Gas Laws and Kinetic Molecular Theory
Kinetic Molecular Theory — Postulates
The Kinetic Molecular Theory explains the macroscopic behaviour of gases through the microscopic motion of their molecules. It was first proposed by Bernoulli (1738) and later developed by Clausius (1857), Maxwell, and Boltzmann. The theory rests on eight fundamental postulates that describe an ideal gas.
Composition: Every gas consists of a large number of very small particles called molecules. Monoatomic gases such as He, Ne, and Ar have single-atom molecules.
Random Motion: Gas molecules move haphazardly, constantly colliding among themselves and with the walls of the container, changing direction after each collision.
Elastic Collisions: All collisions — between molecules and with the walls — are perfectly elastic. Kinetic energy is conserved during every collision.
Large Separation: Molecules of a gas are widely separated from one another, with sufficient empty spaces between them. Most of the volume occupied by a gas is actually empty space.
No Intermolecular Forces: The molecules of a gas exert no forces of attraction or repulsion on each other.
Negligible Molecular Volume: The actual volume of gas molecules is negligible compared to the total volume of the gas.
Gravity Negligible: The effect of gravity on molecular motion is negligible compared to the effect of continued collisions between molecules.
Kinetic Energy and Temperature: The average kinetic energy of gas molecules varies directly as the absolute temperature of the gas.
Eight Postulates at a Glance
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Large number of very small molecules
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Haphazard, random motion of molecules
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Perfectly elastic collisions
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Widely separated molecules with empty spaces
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No intermolecular forces of attraction
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Negligible molecular volume vs container volume
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Gravity effect negligible compared to collisions
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Average KE ∝ absolute temperature
Using these postulates, Clausius derived the Kinetic Equation for an ideal gas. This equation connects the macroscopic properties of pressure and volume to the microscopic properties of molecular mass and speed.
$$PV = \frac{1}{3} mN\overline{c^2}$$
Relates the pressure and volume of a gas to the mass, number, and mean square velocity of its molecules
$P$=Pressure exerted by the gas(atm or Pa)
$V$=Volume of the container(dm³ or m³)
$m$=Mass of one molecule of the gas(kg)
$N$=Total number of molecules in the vessel(dimensionless)
$\overline{c^2}$=Mean square velocity of the molecules(m²/s²)
Mean Square Velocity: All molecules of a gas do not have the same velocity. The quantity $\overline{c^2}$ is the average of the squares of all molecular velocities. Its square root gives the RMS Velocity.
RMS Formula: The root mean square velocity can also be expressed in terms of molar mass and temperature, showing that lighter molecules move faster at the same temperature.
Pressure Origin: Pressure arises from the force of molecular collisions with the container walls divided by the wall area — it is a macroscopic result of microscopic impacts.
The RMS Velocity is a useful measure of molecular speed that follows directly from the Kinetic Equation. It quantitatively links molecular speed to temperature and molar mass.
$$C_{rms} = \sqrt{\frac{3RT}{M}}$$
Gives the root mean square speed of gas molecules in terms of temperature and molar mass
$C_{rms}$=Root mean square velocity(m/s)
$R$=Universal gas constant(8.314 J mol⁻¹ K⁻¹)
$T$=Absolute temperature(K)
$M$=Molar mass of the gas(kg/mol)
$T$ increases
→$C_{rms}$ increases — molecules move faster at higher temperatures
$M$ increases
→$C_{rms}$ decreases — heavier molecules move slower at the same temperature
Temperature Dependence: At higher temperatures, molecules possess greater kinetic energy and therefore move at higher RMS velocities.
Molar Mass Dependence: For two gases at the same temperature, the lighter gas has the higher RMS velocity.
Gas Particle Motion and Pressure
Gas molecules are in a state of continuous, random motion. They travel in straight lines between collisions and change direction abruptly upon each collision. The Pressure a gas exerts on its container is entirely due to these collisions — each impact transfers momentum to the wall, and the summed effect of billions of collisions per second produces the macroscopic pressure we measure.
$$E_k = \frac{3R}{2N_A}T$$
Defines the average translational kinetic energy per molecule of a gas in terms of absolute temperature
$E_k$=Average translational kinetic energy per molecule(J)
$R$=Universal gas constant (8.314 J mol⁻¹ K⁻¹)(J mol⁻¹ K⁻¹)
$N_A$=Avogadro's number (6.022 × 10²³)(mol⁻¹)
$T$=Absolute temperature(K)
$T = 0$ K
→$E_k = 0$ — molecular motion would cease at absolute zero (unattainable in practice)
Elastic Collisions: Both molecule-molecule and molecule-wall collisions conserve total kinetic energy. No energy is lost as heat or deformation during collisions of ideal gas molecules.
Maxwell's Distribution: Molecular velocities are not identical. Molecules group into velocity ranges, with most molecules near the most probable speed. The distribution broadens and shifts to higher speeds as temperature increases.
KE and Temperature Link: From the kinetic equation, comparing $PV = RT$ for one mole with $PV = \frac{2}{3}N_A E_k$ gives $E_k = \frac{3R}{2N_A}T$. This confirms that temperature is directly proportional to the average translational kinetic energy of molecules.
Boyle's Law
Boyle's Law describes the inverse relationship between the pressure and volume of a fixed amount of gas at constant temperature. When you compress a gas (increase pressure), its volume decreases proportionally, and vice versa.
$$P_1 V_1 = P_2 V_2$$
At constant temperature, the product of pressure and volume for a given mass of gas is always constant
$P_1, V_1$=Initial pressure and volume(atm, dm³)
$P_2, V_2$=Final pressure and volume(atm, dm³)
Temperature doubles
→PV remains constant only if T is held fixed — changing T changes the constant k
Inverse Proportionality: $V \propto \frac{1}{P}$ when temperature and number of moles are constant. Doubling the pressure halves the volume.
Constant k: The product $PV = k$ is constant for the same quantity of gas at the same temperature. Different amounts of gas or different temperatures give different k values.
Experimental Basis: When a gas at 2 atm occupying 1 dm³ is compressed to 4 atm, its volume becomes $\frac{1}{2}$ dm³, and at 6 atm it becomes $\frac{1}{3}$ dm³. In each case $PV = 2$ dm³ atm.
Boyle's law can be represented graphically in three ways, each confirming the inverse proportionality between pressure and volume.
V vs P (Isotherm): A plot of volume against pressure gives a hyperbolic curve called an isotherm ('iso' = same, 'therm' = heat). At higher temperatures, the isotherm shifts away from both axes because the gas occupies a larger volume at each pressure.
P vs 1/V: A plot of pressure against $\frac{1}{V}$ gives a straight line passing through the origin. At higher temperatures, this line has a steeper slope (closer to the pressure axis).
PV vs P: A plot of the product $PV$ against pressure gives a horizontal straight line parallel to the pressure axis, confirming that $PV$ is constant at a given temperature.
Boyle's Law can be derived from the Kinetic Equation. Starting from $PV = \frac{2}{3} \times \frac{1}{2}mN\overline{c^2}$ and using the fact that average kinetic energy is proportional to temperature ($\frac{1}{2}mN\overline{c^2} = kT$), we get $PV = \frac{2}{3}kT$. At constant temperature, the right side is a constant, so $PV = k'$ — which is Boyle's law.
Molecular Reasoning: When volume decreases at constant temperature, molecules travel shorter distances between wall collisions and strike the walls more frequently per unit area — hence pressure increases.
Temperature Link: Since $T$ is constant, the average speed of molecules does not change. The pressure increase is purely due to more frequent collisions in a smaller volume.
Charles's Law
Charles's Law describes the direct relationship between the volume and absolute temperature of a fixed amount of gas at constant pressure. As a gas is heated, its volume increases proportionally, and as it is cooled, its volume decreases.
$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$
At constant pressure, the ratio of volume to absolute temperature for a given mass of gas is always constant
$V_1, T_1$=Initial volume and temperature(dm³, K)
$V_2, T_2$=Final volume and temperature(dm³, K)
$T = 0$ K
→Volume would theoretically become zero (absolute zero)
Direct Proportionality: $V \propto T$ when pressure and number of moles are constant. Doubling the absolute temperature doubles the volume.
Constant k: The ratio $\frac{V}{T} = k$ is constant for the same amount of gas at the same pressure.
Volume-Temperature Data: For a gas with volume 546 cm³ at 0 °C (273 K), the ratio $\frac{V}{T} = 2$ cm³/K remains constant whether the temperature is 100 °C (373 K, $V$ = 746 cm³) or −100 °C (173 K, $V$ = 346 cm³).
Charles's Law follows from the Kinetic Equation. From $PV = \frac{2}{3}kT$, at constant pressure we get $V = \frac{2k}{3P}T$. Since $\frac{2k}{3P}$ is a constant when $P$ is fixed, we have $V = k''T$ or $\frac{V}{T} = k''$ — which is Charles's law.
Molecular Reasoning: At constant pressure, when temperature increases, molecules move faster and push the piston outward with greater force, expanding the volume until the internal pressure matches the external pressure again.
Isothermal vs Isobaric: Boyle's law describes isothermal changes (constant T), while Charles's law describes isobaric changes (constant P).
Absolute Zero
Absolute Zero is the lowest possible temperature, theoretically the point where all molecular motion ceases. It is derived by extrapolating the volume-temperature relationship of a gas to the point where volume reaches zero.
$$V_t = V_0 \left(1 + \frac{t}{273}\right)$$
Gives the volume of a gas at any Celsius temperature, based on its volume at 0 °C
$V_t$=Volume of gas at temperature $t$ °C(cm³ or dm³)
$V_0$=Volume of gas at 0 °C(cm³ or dm³)
$t$=Temperature on Celsius scale(°C)
$t = -273$ °C
→$V_t = 0$ — the theoretical point of zero volume, defining absolute zero
1/273 Rule: At constant pressure, the volume of a given mass of gas changes by $\frac{1}{273}$ of its volume at 0 °C for every 1 °C change in temperature.
Graphical Derivation: A plot of volume (y-axis) against Celsius temperature (x-axis) gives a straight line. Extrapolating this line backward, it intersects the temperature axis at −273.16 °C (0 K), where volume would be zero.
Unattainable: Absolute zero (0 K or −273.16 °C) cannot actually be reached because all real gases liquefy before this temperature. It represents a theoretical limit.
Celsius Scale Failure: Charles's law is not obeyed when using Celsius temperatures. For example, $\frac{566}{10} \neq \frac{746}{100}$ in Celsius, but in Kelvin: $\frac{566}{283} = \frac{746}{373} = 2$.
Temperature Scales and STP
Three temperature scales are commonly used in chemistry. The Kelvin Scale (absolute scale) is the standard for gas law calculations because it starts from Absolute Zero and avoids negative values.
Centigrade (Celsius) Scale: Ice melts at 0 °C and water boils at 100 °C at 1 atm. The range is divided into 100 equal parts.
Fahrenheit Scale: Ice melts at 32 °F and water boils at 212 °F at 1 atm. The range is divided into 180 equal parts.
Kelvin (Absolute) Scale: The lowest point is 0 K (absolute zero). Ice melts at 273 K and water boils at 373 K (more precisely 373.16 K) at 1 atm.
Temperature Scale Conversion Formulas
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$K = °C + 273.16$
2
$°C = \frac{5}{9}(°F - 32)$
3
$°F = \frac{9}{5}(°C) + 32$
Standard Temperature and Pressure (STP) is the reference condition used for comparing gas volumes. At STP, one mole of any ideal gas occupies exactly 22.4 dm³.
Standard Temperature: 0 °C (273 K)
Standard Pressure: 1 atmosphere (101.325 kPa or 760 mmHg)
Molar Volume: At STP, 1 mole of any ideal gas = 22.4 dm³. This is a fixed value that applies universally to all ideal gases regardless of their identity.
Practical Use: STP allows direct comparison of gas volumes and simplifies stoichiometric calculations involving gases.
STP Conditions Summary
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Temperature: 0 °C (273 K)
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Pressure: 1 atm (101.325 kPa, 760 mmHg)
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Molar volume of ideal gas: 22.4 dm³/mol