Notes on Heat and Temperature

The earliest definition of temperature

In the earliest times if people put their hands into a fire they would have felt that the fire was hot, if they put their hands onto the ice they would feel cold.

A temperature illusion

This definition of temperature based on what is felt by a human hand can be easily fooled by illusions. Fill a glass with water from the hot water tap, a glass with ice water, and a glass with room temperature water. Dip two fingers from one hand into the hot glass and two from the other hand into the cold glass and hold them their for 15 seconds. Then put both pairs of fingers into the room temperature glass, one hand will report that the room temperature glass is hot while the other reports it is cold.Temperature Illusion

A basic rule of thermodynamics

If two bodies are in contact with each other for a long time they will come to the same temperature.

This is how a thermometer is used to measure the temperature of an object. Put the thermometer into contact with the object and they will come to the same temperature.

A Quantitative definition based on linear expansion

Eventually scientists developed quantitative definitions of temperature. They noticed that when materials were heated, they expanded and when materials were cooled they contracted. The expansion was repeatable and pretty linear. So good glassblowers like Fahrenheit and Celsius could make glass tubes with constant inside diameters, fill the tubes with a liquid such as alcohol or mercury, and measure the length of the expanded liquid. They could mark two places on the scale and define temperatures at these places then linearly interpolate to find temperatures between these reference temperatures. Celsius chose his references to be the boiling point of water at one atmosphere pressure which he called 0, and the freezing point which he called 100. Read that again, it is correct. One year later Jean Pierre Christin in France turned the scale over and invented the scale we use today. So the °C should stand for the Christin scale, not the Celsius scale. By the way, the centigrade scale was abandoned for scientific use in 1948. Fahrenheit used as his reference points the freezing point of salt water which he called 0, and the average temperature of several colleagues which he set to be 100. (Now we know that average human body temperature is 98.6 °F.

A problem in the middle and Gas Thermometers

Unfortunately, there was a problem, if a mercury thermometer calibrated at 0 and 100 Celsius reads a temperature of 50 °C for a water bath, then an alcohol thermometer calibrated at 0 and 100 Celsius will not read 50 °C. The temperature reading is different for thermometers made of different materials because thermal expansion is notexactly linear. In an attempt to find a definition of temperature independent of material scientists turned to gas thermometers. All gas thermometers approach the same temperature reading as the gas in the thermometer is made less and less dense. That is, as it is made closer to an ideal gas.

What is Temperature, 19'th century version

When scientists looked into the behavior of gas they discovered the 19'th century definition of temperature.

The temperature of an ideal gas is the
average
kinetic energy of
random
translational motion
per molecule

Basically, the temperature of an ideal gas is the
kinetic energy per molecule.
This means that you can have a high temperature but only a small amount of energy, if you have only a few molecules. Higher temperature gas means faster molecules.

For an ideal gas, temperature counts only the translational kinetic energy, kinetic energy due to rotation of a molecule is excluded. The kinetic energy of rotation and that of vibration are each related to the kinetic energy of translation, (at high temperatures, e.g. above 1000 K they are linearly proportional).

Each one of these words in the definition is important.

Only the random motion of the molecules counts so that to define temperature requires at least two molecules. The combined motion of the molecules (the motion of their center of mass) is ignored in defining temperature. So, the temperature of one molecule is undefined.

The average refers to the fact that at any given temperature some molecules have high speeds and others have slower speeds. Temperature depends on the average kinetic energy.

Temperature Units

Temperature is measured in kelvins, degrees celsius, or for nonscientific use, in degrees fahrenheit. From its definition you would expect temperature to be measured in joules per molecule. It is not, the units we use are the result of historical accidents. The pioneers of thermodynamics didn't know what temperature was. The kelvin scale has degrees the same size as celsius degrees but, sets its zero at absolute zero. 0 K = -273.15 °C . A gas at 0 K by the above definition would have no kinetic energy per molecule. However quantum mechanics tells us that a collection of molecules can never be at rest due to the Heisenberg uncertainty principle. A collection of molecules can never actually reach absolute zero temperature. However, their temperature can approach absolute zero, low temperature gasses now (2000 AD) reach temperatures of a few nanokelvins.

The thermal energy, of a monatomic (one atom like He) ideal gas is proportional to the temperature, T, by definition, and is
thermal energy = 3/2nkT
where n is the number of atoms,
and k is boltzman's constant = 1.38 x 10
-23 J/K
boltzman's constant is a conversion factor between units of energy and temperature.

(In chemistry we often use the gas constant R = 8.3 J/Mole which is the energy of a mole of gas.)

The temperature of a solid or liquid can be found by bringing them into contact with a dilute gas and then using the above definition for the temperature of the gas.

The Temperature of Space

The temperature of the dilute gas surrounding the spacecraft of an astronaut in orbit is 1000 °C , yet if the astronaut sticks a bare hand out into the hot gas she will not feel the slightest warmth! The energy per molecule is high but there are very few molecules; a mere 1010 per cm3.

The temperature of a bunch of photons

You can even measure the temperature of something as insubstantial as a bunch of photons by putting them in contact with an ideal gas and then measuring the temperature of the ideal gas. If you do this far from any star in intergalactic space you will find the temperature of the photons to be 2.7 kelvins. These photons are from the big bang.

The energy contained in an object can change. The energy change can be stored as potential energy in the bonding between atoms or in the kinetic energy of all types of motion.

A blackbody at any temperature emits a spectrum of light called the blackbody spectrum, the maximum of the blackbody spectrum can be used as a thermometer. Some modern thermometers measure temperatures by looking at an objects emitted electromagnetic radiation spectrum.

A more modern definition

The most modern definition of temperature, T, is that it expresses a relationship between the change in the internal energy, U, and the change in entropy, S, of a system.

1/T = dS/dU

Internal energy, U, is the total of all energies contained within a substance both kinetic and potential.

The entropy measures the disorder of a system. Boltzman discovered that entropy was defined as S = k ln W where k is Boltzman's constant 1.38 x 10-23 J/K, and W is the number of possible ways to arrange the system. So entropy increases as the logarithm of the number of ways to organize a system. For a numerical example see Negative Temperatures.

If you have a system which is an ideal gas and the internal energy increases then the entropy also increases since temperature is almost always positive. (For a system like that found in a laser where the system is in a state known as a "population inversion" this definition leads to negative kelvin temperatures see Negative Temperatures.)

 Scientific Explorations by Paul Doherty © 2001 20 July 2001 revised 14 July 2011