This document was prepared for the middle school
math teachers who are taking part inProject Skymath. http://www.unidata.ucar.edu/staff/blynds/tmp.html It is also hoped
that the general public will find it interesting.
What is Temperature The Development of
Thermometers and Temperature Scales Heat and Thermodynamics The Kinetic Theory Thermal Radiation 3 K The Temperature of the Universe Summary Acknowledgments References Author
In a qualitative manner, we can describe the temperature of an object as that which determines the sensation of warmth or coldness felt from contact with it.
It is easy to demonstrate that when two objects of the same material are placed together (physicists say when they are put in thermal contact), the object with the higher temperature cools while the cooler object becomes warmer until a point is reached after which no more change occurs, and to our senses, they feel the same. When the thermal changes have stopped, we say that the two objects (physicists define them more rigorously as systems) are in thermal equilibrium . We can then define the temperature of the system by saying that the temperature is that quantity which is the same for both systems when they are in thermal equilibrium.
If we experiment further with more than two systems, we find that many systems can be brought into thermal equilibrium with each other; thermal equilibrium does not depend on the kind of object used. Put more precisely, if two systems are separately in thermal equilibrium with a third, then they must also be in thermal equilibrium with each other, and they all have the same temperature regardless of the kind of systems they
are.The statement in italics, called the zeroth law of thermodynamics may be restated as follows:
If three or more systems are in thermal contact with each other and all in equilibrium together, then any two taken separately are in equilibrium with one another. (quote from T. J. Quinn's monograph Temperature)
Now one of the three systems could be an instrument calibrated to measure the temperature - i.e. a thermometer. When a calibrated thermometer is put in thermal contact with a system and reaches thermal equilibrium, we then have a quantitative measure of the temperature of the system. For example, a mercury-in-glass clinical thermometer is put under the tongue of a patient and allowed to reach thermal equilibrium in the patient's mouth - we then see by how much the silvery mercury has expanded in the stem and read the scale of the thermometer to find the patient's temperature.
What is a Thermometer?
A thermometer is an instrument that measures the temperature of a system in a quantitative way. The easiest way to do this is to find a substance having a property that changes in a regular way with its temperature. The most direct 'regular' way is a linear one:
t(x) = ax + b,
where t is the temperature of the substance and changes as the property x of the substance changes. The constants a and b depend on the substance used and may be evaluated by specifying two temperature points on the scale, such as 32° for the freezing point of water and 212° for its boiling point.
For example, the element mercury is liquid in the temperature range of -38.9° C to 356.7° C (we'll discuss the Celsius ° C scale later). As a liquid, mercury expands as it gets warmer, its expansion
rate is linear and can be accurately calibrated.
The mercury-in-glass thermometer illustrated in the above figure contains a bulb filled with mercury that is allowed to expand into a capillary. Its rate of expansion is calibrated on the glass scale.
for you!)
In 1780, J. A. C. Charles, a French physician, showed that for the same increase in temperature, all gases exhibited the same increase in volume. Because the expansion coefficient of gases is so very nearly the same, it
is possible to establish a temperature scale based on a single fixed point rather than the two fixed- point scales, such as the Fahrenheit and Celsius scales. This brings
us back to a thermometer that uses a gas as the thermometric medium.
In a constant volume gas thermometer a large bulb B of gas, hydrogen for example, under a set pressure connects with a mercury-filled "manometer" by means of a tube of very small volume. (The Bulb B is the temperature-sensing portion and should contain almost all of the hydrogen). The level of mercury at C may be adjusted by raising or lowering the mercury reservoir R. The pressure of the hydrogen gas, which is the "x" variable in the linear relation with temperature, is the difference between the levels D and C plus the pressure above D.
P. Chappuis in 1887 conducted extensive studies of gas thermometers with constant pressure or with constant volume using hydrogen, nitrogen, and carbon dioxide as the thermometric medium. Based on his results, the
Comité International des Poids et Mesures adopted the constant-volume hydrogen scale based on fixed points at the ice point (0°C) and the steam point (100°C) as the practical scale for international meteorology.
Experiments with gas thermometers have shown that there is very little difference in the temperature scale for different gases. Thus, it is possible to set up a temperature scale that is independent of the
thermometric medium if it is a gas at low pressure. In this case, all gases behave like an "Ideal Gas" and have a very simple relation between their pressure, volume, and temperature:
pV= (constant)T.
This temperature is called the thermodynamic temperature and is now accepted as the fundamental measure of temperature. Note that there is a naturally-defined zero on this scale - it is the point at which
the pressure of an ideal gas is zero, making the temperature also zero. We will continue a discussion of "absolute zero" in a later section. With this as one point on the scale, only one other fixed point need be defined.
In 1933, the International Committee of Weights and Measures adopted this fixed point as the triple point of water, the temperature at which water, ice, and water vapor coexist in equilibrium); its value is set as 273.16. The unit of temperature on this scale is called the kelvin, after Lord Kelvin (William Thompson), 1824-1907, and its symbol is K (no degree symbol used).
To convert from Celsius to Kelvin, add 273. K = °C + 273.
Thermodynamic temperature is the fundamental temperature; its unit is the kelvin which is defined as the fraction 1 / 273.16 of the thermodynamic temperature of the triple point of water.
Sir William Siemens, in 1871, proposed a thermometer whose thermometric medium is a metallic conductor whose resistance changes with temperature. The element platinum does not oxidize at high temperatures and has a relatively uniform change in resistance with temperature over a large range.
The Platinum Resistance Thermometer is now widely used as a thermoelectric thermometer and covers the temperature range from about -260°C to 1235°C.Several temperatures were adopted as Primary reference points so as to define the International Practical Temperature Scale of 1968.
The International Temperature Scale of 1990 was adopted by the International Committee of Weights and Measures at its meeting in 1989. Between 0.65K and 5.0K, the temperature is defined in terms of the vapor pressure - temperature relations of the isotopes of helium. Between 3.0K and the triple point of neon (24.5561K) the temperature is defined by means of a helium gas thermometer. Between the triple point of hydrogen (13.8033K) and the freezing point of silver (961.78°°C) the temperature is defined by means of platinum resistance thermometers. Above the freezing point of silver the temperature is defined in terms of the Planck radiation law.
T. J. Seebeck, in 1826, discovered that when wires of different metals are fused at one end and heated, a current flows from one to the other. The electromotive force generated can be quantitatively related to the temperature and hence, the system can be used as a thermometer - known as
a thermocouple. The thermocouple is used in industry and
many different metals are used - platinum and platinum/rhodium, nickel-chromium and nickel-aluminum, for example. The National Institute of Standards and
Technology (NIST) maintains
databases for standardizing thermometers.
For the measurement of very low temperatures, the magnetic susceptibility of a paramagnetic substance is used as the thermometric physical quantity. For some substances, the magnetic susceptibility varies inversely as the temperature. Crystals such as cerrous magnesium nitrate and chromic potassium alum have been used to measure temperatures down to 0.05 K; these crystals are calibrated in the liquid helium
range.
This diagram and the last illustration in this text were taken from the Low Temperature Laboratory, Helsinki University of Technology's picture archive.
For these very low, and even lower, temperatures, the thermometer is also the mechanism for cooling. Several
low-temperature laboratories conduct interesting applied and theoretical research on how to reach the lowest
possible temperatures and how work at these temperatures may find application.
Prior to the 19th century, it was believed that the sense of how hot or cold an object felt was determined by how much "heat" it contained. Heat was envisioned as a liquid that flowed from a hotter to a colder object; this weightless fluid was called "caloric", and until the writings of Joseph Black
(1728-1799), no distinction was made between heat and temperature. Black distinguished between the quantity (caloric) and the intensity (temperature) of heat.
Benjamin Thomson, Count Rumford, published a paper in 1798 entitled "An Inquiry Concerning the Source of Heat which is Excited by Friction". Rumford had noticed the large amount of heat generated when a cannon was drilled. He doubted that a material substance was flowing into the cannon and concluded "it appears to me to be extremely difficult if not impossible to form any distinct idea of anything capable of being excited and communicated in the
manner the heat was excited and communicated in these experiments except motion."
But it was not until J. P. Joule published a definitive paper in 1847 that the the caloric idea was abandoned. Joule conclusively showed that heat was a form of energy. As a result of the experiments of Rumford, Joule, and
others, it was demonstrated (explicitly stated by Helmholtz in 1847), that the various forms of energy can be transformed one into another.
When heat is transformed into any other form of energy, or when other forms of energy are transformed into heat, the total amount of energy (heat plus other forms) in the system is constant.
This is the first law of thermodynamics, the conservation of energy. To express it another way: it is in no way possible either by mechanical, thermal, chemical, or other means, to obtain a perpetual motion machine;
i.e., one that creates its own energy (except in the fantasy world of Maurits Escher's
"Waterfall"!)
A second statement may also be made about how machines operate. A steam engine uses a source of heat to produce work. Is it possible to completely convert the heat energy into work, making it a 100% efficient machine? The answer is to be found in the second law of thermodynamics:
No cyclic machine can convert heat energy wholly into other forms of energy. It is not possible to construct a cyclic machine that does nothing but withdraw heat energy and convert it into mechanical energy.
The second law of thermodynamics implies the irreversibility of certain processes - that of converting all heat into mechanical energy, although it is possible to have a cyclic machine that does nothing but convert mechanical
energy into heat!
Sadi Carnot (1796-1832) conducted theoretical studies
of the efficiencies of heat engines (a machine which converts some of its heat into useful work). He was trying to model the most efficient heat engine possible. His theoretical work provided the basis for practical
improvements in the steam engine and also laid the foundations of thermodynamics. He described an ideal engine, called the Carnot engine, that is the most efficient way an engine can be constructed. He showed
that the efficiency of such an engine is given by
efficiency = 1 - T"/T',
where the temperatures, T' and T" , are the hot and cold "reservoirs" , respectively, between which the machine operates. On this temperature scale, a heat engine whose coldest reservoir is zero degrees would operate with 100%
efficiency. This is one definition of absolute zero, and it can be shown to be identical to the absolute zero we discussed previously. The temperature scale is called the absolute, the thermodynamic
, or the kelvin scale.
The way that the gas temperature scale and the thermodynamic temperature scale are shown to be identical is based on the microscopic
interpretation of temperature, which postulates that the
macroscopic measurable quantity called temperature is a result of the random motions of the microscopic particles that make up a system.
This brief summary is abridged from a more detailed discussion to be found in Quinn's "Temperature"
About the same time that thermodynamics was evolving James Clerk Maxwell (1831-1879) and Ludwig Boltzmann (1844-1906) developed a theory describing
the way molecules moved - molecular dynamics.
The molecules that make up a perfect gas move about, colliding with each other like billiard balls and bouncing off the surface of the container holding the gas.
The energy associated with motion is called Kinetic Energy and this kinetic approach to the behavior of ideal gases led to an interpretation of the concept of temperature on a microscopic scale.
The amount of kinetic energy each molecule has is a function of its velocity; for the large number of molecules in a gas (even at low pressure), there should be a range of velocities at any instant of time.
The magnitude of the velocities of the various particles should vary greatly - no two particles should be expected to have the exact same velocity. Some may be moving very fast; others, quite slowly.
Maxwell found that he could represent the distribution of velocities statistically by a function known as the Maxwellian Distribution. The collisions of the molecules with their container gives rise to the pressure of the gas. By considering the average force exerted by the molecular collisions on the wall, Boltzmann was able to show that the average kinetic energy of the molecules was directly comparable to the measured pressure, and the greater the average kinetic energy, the greater the pressure. From Boyles' Law, we know that the pressure is directly proportional to the temperature, therefore, it was shown that the kinetic energy of the molecules related directly to the temperature of the gas. A simple relation holds for this:
average kinetic energy of molecules=3kT/2,
where k is the Boltzmann Constant. Temperature is a measure of the energy of thermal motion and, at a temperature of zero, the energy reaches a minimum (quantum mechanically, the zero-point motion remains at 0 K).
In July, 1995, physicists in Boulder, Colo.achieved a temperature far lower than has ever been produced before and created an entirely new state of matter predicted decades ago by Albert Einsteinand
Satyendra Nath Bose. The press release describes the nature of this experiment and a full description of this phenomenon is described by the University of Colorado's BEC Homepage.
Dealing with a system which contained huge numbers of molecules requires a statistical approach to the problem. About 1902, J. W. Gibbs (1839-1903) introduced statistical mechanics with which he
demonstrated how average values of the properties of a system could be predicted from an analysis of the most probable values of these properties found from a large number of identical systems (called an ensemble). Again,
in the statistical mechanical interpretation of thermodynamics, the key parameter is identified with a temperature which can be directly linked to the
thermodynamic temperature, with the temperature of Maxwell's distribution,and with the perfect gas law.
Temperature becomes a quantity definable either in terms of macroscopic thermodynamic quantities such as heat and work, or, with equal
validity and identical results, in terms of a quantity which characterized the energy distribution among the particles in a system. (Quinn, "Temperature")
With this understanding of the concept of temperature, it is possible to explain how heat (thermal energy) flows from one body to another. Thermal energy is carried by the molecules in the form of their motions and some of it, through molecular collisions, is transferred to molecules of a second object when put in contact with it. This mechanism for transferring thermal energy by contact is called conduction.
A second mechanism of heat transport is illustrated by a pot of water set to boil on a stove - hotter water closest to the flame will rise to mix with cooler water near the top of the pot. Convection involves the bodily
movement of the more energetic molecules in a liquid or gas.
The third way that heat energy can be transferred from one body to another is by radiation; this is the way that the sun warms the earth. The radiation flows from the sun to the earth, where some of it is absorbed, heating the
surface.
A major dilemma in physics since the time of Newton was how to explain the nature of this radiation.
The great question at the turn of the century was to explain the way this total radiant energy emitted by a black body was spread out into the various frequencies or wavelengths of the radiation. Maxwell's "classical"
theory of electromagnetic oscillators failed to explain the observed brightness distribution. It was left to
Max Planck to solve the dilemma by showing that the energy of the oscillators must be quantized, i.e. the energies can not take any value but must change in steps, the size of each step, or quantum, is proportional to the frequency of the oscillator and equal to hv,
where h is the Planck constant. With this assumption, Planck derived the brightness distribution of a black body and showed that it is defined by its temperature. Once the temperature of a black body is specified, the Planck
law can be used to calculate the intensity of the light emitted by the body as a function of wavelength. Conversely, if the brightness distribution of a radiating
body is measured, then, by fitting a
Planck curve to it, its temperature can be determined.
The curves illustrated below show that the hotter the body is, the brighter it is at shorter wavelengths. The surface temperature of the sun is 6000°K, and its Planck curve peaks in the visible wavelength range. For bodies cooler than the sun, the peak of the Planck curve shifts to longer wavelengths, until a temperature is reached such that very little radiant energy is emitted in the visible range.
This figure (adapted from Adkins' "Thermal Physics") shows several Planck curves for black bodies. The Intensity is in units of energy per unit area per unit solid angle per unit time per unit wavelength interval.
The broken line illustrates the variation with wavelength and temperature of the peaks of the curves.
This is a graphical representation of Wien's law, which states:
(max) ~ 0.29/T,
where
(max) is the wavelength of maximum brightness
in cm and T is the absolute temperature of the black body.
The human body has a temperature of about 310°K and radiates primarily in the far infrared. If a photograph of a human is taken with a camera sensitive to this wavelength region, we get a
"thermal" picture. This picture is courtesy of the Infrared Processing and
Analysis Center, Jet Propulsion Laboratory, NASA.
A page developed by
Compix
gives a fine description of thermal images and their uses.
The sun and stars emit thermal radiation covering all wavelengths; other objects in the sky, like the great clouds of gas in the Milky Way, also emit thermal radiation but are much cooler. These objects are best detected by infrared and radio telescopes - telescopes whose detectors are sensitive to the longer wavelengths.
In 1965,Arno Penzias and Robert Wilson were conducting a careful calibration of their radio telescope at the Bell Laboratory at Whippany, New Jersey. The found that their receiver showed a "noise" pattern as if it were inside a container whose temperature was 3K - i.e. as if it were in equilibrium with a black body at 3 K. This "noise" seemed to be coming from every direction. Earlier theoretical predictions by George Gamow and other astrophysicists had predicted the existence of a cosmic 3 K background. Penzias' and Wilson's discovery was the observational confirmation of the isotropic radiation from the Universe, believed to be a relic of the "Big Bang". The enormous thermal energy released during the creation of the universe
began to cool as the universe expanded. Some 12 billion years later, we are in a universe that radiates like a black body now cooled to 3 K. In 1978 Penzias and Wilson were awarded the Nobel prize in physics for this
discovery.
A black body at 3 K emits most of its energy in the microwave wavelength range. Molecules in the earth's atmosphere absorb this radiation so that from the ground, astronomers cannot make observations in this wavelength region. In 1989 the Cosmic Background Explorer (COBE) satellite, developed by NASA's Goddard Space Flight Center, was launched to measure the diffuse infrared and microwave radiation from the early universe. One of its instruments, the Far Infrared Absolute
Spectrophotometer (FIRAS) compared the spectrum of the cosmic microwave background radiation with a precise blackbody.
The cosmic microwave background spectrum was measured with a precision of 0.03% and it fit precisely with a black body of temperature 2.726 K. Even though there are billions of stars in the universe, these precise COBE measurements show
that 99.97% of the radiant energy of the Universe was released within the first year after the Big Bang itself and now resides in this thermal 3 K radiation field.
A more detailed explanation of the origin of the microwave background radiation, and itspossible anisotropy,
may be found here. A new mission selected by NASA is the Microwave
Anisotropy Probe (MAP) will measure the small fluctuations in the background radiation and will yield more
information on the details of the early universe. The European Space Agency has a similar mission
planned.
We can record events (illustration from Low Temperature Laboratory of Helsinki University of Technology)that cover 18 orders of magnitude in the temperature range, and we have one clearly defined lower limit to the temperature,
absolute zero. Because of this 10-with-18-zeros-behind-it range in temperatures, there are many different kinds of thermometers developed to explore it and many different fields of research.
One of the beauties of "publishing" on the web is the interactive element it offers. Joachim Reinhardt has written to point out that the highest temperatures that are accessible on earth (only surpassed
by the early stages of the big bang) occur in high-energy collisions of particles (in particular of heavy ions), during which one sees a "fireball" with a temperature of several hundred MeV (which corresponds to a temperature of 10 to the 12th power k). This fireball cools down by
expanding and by radiating off particles, mostly pions, quite similar to the thermal black-body radiation. For more information on these phenomena, see the Fermi Lab tutorial.
Thermal physics is a field rich in theoretical and practical applications.