How Hot is it?


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.

Line up a dozen people. Ask them to arrange themselves by hand temperature hottest to coldest. Then have the hottest had person and the coldest hand person describe the hand temperature of the person in the middle of the group. One will say the person in the middle has cold hands, the other will say they have warm hands. Feel the Temperature

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

The zeroth law of thermodynamics

If two bodies are in contact 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. This lets us define temperature for any thermometer and then use it to measure the temperature of everything.

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. 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, now we use the Celsius scale. 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.

Thermal expansion of molecules can be made visible by creating a color change, this is done in thermochromic liquid crystal material.
Here is an exploration of temperature change using thermochromic liquid crystal sheets. Thermochromic Liquid Crystals.

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 not exactly 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.

A gas thermometer. Place the opening of a balloon over the mouth of a bottle. A soda bottle will work well.
Run hot water over the bottle and observe what happens, run cold water over the ballon and observe what happens. Place the bottle in the freezer for a minute and observe what happens. You are observing Charles' Law that the volume of a gas is proportional to the temperature of the gas. When they did a quantitative measurement they discovered that the volume of a gas at 0°C decreased by 1/273 when temperature was decreased by 1 °C. Extrapolate this and you find that at -273°C the volume of a gas will be zero. This is an interesting temperature known as absolute zero. -273°C is 0 K, zero kelvins. A better understanding of 0K came about with the atomic theory of gases.

Here is a video of the Exploratorium exhibit "Gas Model"

A simple model of gases can be made by placing ping-pong balls into a wire mesh cage. By shaking the cage you can model solids, liquids, gases, an the effects of temperature on gases. Gas Model

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 proportional to the:
kinetic energy of
translational motion
per molecule

Basically, the temperature of an ideal gas is the
kinetic energy per molecule divided by a constant known as Boltzmann's constant, k with units of joules per kelvin 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 gases. now (2000 AD) reach temperatures of a few nanokelvins.

The thermal energy, of a monatomic (a single 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 Boltzmann's constant = 1.38 x 10
-23 J/K
Boltzmann'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. Boltzmann discovered that entropy was defined as S = k ln W where k is Boltzmann'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.

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.)

Negative Temperatures on Wikipedia

Quantum Gas goes below absolute zero and article from Nature Magazine Negative Absolute Temperatures achieved in the lab.

Schneider and his colleagues reached such sub-absolute-zero temperatures with an ultracold quantum gas made up of potassium atoms. Using lasers and magnetic fields, they kept the individual atoms in a lattice arrangement, a crystal of light. At positive temperatures, the atoms repel, making the configuration stable. The team then quickly adjusted the magnetic fields, causing the atoms to attract rather than repel each other. “This suddenly shifts the atoms from their most stable, lowest-energy state to the highest possible energy state, before they can react,” says Schneider. “It’s like walking through a valley, then instantly finding yourself on the mountain peak.” At positive temperatures, such a reversal would be unstable and the atoms would collapse inwards. But the team also adjusted the trapping laser field to make it more energetically favorable for the atoms to stick in their positions. This result marks the gas’s transition from just above absolute zero to a few billionths of a Kelvin below absolute zero.

Negative absolute temperatures are hotter than absolute zero because when placed in contact energy will spontaneously flow from negative absolute temperatures to absolute zero.

Scientific Explorations by Paul Doherty

© 2013

11 March 2013