The Ideal Gas Law is one of the Equations of State. Although the law describes the behavior of an ideal gas, the equation is applicable to real gases under many conditions, so it is a useful equation to learn to use. The Ideal Gas Law may be expressed as:
PV = NkT
where:
P = absolute pressure in atmospheres
V = volume (usually in liters)
n = number of particles of gas
k = Boltzmann's constant (1.38·10^{−23} J·K^{−1})
T = temperature in Kelvin
The Ideal Gas Law may be expressed in SI units where pressure is in pascals, volume is in cubic meters, N becomes n and is expressed as moles, and k is replaced by R, the Gas Constant (8.314 J·K^{−1}·mol^{−1}):
PV = nRT
Ideal Gases Versus Real Gases
The Ideal Gas Law applies to ideal gases. An ideal gas contains molecules of negligible size that have an average molar kinetic energy that depends only on temperature. Intermolecular forces and molecular size are not considered by the Ideal Gas Law. The Ideal Gas Law applies best to monoatomic gases at low pressure and high temperature. Lower pressure is best because then the average distance between molecules is much greater than the molecular size. Increasing the temperature helps because the kinetic energy of the molecules increases, making the effect of intermolecular attraction less significant.Derivation of the Ideal Gas Law
There are a couple of different ways to derive the Ideal as Law. A simple way to understand the law is to view it as a combination of Avogadro's Law and the Combined Gas Law. The Combined Gas Law may be expressed as:PV / T = C
where C is a constant that is directly proportional to the quantity of the gas or number of moles of gas, n. This is Avogadro's Law:
C = nR
where R is the universal gas constant or proportionality factor. Combining the laws:
PV / T = nR
Multiplying both sides by T yields:
PV = nRT
Ideal Gas Law - Worked Example Problems
Ideal vs Non-Ideal Gas ProblemsIdeal Gas Law - Constant Volume
Ideal Gas Law - Partial Pressure
Ideal Gas Law - Calculating Moles
Ideal Gas Law - Solving for Pressure
Ideal Gas Law - Solving for Temperature
Ideal Gas Equation for Thermodynamic Processes
Process (Constant) |
Known Ratio |
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}) |