Ideal Gas Law

23rd May 2019 @ 10 min read

The Ideal gas law is also known as general gas law. As the name states the law is applicable under the ideal conditions, not to real gases. The law correlates the pressure, volume, temperature, and amount of gas. It was first formulated by French physicist Émile Clapeyron in 1834.

Note: Ideal gases are hypothetical gases that do not exist in the real world. Many gases behave like ideal gases under some extremities like low pressure, high temperature.

Ideal Gas Law Equation

The ideal gas equation is

$PV=nRT$

where:
P is the pressure exerted by an ideal gas,
V is the volume occupied by an ideal gas,
T is the absolute temperature of an ideal gas,
R is universal gas constant or ideal gas constant,
n is the number of moles (amount) of gas.

Derivation of Ideal Gas Law

The ideal gas law can easily be derived from three basic gas laws: Boyle's law, Charles's law, and Avogadro's law.

Ideal gas law = Boyle's law + Charles's law + Avogadro's law — Figure 1: The ideal gas law is the combination of Boyle's law, Charles's law, and Avogadro's law.

Boyle's law states pressure and volume of an ideal gas are in inversely proportional to each other for a fixed amount of the gas at constant temperature.

$V \propto \frac{1}{P} \qquad @ \, \text{constant} \,T,\,n$

Charles's law says the volume of an ideal gas is directly proportional to temperature for a fixed amount of the gas at constant pressure.

$V \propto T \qquad @ \, \text{constant} \,n\, \text{and}\,P$

Avogadro's law says the volume of an ideal gas is directly proportional to amount (moles) of the gas at constant temperature and pressure.

$V \propto n \qquad @ \, \text{constant} \,T\, \text{and}\,P$

All the above three laws are summarised in the below table.

Table 1 Gas Laws: Boyle's Law, Charles's Law, and Avogadro's Law
Parameter	Boyle's law	Charles's law	Avogadro's law
Formula	PV = k₁	^V⁄_T = k₂	^V⁄_n = k₃
Condition	At constant T, n	At constant P, n	At constant P, T

By combining the above expressions, we can arrive at the final expression,

$V \propto \frac{nT}{P}$

By replacing the proportionality,

$V = R\frac{nT}{P}$

where R is ideal gas constant.

Finally, rearranging the above equation,

$PV=nRT$

The ideal gas equation can also be derived from the kinetic theory of gases, but it is not discussed in this article.

The ideal gas equation for two different condition can be written as:

$\frac{P_1V_1}{n_1RT_1}=\frac{P_2V_2}{n_2RT_2}$ $\frac{P_1V_1}{n_1T_1}=\frac{P_2V_2}{n_2T_2}$

This equation is very useful in numerical calculations when there is a change of state.

Other Forms of Ideal Gas Law

Other variations of the ideal gas law are discussed below.

Molar Volume

Molar volume V_m is defined as the volume of gas per unit mole.

$V_\text{m}=\frac{V}{n}=\frac{RT}{P}$ $PV_\text{m}=RT$

Density and Specific Volume

Density (ρ) is mass divided by volume. The mass (m) of any substance is the number of moles (n) times the molecular weight (M_w) of the substance.

$\rho=\frac{m}{V}=\frac{nM_\text{w}}{V}=\frac{PM_\text{w}}{RT}$

Now, R divided by M_w is specific gas constant. It is denoted by R_sp.

$\rho=\frac{P}{R_\text{sp}T}$

The reciprocal of density is specific volume (v_sp).

$v_\text{sp}=\frac{1}{\rho}=\frac{R_\text{sp}T}{P}$

This form of the equation is very useful in mechanical and chemical engineering.

Avogadro's Constant

Avogadro’s Constant (N_A) is the ratio of the total number of molecules (N) to the total moles (n). Its approximate value is 6.022 × 10²³ mol⁻¹.

$N_\text{A}=\frac{N}{n}=\frac{N}{PV \big/ RT}$

Rearranging the above expression,

$PV=\frac{NRT}{N_\text{A}}$

The number density (ρ_n) is the number of molecules per unit volume.

$P=\frac{N}{V}\frac{RT}{N_\text{A}}=\rho_\text{n} \frac{RT}{N_\text{A}}$ $PN_\text{A}=\rho_\text{n}RT$

Boltzmann Constant

Boltzmann constant (k_B) is the ratio of gas constant to Avogadro's constant.

$k_\text{B}=\frac{R}{N_\text{A}}\approx 1.38\times 10^{-23}$

Substituting the above equation in $PV=\frac{NRT}{N_\text{A}}$ ,

$PV=N k_\text{B}T$

This equation is very important particularly in statistical mechanics.

Ideal Gas Constant

Ideal gas constant or universal gas constant is a proportionality constant and denoted by R. It value in SI unit is 8.314 J K⁻¹ mol⁻¹.

$R =\frac{PV}{RT} &=8.314\,\text{J}\,\text{K}^{-1}\,\text{mol}^{-1} \\ &=0.08205\,\text{L}\,\text{atm}\,\text{K}^{-1}\,\text{mol}^{-1}$

Ideal Gas Law with Other Gas Laws

Four important gas laws are Boyle's law, Charles's law, Gay-Lussac's law (or Amontons's law), and Avogadro's law can be easily obtained from the ideal gas equation. This is reversed of what we did in the above derivation section.

The ideal gas law is PV = nRT.

Putting n and T as constant in the ideal gas equation, we have PV = constant. This is Boyle's law.

When P and n are constant, we get Charles's law i.e., V = T × constant.

Gay-Lussac's law is obtained when V and n are constant. The equation is P = T ×constant.

Avogadro's law is the relation between volume and number of moles at constant T and P. The equation is V = n × constant.

We can also derive combined gas law from the ideal gas equation. For constant n, the combined gas law equation is $\frac{PV}{T}=k$ .

Relation between gas laws:Boyle's law, Charles's law, Gay-Lussac's law, and Avogadro's law — Figure 2: Relationships between gas laws

Graph Representation of Ideal Gas Law

The ideal gas law has four variable parameters, P, V, T, and n. The ideal equation will fit into four dimensions, which is impossible to draw on paper. But each of the parameters can be plotted separately. The below figure mentions four main relationships or four main gas laws.

The graph of Boyle's law, Charles's law, Gay-Lussac's law, and Avogadro's law — Figure 3: Graphs of Boyle's Law, Charles's Law, Gay-Lussac's Law, and Avogadro's Law

We can plot the ideal gas equation in three dimensions when one of the four parameters is made constant. So, for constant n, we get the combined gas equation, which is . The surface of this equation is shown in the below figure.

The surface of ideal gas law — Figure 4: Ideal Gas Law Surface for Fixed Amount of Gas. The surface in the figure is ^PV∕_T = k.

As we see from the above graph, the projection of the surface on the pressure-volume plane is Boyle's law. When we project, we make variable temperature T constant in the combined gas equation, and we get PV = k.

Similarly, when we project on the temperature-volume plane, we get Charles's law, T = k × V and for the temperature-pressure plane, we have Gay-Lussac's law, P = k × T.

Limitations of Ideal Gas Law

The limitations are as follows:

The ideal gas law, PV= nRT is applicable only ideal gases. It is a good approximation of real gases under low pressure and/or high temperature.
At high pressure and low temperature, the ideal law equation deviates significantly from the behaviour of real gases. This can be explained because of the increase in intermolecular repulsive forces at these conditions.
For high-density gas, the equation differs significantly.
The equation gives a better result for monatomic and light-weight gases. As the molecular size increases, the deviations also increase.

Assumptions in Ideal Gas Law

The assumptions for the ideal gas law are the same as assumption made in the kinetic theory of gases.

Gas consist of particles which are in constant random motion in straight lines.
The particles of gas do not exert any force among them. Thus, intermolecular forces are zero
The particles do not occupy any space relative to its container.
The molecules of the gas are rigid identical spheres and, all possess the same mass.
The volume of each molecule is negligible in comparison to the size of the container.
All collisions among them and between the molecules and the wall are perfectly elastic in nature. There is no loss of kinetic energy in collisions.
The pressure is a result of collisions among molecules and the wall of the container.
The average kinetic energy of gas molecules is the function of temperature only.

Examples

Example 1

Consider 5.5 mol Helium gas at 30 ℃ and pressure of 1 atm. Calculate the volume?

First, convert the temperature into the kelvin from the celsius.

$T=30+273.15=303.15 \,\text{K}$

Now using 1 atm = 101 235 N m⁻² and R = 8.314 J K⁻¹ mol⁻¹,

$V=\frac{n}{P}\times RT&= \frac{5.5}{101\,325} \times 8.314 \times 303.15 \\&\approx 1.4 \,\text{m}^3$

The volume of gas is 0.14 m³.

Example 2

Carbon dioxide gas undergoes a cooling from an initial temperature of 300 ℃, a pressure of 2.0 atm, and a volume of 20 L to a final temperature of 100 ℃ and a volume of 15 L. Throughout the cooling, the amount of gas remains constant. Calculate the new pressure?

First, convert the temperatures into the kelvin from the celsius.

$T_1=300+273.15=573.15\,\text{K}$ $T_2=100+273.15=373.15\,\text{K}$

From the ideal gas law,

$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$ $P_2=\frac{P_1V_1}{T_1}\times \frac{T_2}{V_2}&=\frac{2.0 \times 20}{573.15} \times \frac{373.15}{15} \\&\approx 1.7\,\text{atm}$

The final pressure of gas is 1.7 atm.

Example 3

How many moles of neon are contained in 12 dm⁻³ at NTP?

At temperature and pressure at NTP are 20 ℃ and 101 325 atm.

Using the ideal gas equation,

$n=\frac{PV}{RT}&=\frac{(101\,325) \times (12 \times 10^{-3})}{8.314 \times (20+273.15)}\\&=\frac{101\,325 \times (12 \times 10^{-3})}{8.314 \times 293.15}\\&\approx 0.50\, \text{mol}$

Therefore, 12 dm⁻³ of neon at NTP contains 0.50 mol.

Example 4

Determine the molar volume and density of air at STP (T = 273.15 K, P = 101 325 Pa).

The molar volume is the ratio of volume to mole.

$V_\text{m} =\frac{V}{n}&=\frac{RT}{P} \\&=\frac{8.314\times 273.15}{101\,325} \\ &=0.022\,41\,\text{m}^3\,\text{mol}^{-1} \\ &=22.41\,\text{m}^{3} \,\text{kmol}^{-1}\\ &=22.41\,\text{dm}^{3} \,\text{mol}^{-1}$

Consider the molecular weight M_w of air 28.84 g mol⁻¹.

The density can be calculated by dividing molar volume to the molar mass of air.

$\rho =\frac{M_\text{w}}{V_\text{m}} =\frac{28.84}{22.41} & \approx{1.287}$

Example 5

In electrolysis of sodium chloride, hydrogen and chlorine are liberated from the aqueous sodium chloride solution. Both the gases are transferred into a container which is maintained at 4 atm and 30 ℃. Calculate the volume of the container if 100 mol of H₂ and 100 mol of Cl₂ are transferred.

The total number of moles transfer be n.

$n=n_{\text{H}_2}+n_{\text{Cl}_2}=100+100=200\,\text{mol}$

The volume by the ideal gas equation is as follows:

$V=\frac{nRT}{P}&=\frac{200\times 8.314 \times 303.15}{4\times 101\,324 } \\&=4.97\,\text{m}^3$

Example 6

If the sample of air of expands from the initial temperature of 25 ℃, initial pressure of 2.2 × 10⁵ Pa, and initial volume of 1.2 m³. The final temperature and volume are 40 ℃. and 5.3 m³. Estimate the final pressure?

Here, the amount of gas is fixed. Using the ideal gas equation,

$\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}$ $P_2 &=\frac{P_1V_1}{T_1} \times \frac{T_2}{V_2}\\ &=\frac{2.2\times 10^5 \times 1.2}{25+273.15} \times \frac{40+273.15}{5.3}\\ &=\frac{2.2\times 10^5 \times 1.2}{298.15} \times \frac{313.15}{5.3}\\ &=5.2\times 10^4\,\text{Pa}$

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