## Quick Reference

A theory, largely the work of Count Rumford, James Prescott Joule, and James Clerk Maxwell, that explains the physical properties of matter in terms of the motions of its constituent particles. In a gas, for example, the pressure is due to the incessant impacts of the gas molecules on the walls of the container. If it is assumed that the molecules occupy negligible space, exert negligible forces on each other except during collisions, are perfectly elastic, and make only brief collisions with each other, it can be shown that the pressure *p* exerted by one mole of gas containing *n* molecules each of mass *m* in a container of volume *V*, will be given by:*p*=**nm¯c**^{2}/3*V*, where *¯c*^{2} is the mean square speed of the molecules. As according to the gas laws for one mole of gas: *pV*=*RT*, where *T* is the thermodynamic temperature, and *R* is the molar gas constant, it follows that:*RT*=**nm¯c**^{2}/3 Thus, the thermodynamic temperature of a gas is proportional to the mean square speed of its molecules. As the average kinetic energy of translation of the molecules is *m¯c*^{2}/2, the temperature is given by:*T*=(*m¯c*^{2}/2)(2*n*/3*R*)The number of molecules in one mole of any gas is the Avogadro constant, *N*_{A}; therefore in this equation *n*=*N*_{A}. The ratio *R*/*N*_{A} is a constant called the Boltzmann constant (*k*). The average kinetic energy of translation of the molecules of one mole of any gas is therefore 3*kT*/2. For monatomic gases this is proportional to the internal energy (*U*) of the gas, i.e.*U*=*N*_{A}3*kT*/2 and as *k*=*R*/*N*_{A}*U*=3*RT*/2 For diatomic and polyatomic gases the rotational and vibrational energies also have to be taken into account (see degrees of freedom).

*p*=**nm¯c**^{2}/3*V*

*RT*=**nm¯c**^{2}/3

*T*=(*m¯c*^{2}/2)(2*n*/3*R*)

*U*=*N*_{A}3*kT*/2

*U*=3*RT*/2

In liquids, according to the kinetic theory, the atoms and molecules still move around at random, the temperature being proportional to their average kinetic energy. However, they are sufficiently close to each other for the attractive forces between molecules to be important. A molecule that approaches the surface will experience a resultant force tending to keep it within the liquid. It is, therefore, only some of the fastest moving molecules that escape; as a result the average kinetic energy of those that fail to escape is reduced. In this way evaporation from the surface of a liquid causes its temperature to fall.

In a crystalline solid the atoms, ions, and molecules are able only to vibrate about the fixed positions of a crystal lattice; the attractive forces are so strong at this range that no free movement is possible.

*Subjects:*
Chemistry — Physics.

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