# Temperature

## Introduction

Temperature is the physical property of a system, which governs the transfer of thermal energy, or heat, between that system and other systems. When two systems are at the same temperature, they are in thermal equilibrium and no heat transfer will occur between the systems. When a temperature difference does exist, heat will tend to move from the higher temperature system to the lower temperature system, until thermal equilibrium is again established. At this point, both systems will have the same temperature. The concepts of "hot" and "cold" are related to temperature and generally the material with the higher temperature is said to be hotter.

Many physical properties of materials including the phase (gas, liquid or solid), density, solubility, vapor pressure, and electrical conductivity depend on the temperature. Temperature also plays an important role in determining the rate and extent to which chemical reactions occur. This is one reason why the human body has several elaborate mechanisms for maintaining the temperature at 37 °C, since temperatures only a few degrees higher can result in harmful reactions with serious consequences. Temperature also controls the type and quantity of thermal radiation emitted from a surface. One application of this effect is the incandescent light bulb, in which a tungsten filament is electrically heated to a temperature at which significant quantities of visible light are emitted.

Temperature is an intrinsic property of a system meaning that it doesn't depend on the system size or the amount of material in the system. Other intrinsic properties include pressure and density. By contrast, mass and volume are extrinsic properties and depend on the amount of material in the system.

## Units of Temperature

The basic unit of temperature is the kelvin (K). One kelvin is formally defined as 1/273.16 of the temperature of the triple point of water (the point at which water, ice and water vapor exist in equilibrium). The temperature 0 K is called Absolute zero and corresponds to the point at which the molecules and atoms have the least possible thermal energy. No macroscopic system can have a temperature less than absolute zero.

For everyday applications, it is often convenient to use the Celsius scale, in which 0 °C corresponds to the temperature at which water freezes and 100 °C corresponds to the boiling point of water at sea level. In this scale a temperature difference of 1 degree is the same as a 1 K temperature difference, so the scale is essentially the same as the Kelvin scale, but offset by the temperature at which water freezes (273.15 K). Thus the following equation can be used to convert from celsius to kelvin.

``` T(K) = T(C) - 273.15
```

In the U.S. the Fahrenheit scale is widely used. On this scale the freezing point of water corresponds to 32 °F and the boiling point to 212 °F. The following formula can be used to convert between fahrenheit and celsius:

```T(C) = 5/9*(T(F) - 32)
```

Other temperature scales include the Rankine and the Reaumur.

## Theoretical foundation of temperature

### Zeroth-Law definition of temperature

While most people have a basic understanding of the concept of temperature, its formal definition is rather complicated. Before jumping to a formal definition, let's consider the concept of thermal equilibrium. If two closed systems with fixed volumes are brought together, so that they are in thermal contact, changes may take place in the properties of both systems. These changes are due to the transfer of heat between the systems. When a state is reached in which no further changes occur, the systems are in thermal equilibrium.

Now a basis for the definition of temperature can be obtained from the zeroth law of Thermodynamics, which states that if two systems, A and B, are in thermal equilibrium and a third system C is in thermal equilibrium with system A then systems B and C will also be in thermal equilibrium. This is an empirical fact, based on observation rather than theory. Since A, B, and C are all in thermal equilibrium, it is reasonable to say each of these systems shares a common value of some property. We call this property temperature.

Generally, it is not convenient to place any two arbitrary systems in thermal contact to see if they are in thermal equilibrium and thus have the same temperature. Therefore, it is useful to establish a temperature scale based on the properties of some reference system. Then, a measuring device can be calibrated based on the properties of the reference system and used to measure the temperature of other systems. One such reference system is a fixed quantity of gas. Boyles law indicates that the product of the Pressure and volume (P*V) of a gas is directly proportional to the temperature. This can be expressed by the Ideal gas law as:

PV = nRT (1)

where T is temperature, n is the amount of gas (number of moles) and R is the Ideal gas constant. Thus, one can define a scale for temperature based on the corresponding pressure and volume of the gas. In practice, such a gas thermometer is not very convenient, but other measuring instruments can be calibrated to this scale.

Equation 1 indicates that for a fixed volume of gas, the pressure increases with increasing temperature. Pressure is just a measure of the force applied by the gas on the walls of the container and is related to the energy of the system. Thus, we can see that an increase in temperature corresponds to an increase in the thermal energy of the system. When two systems of differing temperature are placed in thermal contact, the temperature of the hotter system decreases, indicating that heat is leaving that system, while the cooler system is gaining heat and increasing in temperature. Thus heat always moves from a region of high temperature to a region of lower temperature and it is the temperature difference that drives the heat transfer between the two systems.

### Second-Law definition of temperature

In the previous section temperature was defined in terms of the Zeroth Law of thermodynamics. It is also possible to define temperature in terms of the second law of thermodynamics, which deals with entropy. Entropy is a measure of the disorder in a system. The second law states that any process will result in either no change or a net increase in the entropy of the universe. This can be understood in terms of probability. Consider a series of coin tosses. A perfectly ordered system would be one in which every coin toss would come up either heads or tails. For any number of coin tosses, there is only one combination of outcomes corresponding to this situation. On the other hand, there are multiple combinations that can result in disordered or mixed systems, where some fraction are heads and the rest tails. As the number of coin tosses increases, the number of combinations corresponding to imperfectly ordered systems increases. For a very large number of coin tosses, the number of combinations corresponding to ~50% heads and ~50% tails dominates and obtaining an outcome significantly different than 50/50 becomes extremely unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.

Now, we have stated previously that temperature controls the flow of heat between two systems and we have just shown that the universe, and we would expect any natural system, tends to progress so as to maximize entropy. Thus, we would expect there to be some relationship between temperature and entropy. In order to find this relationship let's first consider the relationship between heat, work and temperature. A Heat engine is a device for converting heat into mechanical work and analysis of the Carnot heat engine provides the necessary relationships we seek. The work from a heat engine corresponds to the difference between the heat put into the system at the high temperature, qH and the heat ejected at the low temperature, qC. The efficiency is the work divided by the heat put into the system or:

efficiency = wcy/qH = (qH-qC)/qH = 1 - qC/qH (2)

where wcy is the work done per cycle. We see that the efficiency depends only on qC/qH. Because qC and qH correspond to heat transfer at the temperatures TC and TH, respectively, qC/qH should be some function of these temperatures:

qC/qH = f(TH,TC) (3)

Carnot's theorem states that all reversible engines operating beteen the same heat reservoirs are equally efficient. Thus, a heat engine operating between T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and T2, and the second between T2 and T3. This can only be the case if:

q13 = q1q2/q2q3

which implies:

q13 = f(T1,T3) = f(T1,T2)f(T2,T3)

Since, the first function is independent of T2, this temperature must cancel on the right side, meaning f(T1,T3) is of the form g(T1)/g(T3) (i.e. f(T1,T3) = f(T1,T2)f(T2,T3) = g(T1)/g(T2)*g(T2)/g(T3) = g(T1)/g(T3)), where g is a function of a single temperature. We can now choose a temperature scale with the property that:

qC/qH = TC/TH (4)

Substituting Equation 4 back into Equation 2 gives a relationship for the efficiency in terms of temperature:

efficiency = 1-qC/qH =1-TC/TH (5)

Notice that for TC=0 K the efficiency is 100% and that efficiency becomes greater than 100% below 0 K. Since an efficiency greater than 100% violates the first law of thermodynamics, this implies that 0 K is the minimum possible temperature. In fact the lowest temperature ever obtained in a macroscopic system was 20 nK, which was achieved in 1995 at NIST. Subtracting the right hand side of Equation 5 from the middle portion and rearranging gives:

qH/TH - qC/TC = 0

where the negative sign indicates heat ejected from the system. This relationship suggests the existence of a state function, S, defined by:

dS = dqrev/T (6)

where the subscript indicates a reversible process. The change of this state function around any cycle is zero, as is necessary for any state function. This function corresponds to the entropy of the system, which we described previously. We can rearranging Equation 6 to get a new definition for temperature in terms of entropy and heat:

T = dqrev/dS (7)

For a system, where entropy S may be a function S(E) of its energy E, the termperature T is given by:

1/T = dS/dE (8)

The reciprocal of the temperature is the rate of increase of entropy with energy.

## Heat capacity

Also see Specific heat capacity.

Temperature is related to the amount of thermal energy or heat in a system. As heat is added to the system, the temperature increases by an amount proportional to the amount of heat being added. The constant of proportionality is called the heat capacity and reflects the ability of the material to store heat.

The heat is stored in a variety of modes, corresponding to the various quantum states accessible to the system. As the temperature increases more quantum states become accessible, resulting in an increase in heat capacity. For a monatomic gas at low temperatures, the only accessible modes correspond to the translational motion of the atoms, so all of the energy is due to movement of the atoms (Actually, a small amount of energy, called the Zero Point Energy arises due to the confinement of the gas into a fixed volume, this energy is present even at 0 K). Since the kinetic energy is related to the motion of the atoms, 0 K corresponds to the point at which all atoms are motionless. For such a system, a temperature below 0 K is not possible, since it is not possible for the atoms to move slower than to be motionless.

At higher temperatures, electronic transitions become accessible, further increasing the heat capacity. For most materials these transitions are not important below 104 K, however for a few common molecules, such transitions are important even at room temperature. At extremely high temperatures (>108 K) nuclear transitions become accessible. In addition to translational, electronic, and nuclear modes, poly atomic molecules also have modes associated with rotation and vibrations along the molecular bonds, which are accessible even at low temperatures. In solids most of the stored heat corresponds to atomic vibrations.

## Negative Temperatures

At low temperatures, particles tend to move to their lowest energy states. As you increase the temperature, particles move into higher and higher energy states. As the temperature becomes infinite, the number of particles in the lower energy states and the higher energy states becomes equal. In some situations, it is possible to create a system in which there are more particles in the higher energy states than in the lower ones. This situation can be described with a negative temperature. A negative temperature is not colder than absolute zero, but rather it is hotter than infinite temperature.

The previous section described how heat is stored in the various translational, vibrational, rotational, electronic, and nuclear modes of a system. The macroscopic temperature of a system is related to the total heat stored in all of these modes and in a normal system thermal energy is constantly being exchanged between the various modes. However, for some cases it is possible to isolate one or more of the modes. In practice the isolated modes still exchange energy with the other modes, but the time scale of this exchange is much slower than for the exchanges within the isolated mode. One example is the case of nuclear spins in a strong external magnetic field. In this case energy flows fairly rapidly among the spin states of interacting atoms, but energy transfer between the nuclear spins and other modes is relatively slow. Since the energy flow is predominately within the spin system, it makes sense to think of a spin temperature that is distinct from the temperature due to other modes.

Based on Equation 7, we can say a positive temperature corresponds to the condition where entropy increases as thermal energy is added to the system. This is the normal condition in the macroscopic world and is always the case for the translational, vibrational, rotational, and non-spin related electronic and nuclear modes. The reason for this is that there are an infinite number of these types of modes and adding more heat to the system increases the number of modes that are energetically accessible, and thus the entropy. However, for the case of electronic and nuclear spin systems there are only a finite number of modes available (often just 2, corresponding to spin up and spin down). In the absense of a magnetic field, these spin states are degenerate, meaning that they correspond to the same energy. When an external magnetic field is applied, the energy levels are split, since those spin states that are aligned with the magnetic field will have a different energy than those that are anti-parallel to it.

In the absense of a magnetic field, one would expect such a two-spin system to have roughly half the atoms in the spin-up state and half in the spin-down state, since this maximizes entropy. Upon application of a magnetic field, some of the atoms will tend to align so as to minimize the energy of the system, thus slightly more atoms should be in the lower-energy state (for the purposes of this example we'll assume the spin-down state is the lower-energy state). It is possible to add energy to the spin system using radio frequency (RF) techniques. This causes atoms to flip from spin-down to spin-up. Since we started with over half the atoms in the spin-down state, initially this drives the system towards a 50/50 mixture, so the entropy is increasing, corresponding to a positive temperature. However, at some point more than half of the spins are in the spin-up position. In this case adding additional energy, reduces the entropy since it moves the system further from a 50/50 mixture. This reduction in entropy with the addition of energy corresponds to a negative temperature. For additional information see .

## Temperature in gases

As mentioned previously for a monatomic ideal gas the temperature is related to the translation motion of the atoms. The Kinetic theory of gases relates this motion to the average kinetic energy of atoms and molecules in the system. For this case 11300 degrees celsius corresponds to an average kinetic energy of one electron volt; to take room temperature (300 kelvin) as an example, the average energy of air molecules is 300/11300 eV, or 0.0273 electron volts. This average energy is independent of particle mass, which seems counterintuitive to many people. Although the temperature is related to the average kinetic energy of the particles in a gas, each particle has its own energy which may or may not correspond to the average. In a gas the distribution of energy of the particles corresponds to the Boltzmann distribution.

An electron volt is a very small unit of energy, on the order of 1.602e-19 joules.

## Temperature Measurement

Many methods have been developed for measuring temperature. Most of these rely on measuring some physical property of a working material that varies with temperature. One of the most common devices for measuring temperature is the glass thermometer. This consists of a glass tube filled with mercury or some other liquid, which acts as the working fluid. Temperature increases cause the fluid to expand, so the temperature can be determined by measuring the volume of the fluid. Such thermometers are usually calibrated, so that one can read the temperature, simply by observing the level of the fluid in the thermometer. Another type of thermometer that is not really used much in practice, but is important from a theoretical standpoint is the gas thermometer mentioned previously.

Other important devices for measuring temperature include:

One must be careful when measuring temperature to ensure that the measuring instrument (thermometer, thermocouple, etc) is really the same temperature as the material that is being measured. Under some conditions heat from the measuring instrument can cause a temperature gradient, so the measured temperature is different from the actual temperature of the system. In such a case the measured temperature will vary not only with the temperature of the system, but also with the heat transfer properties of the system. An extreme case of this effect gives rise to the wind chill factor, where the weather feels colder under windy conditions than calm conditions even though the temperature is the same. What is happening is that the wind increases the rate of heat transfer from the body, resulting in a larger reduction in body temperature for the same ambient temperature.