Ideal gas law

The ideal gas law is useful for calculating temperatures, volumes, pressures or number of moles for many gases over a wide range of temperatures and pressures. However, no gas is an ideal gas so the equation is only approximate and the law fails at low temperatures or high pressures. When the ideal gas law is not accurate enough, one of the "real" gas equations, such as the van der Waals equation must be used. The ideal gas law is the combination of Boyle's law, Charles's law and Avogadro's law and is expressed in any one of the following ways:

$p={\frac {nRT}{V}}.$

$V={\frac {nRT}{p}}.$

$T={\frac {pV}{nR}}.$

$n={\frac {pV}{RT}}.$

where p = pressure, V = volume, n = number of moles, and T = the absolute temperature, in degrees Kelvin. R is the gas constant, R = N_A k, where k is the Boltzmann constant and N_A is Avogadro's constant. R = 8.314472(15) J / mol K

Special cases of the ideal gas law

Amonton's law $\left({\frac {p}{T}}\right)=\mathrm {constant}$ (at a fixed volume and amount of gas)

Avogadro's law $V=\left(nV_{\mathrm {m} }\right).$ (at fixed temperature and pressure)

Boyle's law $\left(pV\right)=\mathrm {constant}$ (at fixed temperature and amount of gas)

Charles's law $\left({\frac {V}{T}}\right)=\mathrm {constant}$ (at fixed pressure and amount of gas)

Boyle's + Charles's $\left({\frac {pV}{T}}\right)=\mathrm {constant}$ (at fixed amount of gas)

An deal gas

To be an ideal gas, several conditions must be met. First, the size of the gas molecules must be negligible compared to the average distance between them. This condition is not true at extremely high pressures or extremely cold temperatures. Second, the intermolecular forces of attraction or repulsion between molecules must be very weak or negligible except during collisions. And third, when the gas molecules do collide, thus must do so in an elastic manner. That is, they bounce right off of each other rather than sticking together.

When the ideal gas law fails

When the ideal gas law fails, a real gas law, such as the van der Waals equation must be used. However, this equation contains constants, $a$ and $b$, that are unique for each gas. This law also fails at extreme high pressures. When the coefficients $a$ and $b$ are set to zero, the van der Waals equation reduces to the ideal gas law.

van der Waals equation :$\left(p+{\frac {n^{2}a}{V^{2}}}\right)\left(V-nb\right)=nRT$

Background

The gas laws started, in the 1660's, with Robert Boyle's law, stating "the volume of a sample of gas at a given temperature varies inversely with the applied pressure, or V = constant/p (at fixed temperature and amount of gas). Then Jacques Alexandre Charles' experiments with hot-air balloons, and additional contributions by John Dalton (1801) and Joseph Louis Gay-Lussac (1802) showed that a sample of gas, at a fixed pressure, increases in volume linearly with the temperature, or V/T = a constant. This is known as Charles's law. Extrapolations of volume/temperature data for many gases, to a volume of zero, all cross at about -273 degrees C, which is absolute zero. Of course, the gases would liquify before reaching this temperature and so the law does not really apply in this temperature region. In 1811 Amedeo Avogadro re-interpreted Gay-Lussac's law of combining volumes (1808) to state Avogadro's law , Equal volumes of any two gases at the same temperature and pressure contain the same number of molecules. The molar volume of gas, at standard temperature ( 0 Celcius) and pressure (1 atm) is 22.4 L.

Example problems

PROBLEM 1) Two liters of gas at 1 atm and 25C is placed under 5 atm of pressure at 25C. What is the final volume of gas?

Using Boyle's law:

Eq. 1.1) $\left(p_{\mathrm {i} }V_{\mathrm {i} }\right)=\left(\mathrm {constant} \right)=\left(p_{\mathrm {f} }V_{\mathrm {f} }\right)$ or

Eq. 1.2) $\left(V_{\mathrm {f} }\right)=\left({\frac {p_{\mathrm {i} }V_{\mathrm {i} }}{p_{\mathrm {f} }}}\right)$

Eq. 1.3) $\left(V_{\mathrm {f} }\right)=\left({\frac {(1\mathrm {atm} )(2\mathrm {L} )}{(5\mathrm {atm} )}}\right)=0.4\mathrm {L}$

Using Ideal gas law:

Eq. 1.4) $n=\left({\frac {p_{\mathrm {i} }V_{\mathrm {i} }}{RT_{\mathrm {i} }}}\right)=\left({\frac {p_{\mathrm {f} }V_{\mathrm {f} }}{RT_{\mathrm {f} }}}\right)$

Because $\left(T_{\mathrm {i} }\right)=\left(T_{\mathrm {f} }\right)$ Eq. 1.4 reduces to Eq. 1.1 shown above.

PROBLEM 2) How many moles of nitrogen are present in a 50L tank at 25C when the pressure is 10 atm? (Note: Kelvin = Celcius + 273.15). Numbers include only 3 significant figures.

Eq 2.1) $n={\frac {pV}{RT}}={\frac {(10.0\mathrm {atm} )(50\mathrm {L} )}{[(0.0821\mathrm {Latm} /(\mathrm {Kmol} )](298\mathrm {K} )}}=20.4mol$

Related topics

Amonton's law

Avogadro's law

Boyle's law

Charles's law

Dalton's law of partial pressure

Gay-Lussac's law

Law of combining volumes

van der Waals equation

References

"General Chemistry, 2nd Ed.", pp 103-117, D. D. Ebbing & M. S. Wrighton, Houghton Mifflin, Boston, 1987. "General Chemistry with Qualitative Analysis, 2nd Ed.", pp. 263-278, Saunders College Publishing, Philadelphia, 1984.