Home › Physics › Thermal Physics › Behaviour of Ideal Gas
Thermal Physics
Behaviour of Ideal Gas
An ideal gas is a theoretical gas composed of a set of randomly moving point particles that interact only through elastic collisions. It strictly obeys the universal gas equation under all conditions of temperature and pressure.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Boyle's Law, Charles's Law, Gay-Lussac's Law, ideal gas equation PV = nRT, universal gas constant, and basic absolute temperature scale.
PGT extension
Kinetic interpretation of temperature, Maxwell-Boltzmann velocity distribution, Van der Waals real gas corrections, and compressibility factor.
Prerequisites
Concept of Temperature
Kinetic theory of gases
Law of equipartition of energy
Postulates of Kinetic Theory of Gases
All gases consist of a very large number of identical, rigid, spherical molecules moving in random directions with all possible speeds.
The actual volume occupied by the gas molecules is negligible compared to the total volume of the container.
Collisions between molecules and with the container walls are perfectly elastic, conserving both total momentum and kinetic energy.
Equation of State for an Ideal Gas
The macroscopic behavior of an ideal gas is summarized by the equation PV = nRT, where n is the number of moles.
Alternatively, in terms of Boltzmann's constant, PV = NkT, where N is the total number of molecules and $k = R/N_A$.
The constant R has a value of approximately 8.314 J mol-1 K-1.
Kinetic Interpretation of Pressure
Pressure arises due to the continuous bombardment of gas molecules against the walls of the container.
The pressure exerted by an ideal gas is given by $P = (1/3)mnv^{2}_{rms}$, where m is molecular mass, n is number density, and v_{rms} is root mean square velocity.
Pressure depends directly on the translational kinetic energy per unit volume of the gas.
Absolute Temperature and Kinetic Energy
Temperature is a direct measure of the average translational kinetic energy of gas molecules.
For a monoatomic gas, the average kinetic energy per molecule is given by $E = (3/2)kT$.
At absolute zero (T = 0 K), the average kinetic energy of gas molecules theoretically drops to zero.
Derivations
Pressure Exerted by an Ideal Gas
Conditions: N identical molecules in a cubical container of side L
Consider a molecule of mass m moving with velocity component v_x along the x-axis inside a cube of volume L3.
The change in momentum when the molecule undergoes an elastic collision with the wall perpendicular to the x-axis is Delta $p_x = 2mv_x$.
The time interval between successive collisions with the same wall is Delta $t = 2L / v_x$.
The force exerted by a single molecule on the wall is $F_x = \delta p_x / \delta t = mv_x^{2} / L$.
Summing over all N molecules and taking the isotropic average $v_x^{2} = (1/3)v^{2}_{rms}$, the total pressure is $P = F / L^{2} = (1/3) (Nm/V) v^{2}_{rms}$.
Result: $P = \frac{1}{3} \rho v_{rms}^2$
Relation between Pressure and Kinetic Energy
Conditions: Ideal gas containing N molecules in volume V
Start from the kinetic theory expression for pressure: $P = (1/3) (Nm/V) v^{2}_{rms}$.
Multiply and divide the right-hand side by 2 to introduce kinetic energy: $P = (2/3) \times (1/2) (Nm v^{2}_{rms}) / V$.
Recognize that the total translational kinetic energy of the gas is $E = (1/2) Nm v^{2}_{rms}$.
Substitute E into the pressure equation to obtain $P = (2/3) (E / V)$.
Result: PV = \frac{2}{3} E
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Ideal Gas Equation
$$PV = nRT$$
P = Pressure, V = Volume, n = Number of moles, R = Universal gas constant, T = Absolute temperature P in Pa, V in m3, T in K
Low pressure and high temperature
Molecular Ideal Gas Equation
$$PV = NkT$$
N = Number of molecules, k = Boltzmann constant k in J K-1
k = Boltzmann constant, T = Absolute temperature J
Valid for monoatomic ideal gas
Boyle's Law
$$P_1 V_1 = P_2 V_2$$
P = Pressure, V = Volume Pa, m3
Constant temperature and mass
Charles's Law
$$\frac{V_1}{T_1} = \frac{V_2}{T_2}$$
V = Volume, T = Temperature m3, K
Constant pressure and mass
Gay-Lussac's Law
$$\frac{P_1}{T_1} = \frac{P_2}{T_2}$$
P = Pressure, T = Temperature Pa, K
Constant volume and mass
Dalton's Law of Partial Pressures
$$P = P_1 + P_2 + P_3 + \dots$$
P = Total pressure, $P_i = \partial pressure of individual gases$ Pa
Non-reacting gas mixtures
Graphs and experiments
Isotherms of an Ideal Gas (P vs V)
Draw: A family of rectangular hyperbolas (curves sloping downward from left to right) for different constant temperatures T_3 > T_2 > T_1.
Interpret: Shows inverse proportionality between pressure and volume at constant temperature according to Boyle's law.
Exam use: Direct identification of isothermal curves and comparison of temperatures from P-V diagrams.
Isobars on P-T Diagram
Draw: Straight lines passing through the origin (or extrapolating to -273.15 °C on Celsius scale) with different slopes for different pressures.
Interpret: Illustrates Charles's law where volume is directly proportional to absolute temperature.
Exam use: Finding slopes representing V/R ratios and determining absolute zero temperature.
Maxwell-Boltzmann Speed Distribution Curve
Draw: Asymmetric bell-shaped curve plotting fraction of molecules against molecular speed, showing distinct peaks for most probable, average, and rms speeds.
Interpret: Demonstrates statistical distribution of molecular speeds at a given temperature.
Exam use: Relating peak position to temperature and distinguishing v_{mp}, v_{avg}, and v_{rms}.
Solved numericals
TGT core
Find the root mean square velocity of oxygen molecules at 27 °C. Given $R = 8.314 J mol^{-1} K^{-1}$ and molar mass of oxygen $M = 32 × 10^{-3} kg mol^{-1}$.
Convert temperature to Kelvin: $T = 27 + 273 = 300 K$.
Calculate the value: $v_{rms} = \sqrt{74826 / 0.032} = \sqrt{2338312.5} \approx 483.5 \text{ m s}^{-1}\text{}$.
Answer: 483.5 m s^-1
PGT extension
Calculate the total translational kinetic energy of 2 moles of an ideal gas at 300 K. Take $R = 8.314 J mol^{-1} K^{-1}$.
Recall the formula for total internal kinetic energy of n moles of an ideal gas: $E = (3/2) nRT$.
Substitute the given values: n = 2, R = 8.314, T = 300.
$E = (3/2) \times 2 \times 8.314 \times 300$.
E = 3 \times 8.314 \times 300 = 7482.6 \text{ J}.
Answer: 7482.6 J
TGT core
A given mass of an ideal gas occupies 400 mL at 27 °C and 700 mm Hg pressure. What will be its volume at standard temperature and pressure (0 °C and 760 mm Hg)?
Assumption: Intermolecular forces are completely zero except during instantaneous elastic collisions. Limitation: Fails at high pressures where molecular volume and attractive forces become significant.
Assumption: Size of molecules is infinitesimally small compared to interparticle distance. Limitation: Fails at low temperatures near liquefaction point where real gases deviate heavily from ideal behavior.
Common misconceptions
Common error: Temperature is a measure of the total kinetic energy of all molecules in a body.
Correct: Temperature is a measure of the average kinetic energy per molecule, not the total sum.
A large pool of water at low temperature has more total kinetic energy than a boiling drop of water, yet its temperature is lower.
Common error: Ideal gas molecules lose energy during wall collisions.
Correct: Collisions with container walls are perfectly elastic.
By definition, kinetic energy is conserved in ideal gas collisions; otherwise, gas pressure would continuously decay over time.
Quick revision
PV = nRT governs macroscopic ideal gas behavior, where R is the universal gas constant.
Ideal Gas Equation: PV = nRT
Molecular Ideal Gas Equation: PV = NkT
Must remember
The macroscopic behavior of an ideal gas is governed by the equation $PV = nRT$.
Collisions between ideal gas molecules and container walls are perfectly elastic, conserving both momentum and kinetic energy.
Absolute temperature ($T$) is a direct measure of the average translational kinetic energy per molecule ($E = \frac{3}{2}kT$).
Root mean square velocity of gas molecules is given by $v_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$.
At absolute zero ($T = 0\text{ K}$), the average kinetic energy of gas molecules theoretically drops to zero.
Glossary
Ideal GasA hypothetical gas that perfectly obeys the gas laws ($PV = nRT$) under all conditions of temperature and pressure, with negligible molecular volume and no intermolecular forces.
Root Mean Square Velocity ($v_{rms}$)The square root of the average of the squares of the speeds of the individual molecules in a gas sample.
Boltzmann's Constant ($k$)The gas constant per molecule, defined as $k = \frac{R}{N_A}$, equal to $1.38 \times 10^{-23}\text{ J K}^{-1}$.
Absolute TemperatureTemperature measured on the Kelvin scale, starting from absolute zero where molecular motion theoretically ceases.
Boyle's Law vs Charles's Law vs Gay-Lussac's Law
Law
Constant Parameter
Mathematical Relation
Boyle's Law
Temperature ($T$) and Mass ($n$)
$P_1 V_1 = P_2 V_2$
Charles's Law
Pressure ($P$) and Mass ($n$)
$\frac{V_1}{T_1} = \frac{V_2}{T_2}$
Gay-Lussac's Law
Volume ($V$) and Mass ($n$)
$\frac{P_1}{T_1} = \frac{P_2}{T_2}$
Practice Quiz
Choose an option to reveal the explanation.
1. Which of the following equations represents the ideal gas law for $N$ molecules?
Explanation: In terms of the total number of molecules $N$ and Boltzmann's constant $k$, the ideal gas equation is written as $PV = NkT$.
2. Collisions between gas molecules and the container walls are assumed to be:
Explanation: Postulates of kinetic theory state that collisions between molecules and container walls are perfectly elastic, conserving both total momentum and kinetic energy.
3. What is the average kinetic energy of a monoatomic ideal gas molecule at absolute temperature $T$?
Explanation: For a monoatomic gas, the average kinetic energy per molecule is given by $E = \frac{3}{2}kT$.
4. Root mean square velocity of gas molecules is directly proportional to:
Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, the root mean square velocity is directly proportional to the square root of absolute temperature ($\sqrt{T}$).
5. Which law states that at constant temperature, the volume of a given mass of gas is inversely proportional to its pressure?
Explanation: Boyle's Law states that $P_1 V_1 = P_2 V_2$, implying that pressure and volume are inversely proportional at constant temperature.
6. What is the value and unit of the universal gas constant $R$?
Explanation: The universal gas constant $R$ has an approximate value of $8.314\text{ J mol}^{-1}\text{ K}^{-1}$.
7. According to the kinetic interpretation of pressure, pressure is directly proportional to:
Explanation: Pressure is given by $P = \frac{1}{3}mnv_{rms}^2 = \frac{2}{3}E_{v}$, showing it depends directly on translational kinetic energy per unit volume.
8. At absolute zero ($T = 0\text{ K}$), what happens to the average kinetic energy of gas molecules?
Explanation: At $T = 0\text{ K}$, the average translational kinetic energy of gas molecules theoretically drops to zero, corresponding to complete cessation of thermal motion.
9. Which of the following conditions favors the validity of the ideal gas equation $PV = nRT$?
Explanation: Gases behave most like ideal gases at low pressures and high temperatures, where intermolecular forces and molecular volumes are negligible.
10. Charles's Law maintains which of the following quantities constant?
Explanation: Charles's Law states that $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ under conditions of constant pressure and mass.
11. Dalton's Law of Partial Pressures states that the total pressure of a mixture of non-reacting gases is equal to:
Explanation: Dalton's Law states that $P = P_1 + P_2 + P_3 + \dots$ for non-reacting gas mixtures.
12. What does temperature measure microscopically in an ideal gas?
Explanation: Temperature is a direct measure of the average translational kinetic energy per molecule, not the total energy.
13. Gay-Lussac's Law relates which two variables at constant volume and mass?
Explanation: Gay-Lussac's Law establishes that $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ when volume and mass are held constant.
14. If the absolute temperature of an ideal gas is quadrupled, its root mean square velocity will:
Explanation: Since $v_{rms} \propto \sqrt{T}$, quadrupling $T$ causes $v_{rms}$ to increase by a factor of $\sqrt{4} = 2$.
15. Which of the following is a primary assumption of the kinetic theory regarding gas molecules?
Explanation: A core postulate of kinetic theory is that the actual volume occupied by gas molecules is negligible compared to the total volume of the container.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The root mean square velocity of hydrogen molecules at $300\text{ K}$ is $v$. At what temperature will the root mean square velocity be double ($2v$)?
Explanation: Since $v_{rms} \propto \sqrt{T}$, doubling the velocity ($v_{rms} \to 2v$) requires quadrupling the absolute temperature ($T \to 4T$). Therefore, $T_2 = 4 \times 300\text{ K} = 1200\text{ K}$? Wait, $2^2 = 4$, so $4 \times 300 = 1200\text{ K}$. Let's verify options: $1200\text{ K}$ is index 2.
2. If the pressure of an ideal gas in a closed container is increased by $0.5\%$ at constant volume, the percentage increase in temperature is:
Explanation: By Gay-Lussac's law at constant volume, $P \propto T$. Therefore, fractional change $\frac{\Delta P}{P} = \frac{\Delta T}{T} = 0.5\%$.
3. The kinetic energy of $1\text{ mole}$ of an ideal monoatomic gas at temperature $T$ is given by:
Explanation: Energy per molecule is $\frac{3}{2}kT$. For one mole containing $N_A$ molecules, total kinetic energy is $N_A \times \frac{3}{2}kT = \frac{3}{2}R T$ since $R = N_A k$.
4. Two identical vessels contain equal masses of helium (molecular weight $4$) and hydrogen (molecular weight $2$) at the same temperature. The ratio of pressure exerted by helium to hydrogen is:
Explanation: Using $PV = \frac{m}{M}RT$, pressure $P \propto \frac{1}{M}$ for equal mass $m$, volume $V$, and temperature $T$. Thus $\frac{P_{He}}{P_{H_2}} = \frac{M_{H_2}}{M_{He}} = \frac{2}{4} = \frac{1}{2}$.
5. At what temperature is the root mean square velocity of an oxygen molecule equal to $1000\text{ m s}^{-1}$? (Given molar mass of oxygen is $32\times 10^{-3}\text{ kg mol}^{-1}$, $R = 8.314\text{ J mol}^{-1}\text{ K}^{-1}$)
6. A gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$. Considering only translational degrees of freedom, the total internal energy of the system is:
Explanation: Both oxygen and argon are treated as ideal gases with translational energy $\frac{3}{2}RT$ per mole. Total moles $n = 2 + 4 = 6$ moles. Total energy $U = n \left(\frac{3}{2}RT\right) = 6 \times \frac{3}{2}RT = 9RT$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The pressure $P$ of an ideal gas is related to its density $\rho$ and root mean square velocity $v_{rms}$ by:
Explanation: Since $P = \frac{1}{3}mnv_{rms}^2$ and mass density $\rho = mn$, substituting gives $P = \frac{1}{3}\rho v_{rms}^2$.
2. Which of the following statements is incorrect regarding ideal gas molecules?
Explanation: An ideal gas has no intermolecular forces, hence its potential energy is zero and internal energy depends entirely on temperature (kinetic energy).
3. If the root mean square speed of gas molecules is $v$ at pressure $P$ and density $\rho$, what happens to $v_{rms}$ if pressure is doubled at constant density?
Explanation: From $P = \frac{1}{3}\rho v_{rms}^2$, $v_{rms} = \sqrt{\frac{3P}{\rho}}$. If $P$ is doubled while $\rho$ is constant, $v_{rms}$ increases by a factor of $\sqrt{2}$.
4. At absolute zero temperature, the root mean square velocity of gas molecules becomes:
Explanation: Since $v_{rms} = \sqrt{\frac{3RT}{M}}$, when $T = 0\text{ K}$, $v_{rms}$ becomes zero.
5. The average kinetic energy of $2\text{ moles}$ of an ideal gas at $27^{\circ}\text{C}$ is: (Given $R = 8.314\text{ J mol}^{-1}\text{ K}^{-1}$)
Explanation: Temperature $T = 27 + 273 = 300\text{ K}$. Total kinetic energy for $n$ moles is $E = \frac{3}{2}nRT = \frac{3}{2} \times 2 \times 8.314 \times 300 = 3 \times 8.314 \times 300 = 7482\text{ J}$? Wait, $3 \times 8.314 = 24.942$, $24.942 \times 300 = 7482\text{ J}$. Let's check index 1 for $7482\text{ J}$.
6. A cylinder contains $5\text{ L}$ of ideal gas at $2\text{ atm}$ pressure. If the volume is compressed to $2\text{ L}$ at constant temperature, the new pressure is: