Temperature

Temperature is the physical property of a system which underlies the common notions of "hot" and "cold"; the material with the higher temperature is said to be hotter. Temperature is a measure of the average kinetic energy of the particles in a sample of matter. In other words, temperature is a measure of activity and the frequency of collisions of molecules.

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 us 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.

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Now a basis for the definition of temperature can be obtained from the so-called 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 (being in thermal equilibrium is a transitive relation; moreover, it is an equivalence relation). 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.

Related Topics:
Zeroth law of thermodynamics - Transitive relation - Equivalence relation

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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. Also, it would only provide an ordinal scale.

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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. The ideal gas law indicates that the product of the pressure and volume (P · V) of a gas is directly proportional to the temperature:

Related Topics:
Ideal gas law - Directly proportional

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:

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P cdot V = n cdot R cdot T

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(1)

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where 'T is temperature, n is the number of moles of gas and R is the gas constant. Thus, one can define a scale for temperature based on the corresponding pressure and volume of the gas: the temperature in kelvins is the pressure in pascals of one mole of gas in a container of one cubic metre, divided by 8.31... In practice, such a gas thermometer is not very convenient, but other measuring instruments can be calibrated to this scale.

Related Topics:
Mole - Gas constant

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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.

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Temperature in gases

As mentioned previously for a monatomic ideal gas the temperature is related to the translational motion or average speed of the atoms. The kinetic theory of gases uses statistical mechanics to relate this motion to the average kinetic energy of atoms and molecules in the system. For this case 7736 K = 7463 degrees Celsius corresponds to an average kinetic energy of one electronvolt; to take room temperature (300 K) as an example, the average energy of air molecules is 300/7736 eV, or 0.0388 electronvolt. 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. However, after an examination of some basic physics equations it makes perfect sense. The second law of thermodynamics states that any two given systems when interacting with each other will later reach the same average energy. Temperature is a measure of the average kinetic energy of a system. The formula for the kinetic energy of an atom is:

Related Topics:
Kinetic theory - Statistical mechanics - 7463 degrees Celsius - One - Electronvolt - 300 K - 0.0388 - Kinetic energy

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: K_t = egin{matrix} rac{1}{2} end{matrix} mv^2

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(Note that a calculation of the kinetic energy of a more complicated object, such as a molecule, is slightly more involved. Additional degrees of freedom are available, so molecular rotation or vibration must be included.)

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Thus, particles of greater mass (say a neon atom relative to a hydrogen molecule) will move slower than lighter counterparts, but will have the same average energy. This average energy is independent of the mass because of the nature of a gas, all particles are in random motion with collisions with other gas molecules, solid objects that may be in the area and the container itself (if there is one). A visual illustration of this from Oklahoma State University makes the point more clear. Not all the particles in the container have different velocities, regardless of whether there are particles of more than one mass in the container, but the average kinetic energy is the same because of the ideal gas law. In a gas the distribution of energy (and thus speeds) of the particles corresponds to the Boltzmann distribution.

Related Topics:
Neon - Hydrogen - Gas - Ideal gas law - Boltzmann distribution

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An electronvolt is a very small unit of energy, approximately 1.602×10-19 joule.

Related Topics:
Electronvolt - Joule

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Temperature of the vacuum

When a satellite in empty space is heated by sunshine and cooled by radiating energy away it is not in thermodynamic equilibrium and has no well-defined temperature.

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A system in a vacuum will radiate its thermal energy, i.e. convert heat into electromagnetic waves. If vacuum is filled with electromagnetic waves (say, radiation from walls of vacuum chamber, or relic microwave radiation in space) then the system will exchange by energy with these waves and thermally equilibrates at some finite (non zero) temperature.

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Cosmic microwave background radiation being remnant of radiation of hot early universe when radiation was in thermal equilibrium with matter has Planck spectrum (black body spectrum) with the temperature (at present) of about 2.7 K.

Related Topics:
Planck spectrum - Black body spectrum

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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 from 50/50 becomes extremely unlikely. Thus the system naturally progresses to a state of maximum disorder or entropy.

Related Topics:
Second law of thermodynamics - Entropy

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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:

Related Topics:
Heat engine - Carnot heat engine

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:

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extrm{efficiency} = rac {w_{cy}}{q_H} = rac{q_H-q_C}{q_H} = 1 - rac{q_C}{q_H}

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(2)

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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:

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:

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rac{q_C}{q_H} = f(T_H,T_C)

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(3)

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Carnot's theorem states that all reversible engines operating between 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:

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:

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q_{13} = rac{q_1 q_2} {q_2 q_3}

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which implies:

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:

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q_13 = f(T_1,T_3) = f(T_1,T_2)f(T_2,T_3)

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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:

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:

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rac{q_C}{q_H} = rac{T_C}{T_H}

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(4)

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

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:

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extrm{efficiency} = 1 - rac{q_C}{q_H} = 1 - rac{T_C}{T_H}

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(5)

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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:

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:

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rac {q_H}{T_H} - rac{q_C}{T_C} = 0

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

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:

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dS = rac {dq_mathrm{rev}}{T}

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(6)

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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:

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:

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T = rac{dq_mathrm{rev}}{dS}

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(7)

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

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:

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rac{1}{T} = rac{dS}{dE}

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(8)

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

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