Equation of State
An equation of state is a quantitative relationship among the variables that determine the state of a given system, that is, whether it is solid, liquid, or vapor, its enthalpy or heat content, etc. The state of a system is its condition or situation, at a given instant, and is determined by its properties. Such an equation can be used to calculate the value of one state variable, for example, pressure, if the values of the other variable are known.
Scientific studies generally involve the examination of changes in the states of systems. A system is the portion of the physical world which we set apart for study, e.g., a chemical reaction or a container of gas. The values of a specific number of the properties or variables of the system must be specified in order to completely define its state.
Variables of state, also known as state functions or properties of state, are those properties of a system which can be determined without reference to the history of the system. They depend only on the present state of the system and can be measured or calculated directly from the system in its present condition. Regardless of the previous states that the system has passed through, regardless of the values its properties have had previously, and regardless of the means used to bring it to its present condition, the present values of its state variables (functions or properties) will be the same. The change in a function of state depends only on the beginning state and the ending state, not on the path taken between the two states.
Pressure (P), temperature (T), volume (V) and the amount of material (n) are especially important state properties for many systems. Knowledge of the values of these variables determines the state of the system and facilitates the calculation of the values of other properties.
The term state is also used to designate the physical phase of the system, i.e., solid, liquid, or gas. In some cases it is necessary to give the phase of a substance, in addition to values of such properties as P, T, V, and n, in order to completely and unequivocally designate its state. For instance, water can exist as a supercooled liquid below the normal melting point, and we must stipulate whether we are dealing with ice or liquid water at that temperature. There are also cases for which a particular solid form must be designated in order to establish a material's state completely. For instance, solid carbon exists as both graphite and diamond at a wide range of pressure, volume, and temperature.
As stated above, an equation which gives the relationship among functions of state for a given system is called its equation of state. Such an equation can be used to calculate the values of a state variable using the known values of other variables.
Of particular interest are the equations of state for gases. Based on experimental measurements, Robert Boyle developed an equation in which there is a reciprocal relationship between the pressure and the volume of gases, when the temperature and the amount of gas are kept constant. The work of Jacques Charles in 1787 and of Joseph-Louis Gay-Lussac in 1808 demonstrated that the volume of a given amount of gas is directly proportional to the temperature, when its pressure remains constant.
Amedeo Avogadro proposed in 1811 that equal volumes of all gases, at the same pressure and temperature, contain the same number of molecules. In other words, an equal number of moles of all gases occupy the same volume at constant temperature and pressure. This proposal, combined with the relationships discovered by Boyle, Charles, and Gay-Lussac led to an equation of state for gases: PV=nRT. T is the absolute temperature in degree Kelvin (equal to the temperature in C plus 273°) and R is the universal gas constant (equal to 0.08, when V is is measured in liters, P in atmospheres, and n in moles; or equivalently, 8.3 joules per degree mole.)
This equation is now known as the equation of state for ideal gases or the ideal gas law. Although the behavior of real gases closely follows the ideal gas equation under circumstances of low pressure and reasonably high temperature, significant deviations from the ideal expectations do occur. Studies of the departures from ideal behavior by gases has led to a better understanding of atoms and molecules and their interactions in the gaseous state.
The most widely used equation of state for real gases is that of Johannes van der Waals. Instead of using the measured value of pressure in the equation of state for ideal gases, he substituted a term which takes into consideration the reduced pressure on the walls of a container due to the attractive forces. This magnitude of the pressure term increases with the density of molecules: when they are closer together, the molecules are more constrained by their mutual attraction. The van der Waals equation also corrects the volume term in the ideal gas equation by subtracting the space that is taken up by the molecules themselves. This molecular volume is not available for other molecules to move about in. The study of real gases in relationship to the van der Waals equation yields valuable information about intermolecular forces and the volume of molecules.
Although the van der Waals equation of state provides helpful information, it does not fit the behavior of gases at all values of pressure and temperature. A number of other theoretical and empirical equations of state have been proposed. One of these is the virial equation of state suggested by Heike Kammerlingh-Onnes (1853-1926) in 1901. It adds correction terms to the ideal gas equation by multiplying the right side of the equation (nRT) by the mathematical progression (1 + B/V + C/ V2 + D/V3 + E/V4 +...) where the constants A, B, C, etc. are temperature dependent. Computers are used to fit this equation to the experimental data, and values for the constants B, C, D, E, etc. are determined for the particular gas. Although this is a purely empirical procedure, there has been success in using the values of the constants to understand the behavior of gases.
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