale and uses the same unitary value for single degrees as the modern Celsius scale. So a one-degree change as measured by the Kelvin scale represents the same change in temperature as a one-degree change as measured using the Celsius scale.

The zero-point for the Kelvin scale, however isn't an arbitrary one (eg the freezing point of water) but the absolute one.

Absolute zero is, as it happens, equivalent to -273.15 C, so converting between K and C is a simple matter of addition or subtraction:

C = K - 273.15
K = C + 273.15

So 0 degrees Celsius is 273.15 Kelvin. Using standard notation for each scale we can re-state this sentence thus:

0°C = 273.15K

Note there is no degree symbol used when denoting a temperature in Kelvin. And, just as there is no degree symbol, the word isn't used either. The phrase 'degrees Kelvin' is incorrect: just use the word 'Kelvin.'

Which brings us, finally, to explaining how I arrived at the temperature I listed at the beginning of this article. As I noted above, the Celsius and Fahrenheit scales are relative scales, so you can't compare two different temperatures measured using these scales absolutely.

20°C is not twice as warm as 10°C, since both are a measure relative to the triple point of water.

The Kelvin scale, however, is an absolute scale. Different values measured using this scale are related in absolute comparative terms. 20K is twice as warm as 10K (although both values are pretty damned cold relative to what you or I are comfortable with).

So, to find out what temperature (in degrees Celsius) would be 'twice as cold' (ie half the temperature) of 0°C I simply converted the value to Kelvin:

0 + 273.15 = 273.15K

Divided this value by 2:

273.15/2 = 136.575 K

and converted it back to degrees Celsius:

136.575 - 273.15 = -136.575°C

Working in the other direction, twice as hot as 0°C is easy to calculate. It's 273.15°C. Which is rather hotter than any human can handle.

If nothing else, this demonstrates how narrow a range of temperatures suit human beings. Let's presume -10°C – 50°C is a useful range of liveable temperatures for human beings.

I'm being generous with this range. The low is, in human terms, well below the freezing point of water. And the high is, again in human terms, a long way above blood temperature. This range is only acceptable as a liveable range if we assume 1) the range refers to measured temperatures and 2) we have technology capable of keeping experienced temperatures (eg, in a dwelling or next to human skin) from reaching these extremes of heat and cold.

Converting this to Kelvin, we have a range of 263K – 323K. (I'm leaving the 0.15 off: it doesn't change the arithmetic, other than to needlessly complicate things.)

The lowest temperature in this range is 81% of the highest temperature in this range. 323K (50°C) is only 19% warmer than 263K (-10°C).

Change the liveable range to 0°C – 40°C (a range more genuinely liveable, especially if we assume only basic available technology) and the hottest we can reasonably handle is only 13% warmer than the coldest we can live with.

Be even more conservative, and restrict the range so it runs roughly through the human comfort zone: 10°C – 35°C (a range that goes from 'cold if you don't have warm clothing' through to 'hot in the sun but bearable if there's almost any sort of breeze') and the hottest weather we can comfortably manage is only 9% warmer than the coldest most of us are willing to deal with.

No wonder folk are concerned about a 0.6°C increase in global surface temperatures over the last 100 years.

An Alternative History of Australian Football

The Year is 1997. The month is September. Finals time. And a more hotly contested, more widely open to upset finals series is not within living memory. For the first time since the inception of the National Football League almost six years ago the final five consists of teams from five different states.

The West Adelaide Eagles have scraped into the minor premier's spot with a 3 point victory over Launceston in the final minor round game. The win edges them ahead of Freemantle, now relegated to second position on percentage. In third spot, only a game away, is the Northern Territory, rejuvenated in the season's second half by two key players returning after injury. Hobart sits in fourth place two games further behind. But the real excitement is the first appearance in the final five of a Victorian team. In fifth spot, a point behind Hobart and a point ahead of Port Adelaide, is Williamstown Ports, created from the merger, in '95, of Williamstown and Port Melbourne from the old VFA.

The Directors of the NFL could not be happier. Back in 1990 the alternatives facing them are either a slow death in isolation (as the old SANFL was suffering) or a quicker one in capitulation (as the old WAFL was undergoing) The bold notion of actually taking on the AFL seems to be, at the very least, a way of going down fighting. Now, in 1997, an almost complete victory seems within sight.

Back in '94 the decision to invite the VFA to join the NFL is widely held to be more provocative and more dangerous than either the initial step, in 1992, of combining the WAFL, the TFL and the SANFL into the NFL, or the further expansion in '93 to include composite clubs drawn from the district leagues of Sydney and Brisbane and a composite team from the Northern Territory The prospect of taking the battle for Australian Football right into the heart of AFL territory is, however, too grand to resist.

With the success of Williamstown Ports the NFL now has the very real prospect of a Finals game in Melbourne. Already negotiations are underway with the Melbourne Cricket Club. If Williamstown Ports makes it through the Elimination Final in Hobart, the NFL wants the First Semi-Final played at the MCG. For something this big the rules which gave home ground rights in finals to the team placed higher on the ladder can and will be pushed aside.

The stage is set for the most exciting finals series in the history of Australian Football. The Managing Director of the NFL sits in his office and allows himself an almost wicked smile. He is about to put a call through to his counterpart in the AFL; this year it will be their job to adjust their fixtures, to play their finals on Sundays and Monday nights. The phone rings. It is the AFL chief calling; they have to talk, he's heard rumours that one of the AFL teams is considering an offer to join the NFL, he's worried about the future of the two leagues in general, neither side can afford the bidding war currently raging, both need to consider opening up a dialogue and working towards some sort of negotiated settlement, maybe even a coalition approach with an end of season Super-Finals series until the TV contracts run out followed by a fair and equal merger. The NFL chief leans back in his chair and puts his feet up on the desk. By this time the smile is one of utter triumph.

No this is not just some blind fever dream. All of the above might sound fanciful, even impossible. Nonetheless, in another time and another place, under a different code and in a different country, most of what is described above actually took place.

The year is 1959, the country is the United States, the national winter sport is Gridiron and The National Football League, or NFL, comprising sixteen teams, is its governing body. Al Davis from Oakland California, Lamar Hunt of Dallas, and others, want to expand the league and acquire NFL franchises in cities and states where it doesn't currently operate but the NFL doesn't want to know. So on August 14th 1959 Lamar Hunt announces his intention to form a second professional football league, to be called the American Football League, or AFL. Their first season is to begin in 1960 and with the Oakland franchise delivered to Al Davis on January 30th of that year they begin with nine teams.

There is no question that the beginnings are shaky. Out in East Oakland games are played on open fields turned into stadiums by the simple expedient of surrounding them with portable stands. There is no fence or gate and admission is collected by passing around a bucket.

But on June the ninth 1960 the AFL signs a five-year, $1.6 million dollar contract with the American T.V. network ABC. The AFL is now in competion with the NFL not only on the field but in the living rooms of America.

The bidding war begins. On January 30th 1961 the NFL enters into an agreement with the CBS network for the rights to all regular-season games at a cost of $4,650,000 a year. On January 29th 1964 NBC pays $36 million for a five-year contract with the AFL for telecasting rights beginning with the 1965 season. On January 24th of the same year CBS pays $14.1 million for regular season rights with the NFL for the 1964-65 seasons, on April 17th they pay a further $1.8 million for the rights to the championship games.

On the playing field a similar bidding battle is taking place. Beginning in 1959 when the infant AFL secures the entire first round college draft choice (rather like the AFL in Australia actually acquiring every SANFL player named in a given year's draft) it culminates in 1965 when the New York Jets pay $400,000 for a then unknown player from Alabama; Joe Namath. In 1965 secret talks are held between the AFL and the NFL and on June 8th it is announced that the two leagues will merge into an expanded league of 24 teams, expanding further to 26 in 1968 and 28 in 1970. To run out existing TV contracts the teams will keep separate fixtures until 1970 but will inaugurate an annual inter-league championship, whch comes to be known as the Superbowl, in January 1967.

Meanwhile back in our what might have been 1997 the Director of the Australian NFL is deep into some hard talking with his now concilliatory rival. The problem of having too many teams for a single league is being discussed. Although mergers have occured there is real grassroots opposition to them going any further. The feeling is that too much merging is going to create clubs and teams with no local identity.

The NFL director smiles yet again, this time almost mischeviously, and asks the AFL chief whether he has considered the option presented by English Soccer. Back in 1892, when faced with a similar problem, they formed what we know know as the Second Division.