The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron and independently of him, Dmitry Mendeleev in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. Facsimile at the Bibliothèque nationale de France (pp. 153–90). The ideal gas law is often written in an empirical form:
where , , and are respectively the pressure, volume, and temperature, is the amount of substance, and is the ideal gas constant. It can also be derived from the microscopic kinetic theory, as was achieved (independently) by August Krönig in 1856 Facsimile at the Bibliothèque nationale de France (pp. 315–22). and Rudolf Clausius in 1857. Facsimile at the Bibliothèque nationale de France (pp. 353–79).
Formulations
The state of an amount of gas is determined by its pressure, volume, and temperature. The modern form of the equation relates these simply in two main forms. The temperature used in the equation of state is an absolute temperature: the appropriate SI unit is the kelvin.
Common forms
The most frequently introduced forms are:where: is the absolute pressure of the gas, is the volume of the gas, is the amount of substance of gas (also known as number of moles), is the ideal, or universal, gas constant, equal to the product of the Boltzmann constant and the Avogadro constant, is the Boltzmann constant, is the Avogadro constant, is the absolute temperature of the gas, is the number of particles (usually atoms or molecules) of the gas.
In SI units, p is measured in pascals, V is measured in cubic meters, n is measured in moles, and T in kelvins (the Kelvin scale is a shifted Celsius scale, where 0 K = −273.15 °C, the lowest possible temperature). R has for value 8.314 J/(mol·K) = 1.989 ≈ 2 cal/(mol·K), or 0.0821 L⋅atm/(mol⋅K).
Molar form How much gas is present could be specified by giving the mass instead of the chemical amount of gas. Therefore, an alternative form of the ideal gas law may be useful. The chemical amount, n (in moles), is equal to total mass of the gas (m) (in kilograms) divided by the molar mass, M (in kilograms per mole):
By replacing n with m/M and subsequently introducing density ρ = m/V, we get:
Defining the specific gas constant Rspecific as the ratio R/M,
This form of the ideal gas law is very useful because it links pressure, density, and temperature in a unique formula independent of the quantity of the considered gas. Alternatively, the law may be written in terms of the specific volume v, the reciprocal of density, as
It is common, especially in engineering and meteorological applications, to represent the specific gas constant by the symbol R. In such cases, the universal gas constant is usually given a different symbol such as or to distinguish it. In any case, the context and/or units of the gas constant should make it clear as to whether the universal or specific gas constant is being used.
Statistical mechanics In statistical mechanics, the following molecular equation (i.e. the ideal gas law in its theoretical form) is derived from first principles:
where is the absolute pressure of the gas, is the number density of the molecules (given by the ratio , in contrast to the previous formulation in which is the number of moles), is the absolute temperature, and is the Boltzmann constant relating temperature and energy, given by:
where is the Avogadro constant. The form can be further simplified by defining the kinetic energy corresponding to the temperature:
so the ideal gas law is more simply expressed as:
From this we notice that for a gas of mass , with an average particle mass of times the atomic mass constant, , (i.e., the mass is Da) the number of molecules will be given by
and since , we find that the ideal gas law can be rewritten as
In SI units, is measured in pascals, in cubic metres, in kelvins, and in SI units.
Combined gas law Combining the laws of Charles, Boyle, and Gay-Lussac gives the combined gas law, which can take the same functional form as the ideal gas law. This form does not specify the number of moles, and the ratio of to is simply taken as a constant:
where is the pressure of the gas, is the volume of the gas, is the absolute temperature of the gas, and is a constant. More commonly, when comparing the same substance under two different sets of conditions, the law is written as:
Energy associated with a gas According to the assumptions of the kinetic theory of ideal gases, one can consider that there are no intermolecular attractions between the molecules, or atoms, of an ideal gas. In other words, its potential energy is zero. Hence, all the energy possessed by the gas is the kinetic energy of the molecules, or atoms, of the gas.
This corresponds to the kinetic energy of n moles of a monoatomic gas having 3 degrees of freedom: x, y, z. The table here below gives this relationship for different amounts of a monoatomic gas.
Energy of a monoatomic gasMathematical expressionEnergy associated with one moleEnergy associated with one gramEnergy associated with one atom
Applications to thermodynamic processes The table below essentially simplifies the ideal gas equation for a particular process, making the equation easier to solve using numerical methods.
A thermodynamic process is defined as a system that moves from state 1 to state 2, where the state number is denoted by a subscript. As shown in the first column of the table, basic thermodynamic processes are defined such that one of the gas properties (P, V, T, S, or H) is constant throughout the process.
For a given thermodynamic process, in order to specify the extent of a particular process, one of the properties ratios (which are listed under the column labeled "known ratio") must be specified (either directly or indirectly). Also, the property for which the ratio is known must be distinct from the property held constant in the previous column (otherwise the ratio would be unity, and not enough information would be available to simplify the gas law equation).
In the final three columns, the properties (p, V, or T) at state 2 can be calculated from the properties at state 1 using the equations listed.
Process Constant Known ratio or delta p2 V2 T2Isobaric processPressure V2/V1 p2 = p1 V2 = V1(V2/V1) T2 = T1(V2/V1) T2/T1 p2 = p1 V2 = V1(T2/T1) T2 = T1(T2/T1) Isochoric process(Isovolumetric process)(Isometric process)Volume p2/p1 p2 = p1(p2/p1) V2 = V1 T2 = T1(p2/p1) T2/T1 p2 = p1(T2/T1) V2 = V1 T2 = T1(T2/T1) Isothermal process Temperature p2/p1 p2 = p1(p2/p1) V2 = V1(p1/p2) T2 = T1 V2/V1 p2 = p1(V1/V2) V2 = V1(V2/V1) T2 = T1 Isentropic process(Reversible adiabatic process) p2/p1 p2 = p1(p2/p1) V2 = V1(p2/p1)(−1/γ) T2 = T1(p2/p1)(γ − 1)/γ V2/V1 p2 = p1(V2/V1)−γ V2 = V1(V2/V1) T2 = T1(V2/V1)(1 − γ) T2/T1 p2 = p1(T2/T1)γ/(γ − 1) V2 = V1(T2/T1)1/(1 − γ) T2 = T1(T2/T1) Polytropic processP Vn p2/p1 p2 = p1(p2/p1) V2 = V1(p2/p1)(−1/n) T2 = T1(p2/p1)(n − 1)/n V2/V1 p2 = p1(V2/V1)−n V2 = V1(V2/V1) T2 = T1(V2/V1)(1 − n) T2/T1 p2 = p1(T2/T1)n/(n − 1) V2 = V1(T2/T1)1/(1 − n) T2 = T1(T2/T1) Isenthalpic process(Irreversible adiabatic process) p2 − p1 p2 = p1 + (p2 − p1) T2 = T1 + μJT(p2 − p1) T2 − T1 p2 = p1 + (T2 − T1)/μJT T2 = T1 + (T2 − T1)
a. In an isentropic process, system entropy (S) is constant. Under these conditions, p1V1γ = p2V2γ, where γ is defined as the heat capacity ratio, which is constant for a calorifically perfect gas. The value used for γ is typically 1.4 for diatomic gases like nitrogen (N2) and oxygen (O2), (and air, which is 99% diatomic). Also γ is typically 1.6 for mono atomic gases like the noble gases helium (He), and argon (Ar). In internal combustion engines γ varies between 1.35 and 1.15, depending on constitution gases and temperature.
b. In an isenthalpic process, system enthalpy (H) is constant. In the case of free expansion for an ideal gas, there are no molecular interactions, and the temperature remains constant. For real gases, the molecules do interact via attraction or repulsion depending on temperature and pressure, and heating or cooling does occur. This is known as the Joule–Thomson effect. For reference, the Joule–Thomson coefficient μJT for air at room temperature and sea level is 0.22 °C/bar.
Deviations from ideal behavior of real gases The equation of state given here (PV = nRT) applies only to an ideal gas, or as an approximation to a real gas that behaves sufficiently like an ideal gas. There are in fact many different forms of the equation of state. Since the ideal gas law neglects both molecular size and intermolecular attractions, it is most accurate for monatomic gases at high temperatures and low pressures. The molecular size becomes less important for lower densities, i.e. for larger volumes at lower pressures, because the average distance between adjacent molecules becomes much larger than the molecular size. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy, i.e., with increasing temperatures. More detailed equations of state, such as the van der Waals equation, account for deviations from ideality caused by molecular size and intermolecular forces.
Derivations
Empirical The empirical laws that led to the derivation of the ideal gas law were discovered with experiments that changed only 2 state variables of the gas and kept every other one constant.
All the possible gas laws that could have been discovered with this kind of setup are: Boyle's law () Charles's law () Avogadro's law () Gay-Lussac's law ()
where P stands for pressure, V for volume, N for number of particles in the gas and T for temperature; where are constants in this context because of each equation requiring only the parameters explicitly noted in them changing.
To derive the ideal gas law one does not need to know all 6 formulas, one can just know 3 and with those derive the rest or just one more to be able to get the ideal gas law, which needs 4.
Since each formula only holds when only the state variables involved in said formula change while the others (which are a property of the gas but are not explicitly noted in said formula) remain constant, we cannot simply use algebra and directly combine them all. This is why: Boyle did his experiments while keeping N and T constant and this must be taken into account (in this same way, every experiment kept some parameter as constant and this must be taken into account for the derivation).
Keeping this in mind, to carry the derivation on correctly, one must imagine the gas being altered by one process at a time (as it was done in the experiments). The derivation using 4 formulas can look like this:
at first the gas has parameters
Say, starting to change only pressure and volume, according to Boyle's law (), then:
After this process, the gas has parameters
Using then equation () to change the number of particles in the gas and the temperature,
After this process, the gas has parameters
Using then equation () to change the pressure and the number of particles,
After this process, the gas has parameters
Using then Charles's law (equation 2) to change the volume and temperature of the gas,
After this process, the gas has parameters
Using simple algebra on equations (), (), () and () yields the result: or where stands for the Boltzmann constant.
Another equivalent result, using the fact that , where n is the number of moles in the gas and R is the universal gas constant, is: which is known as the ideal gas law.