The **ideal gas law** is the equation of state for an ideal gas that relates pressure, volume, gas quantity, and absolute temperature. Although the law describes the behavior of an ideal gas, it approximates real gas behavior in many cases. Uses of the ideal gas law including solving for an unknown variable, comparing initial and final states, and finding partial pressure. Here is the ideal gas law formula, a look at its units, and a discussion of its assumption and limitations.

### Ideal Gas Formula

The ideal gas formula takes a couple of forms. The most common one uses the ideal gas constant:

**PV = nRT**

where:

- P is gas pressure.
- V is the volume of gas.
- n is the number of moles of gas.
- R is the ideal gas constant, which is also the universal gas constant or the product of the Boltzmann constant and Avogadro’s number.
- T is the absolute temperature.

There are other formulas for the ideal gas equation:

**P = ρRT/M**

Here, P is pressure, ρ is density, R is the ideal gas constant, T is absolute temperature, and M is molar mass.

**P = k _{B}ρT/μM_{u}**

Here, P is pressure, k_{B} is Boltzmann’s constant, ρ is density, T is absolute temperature, *μ* is the average particle mass, and M_{u} is the atomic mass constant.

### Units

The value of the ideal gas constant, R, depends on the other units chosen for the formula. The SI value of R is exactly 8.31446261815324 J⋅K^{−1}⋅mol^{−1}. Other SI units are pascals (Pa) for pressure, cubic meters (m^{3}) for volume, moles (mol) for gas quantity, and kelvin (K) for absolute temperature. Of course, other units are fine, so long as they agree with one another and you remember the T is absolute temperature. In other words, convert Celsius or Fahrenheit temperatures to Kelvin or Rankine.

To summarize, here are the two most common sets of units:

- R is 8.314 J⋅K
^{−1}⋅mol^{−1} - P is in pascals (Pa)
- V is in cubic meters (m
^{3}) - n is in moles (mol)
- T is in kelvin (K)

or

- R is 0.08206 L⋅atm⋅K
^{−1}⋅mol^{−1} - P is in atmospheres (atm)
- V is in liters (L)
- n is in moles (mol)
- T is in kelvin (K)

### Assumptions Made in the Ideal Gas Law

The ideal gas law applies to ideal gases. What this means is that the gas has the following properties:

- Particles in a gas move randomly.
- Atoms or molecules have no volume.
- The particles do not interact with one another. They are neither attracted to one another nor repelled by each other.
- Collisions between gas particles and between the gas and the container wall are perfectly elastic. No energy is lost in a collision.

### Ideal Gas Law Uses and Limitations

Real gases do not behave exactly the same as ideal gases. However, the ideal gas law accurately predicts the behavior of monatomic gases and most real gases at room temperature and pressure. In other words, you can use the ideal gas law for most gases at relatively high temperatures and low pressures.

The law does not apply when mixing gases that react with one another. The approximation deviates from true behavior at very low temperatures or high pressures. When temperature is low, kinetic energy is low, so there is a higher likelihood of interactions between particles. Similarly, at high pressure, there are so many collisions between particles that they don’t behave ideally.

### Ideal Gas Law Examples

For example, there are 2.50 g of XeF_{4} gas in a 3.00 liter container at 80°C. What is the pressure in the container?

PV = nRT

First, write down what you know and convert units so they work together in the formula:

P=?

V = 3.00 liters

n = 2.50 g XeF_{4} x 1 mol/ 207.3 g XeF_{4} = 0.0121 mol

R = 0.0821 l·atm/(mol·K)

T = 273 + 80 = 353 K

Plugging in these values:

P = nRT/V

P = 00121 mol x 0.0821 l·atm/(mol·K) x 353 K / 3.00 liter

Pressure = 0.117 atm

Here are more examples:

- Solve for the number of moles.
- Find the identity of an unknown gas.
- Solve for density using the ideal gas law.

### History

French engineer and physicist Benoît Paul Émile Clapeyron gets credit for combining Avogadro’s law, Boyle’s law, Charles’s law, and Gay-Lussac’s law into the ideal gas law in 1834. August Krönig (1856) and Rudolf Clausius (1857) independently derived the ideal gas law from kinetic theory.

### Formulas for Thermodynamic Processes

Here are some other handy formulas:

Process(Constant) | KnownRatio | P_{2} | V_{2} | T_{2} |

Isobaric (P) | V_{2}/V_{1}T _{2}/T_{1} | P_{2}=P_{1}P _{2}=P_{1} | V_{2}=V_{1}(V_{2}/V_{1})V _{2}=V_{1}(T_{2}/T_{1}) | T_{2}=T_{1}(V_{2}/V_{1})T _{2}=T_{1}(T_{2}/T_{1}) |

Isochoric (V) | P_{2}/P_{1}T _{2}/T_{1} | P_{2}=P_{1}(P_{2}/P_{1})P _{2}=P_{1}(T_{2}/T_{1}) | V_{2}=V_{1}V _{2}=V_{1} | T_{2}=T_{1}(P_{2}/P_{1})T _{2}=T_{1}(T_{2}/T_{1}) |

Isothermal (T) | P_{2}/P_{1}V _{2}/V_{1} | P_{2}=P_{1}(P_{2}/P_{1})P _{2}=P_{1}/(V_{2}/V_{1}) | V_{2}=V_{1}/(P_{2}/P_{1})V _{2}=V_{1}(V_{2}/V_{1}) | T_{2}=T_{1}T _{2}=T_{1} |

isoentropic reversible adiabatic (entropy) | P_{2}/P_{1}V _{2}/V_{1}T _{2}/T_{1} | P_{2}=P_{1}(P_{2}/P_{1})P _{2}=P_{1}(V_{2}/V_{1})^{−γ}P _{2}=P_{1}(T_{2}/T_{1})^{γ/(γ − 1)} | V_{2}=V_{1}(P_{2}/P_{1})^{(−1/γ)}V _{2}=V_{1}(V_{2}/V_{1})V _{2}=V_{1}(T_{2}/T_{1})^{1/(1 − γ)} | T_{2}=T_{1}(P_{2}/P_{1})^{(1 − 1/γ)}T _{2}=T_{1}(V_{2}/V_{1})^{(1 − γ)}T _{2}=T_{1}(T_{2}/T_{1}) |

polytropic (PV ^{n}) | P_{2}/P_{1}V _{2}/V_{1}T _{2}/T_{1} | P_{2}=P_{1}(P_{2}/P_{1})P _{2}=P_{1}(V_{2}/V_{1})^{−n}P _{2}=P_{1}(T_{2}/T_{1})^{n/(n − 1)} | V_{2}=V_{1}(P_{2}/P_{1})^{(-1/n)}V _{2}=V_{1}(V_{2}/V_{1})V _{2}=V_{1}(T_{2}/T_{1})^{1/(1 − n)} | T_{2}=T_{1}(P_{2}/P_{1})^{(1 – 1/n)}T _{2}=T_{1}(V_{2}/V_{1})^{(1−n)}T _{2}=T_{1}(T_{2}/T_{1}) |

### References

- Clapeyron, E. (1834). “Mémoire sur la puissance motrice de la chaleur.”
*Journal de l’École Polytechnique*(in French). XIV: 153–90. - Clausius, R. (1857). “Ueber die Art der Bewegung, welche wir Wärme nennen”.
*Annalen der Physik und Chemie*(in German). 176 (3): 353–79. doi:10.1002/andp.18571760302 - Davis; Masten (2002).
*Principles of Environmental Engineering and Science*. New York: McGraw-Hill. ISBN 0-07-235053-9. - Moran; Shapiro (2000).
*Fundamentals of Engineering Thermodynamics*(4th ed.). Wiley. ISBN 0-471-31713-6. - Raymond, Kenneth W. (2010).
*General, Organic, and Biological Chemistry: An Integrated Approach*(3rd ed.). John Wiley & Sons. ISBN 9780470504765.