Here's an example.
To see where to go from here, let's look at a concrete example, using the augmented matrix above.
At the bottom is a row of zeroes, which translates into the equation "0 = 0". This is always true, so it contributes no information about the solutions of the system. You can just ignore this row.
To solve a linear system when its augmented matrix is in reduced row-echelon form
- If there's a leading 1 in the last column, stop: there is no solution.
- Otherwise, identify the leading variables and the free variables. Assign parameter values to the free variables.
- Translate the non-zero rows of the matrix back into equations.
- Solve each equation for its leading variable in terms of the parameters.
When you apply row operations to an augmented matrix, you generally want to end up with a matrix in a very specific form. Some terminology: the first non-zero number in any row of your augmented matrix is called the leading entry, or pivot entry for that row.
A matrix is in reduced row-echelon form when
Using row operations to transform a matrix into reduced row-echelon form is called row-reducing the matrix (to reduced row-echelon form). Row reduction is a very common process in linear algebra used for many processes besides solving linear systems. For how to do it, see the learning object How to Row Reduce a Matrix
Once your augmented matrix has been transformed into reduced row-echelon form, there's a systematic way to solve the linear system. Each row of your new matrix corresponds to an equation, and the new system of equations has the same solutions as the old one.
x1 – 2x2 + 3x5 = 2, or x1 = 2 + 2s – 3t
x3 – 5x5 = -3, or x3 = –3 +5t
x4 + x5 = 7, or x4 = 7 - t .
The full solution to the system is then
x1 =2 + 2s – 3t
x2 = s
x3 = -3 + 5t for parameters s and t.
x4 =7 – t
x5 = t
In other words, the parameters s and t can take on any values, but once you choose those values, the values of the other variables are completely determined.
- no solution (in which case it's said to be inconsistent),
- a single solution (with no parameters, called a unique solution) or
- infinitely many solutions (i.e. a solution with one or more parameters, each of which can take on any real number as value).
A system with 5 equations in three variables has the unique solution x = 1, y = 2 and z = 3. What does the reduced row-echelon form of its augmented matrix look like?
Can a linear system with more variables than equations have a unique solution? Why or why not?
Suppose first that your reduced augmented matrix has a "bad row" - a string of zeroes with a 1 in the very last column. The bad row translates into the equation "0 = 1". This is impossible - no values of the variables can satisfy this equation. Your system has no solution and no further calculation will change that, so you stop.
For this system, the leading variables are x1, x3 and x4 and the free variables are x2 and x5. Assign arbitrary parameter variables to these free variables to indicate their freedom: set set x2 = s and x5 = t.
For the use of this in cosmology, see Equation of state (cosmology). For the use of this concept in optimal control theory, see Optimal control § General method.
In physics and thermodynamics, an equation of state is a thermodynamic equation relating state variables which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature (PVT), or internal energy. Equations of state are useful in describing the properties of fluids, mixtures of fluids, solids, and the interior of stars.
The most prominent use of an equation of state is to correlate densities of gases and liquids to temperatures and pressures. One of the simplest equations of state for this purpose is the ideal gas law, which is roughly accurate for weakly polar gases at low pressures and moderate temperatures. However, this equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid. Therefore, a number of more accurate equations of state have been developed for gases and liquids. At present, there is no single equation of state that accurately predicts the properties of all substances under all conditions.
Measurements of equation-of-state parameters, especially at high pressures, can be made using lasers.
In addition, there are also equations of state describing solids, including the transition of solids from one crystalline state to another. There are equations that model the interior of stars, including neutron stars, dense matter (quark–gluon plasmas) and radiation fields. A related concept is the perfect fluidequation of state used in cosmology.
In practical context, the equations of state are instrumental for PVT calculation in process engineering problems and especially in petroleum gas/liquid equilibrium calculations. A successful PVT model based on a fitting equation of state can be helpful to determine the state of the flow regime, the parameters for handling the reservoir fluids, piping and sizing.
Boyle's law (1662)
Boyle's Law was perhaps the first expression of an equation of state. In 1662, the Irish physicist and chemist Robert Boyle performed a series of experiments employing a J-shaped glass tube, which was sealed on one end. Mercury was added to the tube, trapping a fixed quantity of air in the short, sealed end of the tube. Then the volume of gas was measured as additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the tube and that in the long, open end. Through these experiments, Boyle noted that the gas volume varied inversely with the pressure. In mathematical form, this can be stated as:
The above relationship has also been attributed to Edme Mariotte and is sometimes referred to as Mariotte's law. However, Mariotte's work was not published until 1676.
Charles's law or Law of Charles and Gay-Lussac (1787)
In 1787 the French physicist Jacques Charles found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to roughly the same extent over the same 80-kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments, indicating a linear relationship between volume and temperature:
Dalton's law of partial pressures (1801)
Dalton's Law of partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of the constituent gases alone.
Mathematically, this can be represented for n species as:
The ideal gas law (1834)
In 1834 Émile Clapeyron combined Boyle's Law and Charles' law into the first statement of the ideal gas law. Initially, the law was formulated as pVm = R(TC + 267) (with temperature expressed in degrees Celsius), where R is the gas constant. However, later work revealed that the number should actually be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K, giving:
Van der Waals equation of state (1873)
In 1873, J. D. van der Waals introduced the first equation of state derived by the assumption of a finite volume occupied by the constituent molecules. His new formula revolutionized the study of equations of state, and was most famously continued via the Redlich–Kwong equation of state and the Soave modification of Redlich-Kwong.
Major equations of state
For a given amount of substance contained in a system, the temperature, volume, and pressure are not independent quantities; they are connected by a relationship of the general form:
In the following equations the variables are defined as follows. Any consistent set of units may be used, although SI units are preferred. Absolute temperature refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.
- = pressure (absolute)
- = volume
- = number of moles of a substance
- = = molar volume, the volume of 1 mole of gas or liquid
- = absolute temperature
- = ideal gas constant (8.3144621 J/(mol·K))
- = pressure at the critical point
- = molar volume at the critical point
- = absolute temperature at the critical point
Classical ideal gas law
The classical ideal gas law may be written:
In the form shown above, the equation of state is thus
If the calorically perfect gas approximation is used, then the ideal gas law may also be expressed as follows
where is the density, is the adiabatic index (ratio of specific heats), is the internal energy per unit mass (the "specific internal energy"), is the specific heat at constant volume, and is the specific heat at constant pressure.
Cubic equations of state
Cubic equations of state are called such because they can be rewritten as a cubic function of Vm.
Van der Waals equation of state
The Van der Waals equation of state may be written:
where is molar volume. The substance-specific constants and can be calculated from the critical properties and (noting that is the molar volume at the critical point) as:
Also written as
Proposed in 1873, the van der Waals equation of state was one of the first to perform markedly better than the ideal gas law. In this landmark equation is called the attraction parameter and the repulsion parameter or the effective molecular volume. While the equation is definitely superior to the ideal gas law and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the liquid forms. While the van der Waals equation is commonly referenced in text-books and papers for historical reasons, it is now obsolete. Other modern equations of only slightly greater complexity are much more accurate.
The van der Waals equation may be considered as the ideal gas law, "improved" due to two independent reasons:
- Molecules are thought as particles with volume, not material points. Thus cannot be too little, less than some constant. So we get () instead of .
- While ideal gas molecules do not interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the ideal gas law), so we write ( something) instead of . To evaluate this ‘something’, let's examine an additional force acting on an element of gas surface. While the force acting on each surface molecule is ~, the force acting on the whole element is ~~