Ideal Gas Equation and Real Gases
Derivation of the Ideal Gas Equation
The Ideal Gas Equation is obtained by combining Boyle's Law, Charles's Law, and Avogadro's Law into a single relationship. Boyle's law states that volume is inversely proportional to pressure when temperature and moles are constant ($V \propto \frac{1}{P}$). Charles's law states that volume is directly proportional to temperature when pressure and moles are constant ($V \propto T$). Avogadro's law states that volume is directly proportional to the number of moles when pressure and temperature are constant ($V \propto n$). When none of the variables are held constant, all three relationships merge:
$$V \propto \frac{nT}{P}$$
Combining the proportions: Replacing the proportionality sign with a constant gives $V = \text{Constant} \times \frac{nT}{P}$. The constant is the Universal Gas Constant, denoted by $R$.
Rearranging: Multiplying both sides by $P$ gives $PV = nRT$ — the ideal gas equation. It states that for any quantity of an ideal gas, the product of pressure and volume equals the product of moles, the gas constant, and absolute temperature.
Reduction to individual laws: When $T$ and $n$ are constant, $PV = nRT$ becomes $PV = k$ (Boyle's Law). When $P$ and $n$ are constant, it becomes $V = kT$ (Charles's Law). When $P$ and $T$ are constant, it becomes $V = kn$ (Avogadro's Law).
For practical calculations where a fixed sample of gas changes from one set of conditions ($P_1$, $V_1$, $T_1$) to another ($P_2$, $V_2$, $T_2$), the ideal gas equation takes a convenient ratio form. For one mole of gas, $\frac{PV}{T} = R$, which is constant. Therefore:
$$\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$$
When to use: This combined form is used whenever a gas undergoes a change in two or more of its parameters (pressure, volume, temperature) and the amount of gas remains constant.
Unit consistency: Always convert temperature to Kelvin and ensure pressure and volume use compatible units on both sides of the equation.
Density form: By substituting $n = \frac{m}{M}$ (where $m$ is mass and $M$ is molar mass), the equation becomes $PV = \frac{m}{M}RT$. Rearranging gives $d = \frac{PM}{RT}$, where $d$ is the density of the gas. This shows density is directly proportional to molar mass and pressure, and inversely proportional to temperature.
The Universal Gas Constant R
The value of $R$ is calculated using the fact that one mole of an ideal gas occupies 22.414 dm³ at STP (1 atm pressure, 273.16 K). Substituting these into $R = \frac{PV}{nT}$:
$$R = \frac{1 \text{ atm} \times 22.414 \text{ dm}^3}{1 \text{ mol} \times 273.16 \text{ K}} = 0.0821 \text{ dm}^3 \text{ atm K}^{-1} \text{ mol}^{-1}$$
R is a universal constant whose value depends only on the units chosen for pressure, volume, and temperature. Its physical meaning: one mole of ideal gas absorbs 0.0821 dm³·atm of energy when heated by 1 K at constant pressure.
$R$=Universal gas constant — same for all ideal gases(varies with unit system)
$P$ in atm, $V$ in dm³
→$R = 0.0821$ dm³ atm K⁻¹ mol⁻¹
$P$ in torr, $V$ in dm³
→$R = 62.4$ dm³ torr K⁻¹ mol⁻¹
$P$ in torr, $V$ in cm³
→$R = 62400$ cm³ torr K⁻¹ mol⁻¹
$P$ in Nm⁻², $V$ in m³
→$R = 8.314$ J K⁻¹ mol⁻¹
Energy in calories
→$R = 1.989$ cal K⁻¹ mol⁻¹
Atmosphere to torr conversion: $1 \text{ atm} = 760 \text{ torr}$, so $R = 0.0821 \times 760 = 62.4$ dm³ torr K⁻¹ mol⁻¹. Since $1 \text{ mm Hg} = 1 \text{ torr}$, these units are interchangeable.
dm³ to cm³ conversion: $1 \text{ dm}^3 = 1000 \text{ cm}^3$, so $R = 62.4 \times 1000 = 62400$ cm³ torr K⁻¹ mol⁻¹.
SI calculation: Using $P = 101325 \text{ Nm}^{-2}$, $V = 0.022414 \text{ m}^3$, $T = 273.16 \text{ K}$, and $1 \text{ Nm} = 1 \text{ J}$, we get $R = 8.314$ J K⁻¹ mol⁻¹.
Calorie conversion: Since $1 \text{ cal} = 4.18 \text{ J}$, $R = \frac{8.314}{4.18} = 1.989$ cal K⁻¹ mol⁻¹.
Common Values of R
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$0.0821$ dm³ atm K⁻¹ mol⁻¹ (most common in chemistry problems)
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$62.4$ dm³ torr K⁻¹ mol⁻¹ (when pressure is in torr/mm Hg)
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$62400$ cm³ torr K⁻¹ mol⁻¹ (when volume is in cm³)
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$8.314$ J K⁻¹ mol⁻¹ (SI units)
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$1.989$ cal K⁻¹ mol⁻¹ (energy in calories)
Ideal vs Real Gases
An Ideal Gas is a theoretical gas that perfectly obeys all gas laws under all conditions of temperature and pressure. Its behaviour is described by two key postulates of the Kinetic Molecular Theory: there are no forces of attraction between gas molecules, and the actual volume of gas molecules is negligible compared to the volume of the container. A Real Gas, on the other hand, shows measurable deviations from ideal behaviour because these assumptions break down under certain conditions.
No intermolecular forces (ideal): Ideal gas molecules do not attract or repel each other. Real gas molecules do experience intermolecular forces — the strength depends on the nature of the gas.
Negligible molecular volume (ideal): Ideal gas molecules are treated as point masses with zero volume. Real gas molecules occupy a finite, non-negligible volume, especially at high pressure.
Compressibility factor: The ratio $\frac{PV}{nRT}$ equals exactly 1 for an ideal gas at all conditions. For real gases, this ratio deviates from 1 — it is called the Compressibility Factor. A value above 1 means the gas is less compressible than ideal; below 1 means more compressible.
Ideal vs Real Gas Behaviour
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Ideal: $PV = nRT$ holds at all $P$ and $T$ — Real: deviates at high $P$ and low $T$
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Ideal: no intermolecular forces — Real: attractive forces reduce observed pressure
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Ideal: zero molecular volume — Real: molecules have finite volume (excluded volume $b$)
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Ideal: compressibility factor $Z = 1$ always — Real: $Z$ varies with $P$ and $T$
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Ideal: obeys Boyle's, Charles's, and Avogadro's laws exactly — Real: deviates especially at extreme conditions
Experimental observations with gases like $H_2$, $He$, $N_2$, and $CO_2$ reveal two clear trends about when real gases behave ideally and when they do not. The Compressibility Factor graphs ($PV/RT$ vs pressure) show deviations that shift with temperature.
Low pressure = ideal behaviour: At low pressures, gas molecules are far apart, intermolecular forces are negligible, and molecular volume is insignificant compared to the container — the gas behaves ideally.
High pressure = non-ideal behaviour: At high pressures, molecules are forced close together. Intermolecular attractive forces become significant and the finite volume of molecules can no longer be ignored — the gas deviates from ideality.
High temperature = ideal behaviour: At elevated temperatures, molecules move fast enough that intermolecular attractions are overcome by kinetic energy. At 100°C, deviations are much smaller than at 0°C for the same gas.
Low temperature = non-ideal behaviour: At low temperatures, molecules move slower and intermolecular forces have a greater relative effect, causing larger deviations from the ideal gas equation.
van der Waals Equation for Real Gases
van der Waals (1873) identified the two postulates of the kinetic molecular theory that fail for real gases. He introduced corrections for both pressure and volume to modify the ideal gas equation into an equation that describes real gas behaviour. The two causes of deviation from ideality are intermolecular attractive forces and the finite volume of gas molecules.
Pressure correction: A molecule striking the container wall is pulled inward by neighbouring molecules, reducing its impact. The observed pressure $P$ is therefore less than the ideal (kinetic) pressure $P_i$. To compensate, a correction term $\frac{a}{V^2}$ is added, where $a$ measures the strength of intermolecular attraction.
Volume correction: Real gas molecules occupy space, so the volume available for free movement is less than the container volume. An exclusion term $b$ (the Excluded Volume) is subtracted from the measured volume $V$. The excluded volume is approximately four times the actual volume of the molecules: $b = 4V_m$.
Nature of constants: The constant $a$ reflects the strength of intermolecular forces — gases with stronger attractions (like $SO_2$, $Cl_2$) have higher $a$ values. The constant $b$ reflects the size of molecules — larger molecules have higher $b$ values.
van der Waals Constants for Common Gases
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$H_2$: $a = 0.245$ atm dm⁶ mol⁻², $b = 0.0266$ dm³ mol⁻¹
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$O_2$: $a = 1.360$ atm dm⁶ mol⁻², $b = 0.0318$ dm³ mol⁻¹
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$N_2$: $a = 1.390$ atm dm⁶ mol⁻², $b = 0.0391$ dm³ mol⁻¹
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$CO_2$: $a = 3.590$ atm dm⁶ mol⁻², $b = 0.0428$ dm³ mol⁻¹
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$NH_3$: $a = 4.170$ atm dm⁶ mol⁻², $b = 0.0371$ dm³ mol⁻¹
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$SO_2$: $a = 6.170$ atm dm⁶ mol⁻², $b = 0.0564$ dm³ mol⁻¹
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$Cl_2$: $a = 6.493$ atm dm⁶ mol⁻², $b = 0.0562$ dm³ mol⁻¹
Applying both corrections to the ideal gas equation $PV = nRT$ gives the van der Waals Equation. For one mole of gas, the corrected pressure $(P + \frac{a}{V^2})$ replaces $P$ and the corrected volume $(V - b)$ replaces $V$:
$$(P + \frac{a}{V^2})(V - b) = RT$$
This equation modifies the ideal gas law to account for intermolecular forces (pressure correction) and molecular volume (volume correction) in real gases.
$P$=Observed (measured) pressure of the real gas(atm or Nm⁻²)
$a$=van der Waals constant for intermolecular attraction — larger for gases with stronger forces(atm dm⁶ mol⁻² or Nm⁴ mol⁻²)
$V$=Measured volume of the container(dm³ or m³)
$b$=Excluded volume — the volume occupied by 1 mole of molecules in compressed state (≈ 4 × actual molecular volume)(dm³ mol⁻¹ or m³ mol⁻¹)
$R$=Universal gas constant(matches units of P and V)
$T$=Absolute temperature(K)
$n$ moles of gas (not 1 mole)
→$(P + \frac{n^2a}{V^2})(V - nb) = nRT$
$a = 0$ and $b = 0$
→Reduces to the ideal gas equation $PV = nRT$
Derivation of pressure correction: The inward pull on a wall-striking molecule is proportional to the concentration of gas molecules ($\propto n/V$). Since both the striking molecule and the pulling molecules contribute, the correction is proportional to $(n/V)^2$. Thus $P' = \frac{an^2}{V^2}$, and for one mole, $P' = \frac{a}{V^2}$.
For n moles: The full van der Waals equation for $n$ moles is $(P + \frac{n^2a}{V^2})(V - nb) = nRT$. Note that the pressure correction scales with $n^2$ while the volume correction scales with $n$.