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Ideal Gas Equation

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Introduction

Also known as the ideal gas law, the ideal-gas equation is the simplest and best-known equation of state for substances in the gas phase.

At very low densities the average distances between molecules is so large that the intermolecular potential energy may effectively be neglected. In such a case, the particles would be independent of one another, a situation referred to as an ideal gas. Under this approximation, it has been observed experimentally that, to a close degree, at very low density gas behaves according to the ideal gas equation of state.


Visualization

The following figure shows a hypothetical contraption which can be used to control the temperature, volume, and amount of molecules of a gas. Measurements can be made and compared to results of the ideal-gas equation.



Measurements of the behavior of various gases lead to several conclusions. First, the volume V is proportional to the number of moles n. For example, if the number of moles is doubled, keeping the pressure and temperature constant, the volume doubles.

Second, the volume varies inversely with the absolute pressure P. If the pressure is doubled while holding the temperature T and number of moles n constant, the gas compresses to one half of its initial volume. In other words, PV = constant when n and T are constant.

Third, the pressure is proportional to the absolute temperature. If the absolute temperature is doubled, keeping the volume and number of moles constant, the pressure doubles. In other words, P = (constant)T when n and V are constant.

These three relationships can be combined neatly into a single equation, called the ideal-gas equation:


Nomenclature

symbol	description
m	mass
M	molar mass
n	number of moles (in kmol) of gas
P	pressure
R	unique gas constant for each gas
	universal gas constant
T	temperature
v	specific volume
V	volume
	?
ρ	density

Equations

The ideal-gas equation can be expressed in many different forms. One example is:

Pv = RT

or

P = ρRT

where P is the absolute pressure, v is the specific volume, T is the thermodynamic (absolute) temperature, ρ is the density, and R is the gas constant.

PV = nT

or

P = T

where n is the number of kmol of gas, and V is the volume

is the universal gas constant, the value of which is, for any gas, 8.3145 kJ/kmolK. Note that, as indicated by the nomenclature above, that is different from R. R is a unique value, which is different for each gas.

and T is the absolute (ideal gas scale) temperature in kelvins (i.e., T(K) = T(°C) + 273.15). It is important to note that T must always be the absolute temperature whenever it is being used to multiply or divide in an equation. For U.S. Customary units:

The ideal gas equation of state can be written in the form:

PV = mRT

or,

Pv = RT

or,

PV = nT

where,

R =

M

in which R is a different constant for each particular gas.

The gas can be described in terms of the number of moles n, rather than the mass. The molar mass M of a compound (sometimes called molecular weight) is the mass per mole, and the total mass m_tot of a given quantity of that compound is the number of moles n times the mass per mole (or molar mass) M:

m_tot = nM

PV = nRT

where R is a proportionality constant. An ideal gas is one for which the ideal-gas equation holds precisely for all pressures and temperatures. This is an idealized model; it works best at very low pressures and high temperatures, when the gas molecules are far apart and in rapid motion. It is reasonably good (within a few percent) at moderate pressures (such as a few atmmospheres) and at temperatures well above those at which the gas liquefies.

It might be expected that the constant R in the ideal-gas equation would have different values for different gases, but it turns out to have the same value for all gases, at least at sufficiently high temperature and low pressure. It is called the gas constant (or ideal-gas constant or universal gas constant). The numerical value of R depends on the units of P, V, and T. In SI units, in which the unit of P is Pa (1 Pa = 1 N/m²) and the unit of V is m³, the current best numerical value of R is

R = 8.31447 J/mol K

Note that the units of pressure times volume are the same as units of work and energy (for example, N/m² times m³); that's why R has units of energy per mole per unit of absolute temperature. In chemical calculations, volumes are often expressed in liters (L) and pressures in atmospheres (atm). In this system, to four significant digits:

R = 0.08216 (L atm)/(mol K)

The ideal-gas equation can be expressed in terms of the mass m_tot of gas, using m_tot = nM from Eq1:

PV =

mtot M

RT

From this an expression for the density ρ = m_tot/V of the gas can be obtained:

ρ =

PM RT

For a constant mass (or constant number of moles) of an ideal gas the product nR is constant, so the quantity PV/T is also constant. If the subscripts 1 and 2 refer to any two states of the same mass of a gas, then

P1V1 T1

=

P2V2 T2

= constant

The proportionality of pressure to absolute temperature is familiar; in fact, the temperature scale can be defined in terms of pressure in a constant-volume gas thermometer. That may make it seem that the pressure-temperature relation in the ideal-gas equation is just a result of the way temperature is defined. But the equation may reveal what happens when the volume or amount of substance is changed as well.

The ideal gas equation is used mainly in fluid mechanics and thermodynamics.


Conclusion

Because of its simplicity, the ideal-gas equation of state is very convenient to use in thermodynamic calculations. However, two questions are now appropriate. The ideal-gas equation of state is a good approximation at low density. But what constitutes low density? Or, over what range of density will the ideal-gas equation of state hold with accuracy? The second question is, how much does an actual gas at a given pressure and temperature deviate from ideal-gas behavior? Temperature-volume (T-v)diagrams can help answer these questions.

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The behavior of gases can be described by three laws:

Boyle's law	Charles's law	Avogadro's law
V = constantB/P	V = (constantC )(T)	V = (constantA)(n)
with T and n held constant	with P and n held constant	with T and P held constant

Because V is proportional to (1/P), T, and n, it would seem that V should be proportional to all three:

V = a constant

nT P

experiments show that the above equation is correct. This equation is usually written:

PV = nRT

where P, V, and T have their usual meanings; n is the number of moles of gas; and R is a proportionality constant.

The ideal gas equation states that:

PV = nRT

where P is the pressure of the gas, V is the volume of the gas, n is the number of moles of gas, R is a proportionality constant, and T is the Kelvin temperature.

The value of the proportionality constant R, which is called the gas constant, can be calculated from the STP molar volume, 22.4141 L/mol, which is the volume of gas at 0°C (273.15 K) and 1 atm of pressure. Solution of the ideal gas equation for R gives:

R =

PV nT

=

(1 atm)(22.4141 L) (1 mol)(273.15 K)

= 0.082058

atm L mol K

Note that the value of R depends on the units used for pressure and volume.

A mathematical relation between the temperature, pressure, and volume of a given quantity of material such as the ideal gas equation is called an equation of state. A gas that behvaves exactly as described by the ideal gas equation is called an ideal gas. No real gases are ideal. But this one simple equation of state describes the behavior of most real gases to within a few percent at temperatures around room temperature and above, and under pressures of about one atmosphere or less. Liquids and solids are different from gases in this respect. The equations of state for liquids and solids are more complicated and differ from one substance to another.