19Gas Laws and Kinetic theory of Gases

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GAS LAWS AND KINETIC THEORY OF GASES
▢ Gas Laws:

Table 1: Boyle, Gay-Lussac and Charles Law

Law
Constant
Statement
Formula
Boyle's law
Temperature \(T\)
For given mass of gas at constant temperature, volume is inversely proportional to pressure
\(V\propto\frac{1}{P}\), \(PV=constant\), \(P_1V_1=P_2V_2\)
Gay-Lussac's law / Pressure law
Volume \(V\)
For given mass of ideal gas at constant volume, pressure is directly proportional to absolute temperature
\(P\propto T\), \(\frac{P}{T}=constant\), \(\frac{P_1}{T_1}=\frac{P_2}{T_2}\)
Charles law
Pressure \(P\)
For given mass of gas at constant pressure, volume is directly proportional to absolute temperature
\(V\propto T\), \(\frac{V}{T}=constant\), \(\frac{V_1}{T_1}=\frac{V_2}{T_2}\)
❖ Important Points:
  • •
    Real gases obey Boyle's law at low pressure and high temperature
  • •
    Unsaturated vapour obeys Boyle's law
  • •
    Saturated vapour does not obey Boyle's law
  • •
    At constant volume: \(P_t=P_0\left(1\pm\frac{t}{273}\right)\)
  • •
    At constant pressure: \(V=V_0(1+\alpha t)\), where \(\alpha=\frac{1}{273}\)
▢ Combined Gas Equation:
❖ Formula: \(\frac{PV}{T}=constant\)
❖ Two-State Form: \(\frac{P_1V_1}{T_1}=\frac{P_2V_2}{T_2}\)
▢ Avogadro's Law:
❖ Statement: At same temperature and pressure, equal volumes of all gases contain equal number of molecules
❖ Formula: \(N_1=N_2\) if \(P,V,T\) are same
▢ Ideal Gas Equation:
❖ Definition: Relation between \(P,V,T\) describing state of ideal gas
❖ Based On:
  • •
    Boyle's law
  • •
    Charles law
  • •
    Avogadro's law

Table 1: Forms of Ideal Gas Equation

Condition
Equation
Point
For 1 mole
\(PV=RT\)
\(R\) = universal gas constant
For \(n\) moles
\(PV=nRT\)
Standard equation
For 1 kg gas
\(PV=rT\)
\(r=\frac{R}{M}\) = specific gas constant
For \(m\) kg gas
\(PV=mrT\)
Mass form
For 1 molecule
\(PV=kT\)
\(k=\frac{R}{N_A}\) = Boltzmann constant
For \(N\) molecules
\(PV=NkT\)
Molecular form
For number density \(n\)
\(P=nkT\)
\(n\) = molecules per unit volume
❖ Specific Gas Constant:
  • •
    Universal gas constant \(R\) is same for all gases
  • •
    Specific gas constant \(r=R/M\) is different for different gases
  • •
    \(r\) is maximum for hydrogen gas
▢ Van der Waals Equation:
❖ Need: Real gases deviate from ideal behaviour at low temperature and high pressure due to molecular volume and intermolecular attraction
❖ Ideal Behaviour: Real gases behave nearly ideal at high temperature and low pressure

Table 1: Van der Waals Equation

Condition
Equation
For 1 mole
\(\left(P+\frac{a}{V^2}\right)(V-b)=RT\)
For \(n\) moles
\(\left(P+\frac{n^2a}{V^2}\right)(V-nb)=nRT\)
\(a\)
Measure of attractive force between gas molecules
\(b\)
Measure of size / volume of molecules
▢ Critical Constants:

Table 1: Critical Constants of Real Gas

Quantity
Meaning
Formula
Critical temperature \(T_c\)
Temperature below which gas can be liquefied by pressure alone
\(T_c=\frac{8a}{27Rb}\)
Critical pressure \(P_c\)
Minimum pressure required to liquefy gas at critical temperature
\(P_c=\frac{a}{27b^2}\)
Critical volume \(V_c\)
Volume occupied by gas at critical temperature and pressure
\(V_c=3b\)
Boyle's temperature \(T_B\)
Temperature at which real gas behaves like ideal gas over appreciable pressure range
\(T_B=\frac{a}{Rb}=\frac{27}{8}T_c\)
❖ Important Points:
  • •
    Below critical temperature, gaseous state is called vapour
  • •
    Gas with high critical temperature can be liquefied easily
▢ Graham's Law of Diffusion:
❖ Statement: At constant temperature and pressure, rate of diffusion of a gas is inversely proportional to square root of its density
❖ Formula: \(r\propto\frac{1}{\sqrt\rho}\)

Table 1: Rate of Diffusion

Relation
Formula
Two gases
\(\frac{r_1}{r_2}=\sqrt{\frac{\rho_2}{\rho_1}}=\sqrt{\frac{M_2}{M_1}}\)
Using volume and time
\(\frac{r_1}{r_2}=\frac{V_1/t_1}{V_2/t_2}\)
Using pressure
\(\frac{r_1}{r_2}=\frac{P_1}{P_2}\)
❖ Symbols:
  • •
    \(M\) = molecular mass
  • •
    \(V\) = volume diffused
  • •
    \(t\) = time
  • •
    \(\rho\) = density
▢ Dalton's Law of Partial Pressure:
❖ Statement: Pressure exerted by a mixture of non-reacting gases at constant temperature equals sum of partial pressures of components
❖ Formula: \(P=P_1+P_2+...+P_n\)
❖ Partial Pressure: \(Partial\ pressure=Mole\ fraction\times Total\ pressure\)
▢ Mean Free Path:
❖ Definition: Average distance travelled by a gas molecule between two successive collisions

Table 1: Mean Free Path

Quantity
Formula / Value
Mean free path
\(\lambda=\frac{1}{\sqrt2\pi d^2n}\)
\(d\)
Diameter of molecule
\(n\)
Number of molecules per unit volume
At NTP for air molecules
\(\lambda\approx10\ nm\)
▢ Assumptions of Kinetic Theory of Gases:
  • •
    Gas consists of large number of identical, tiny, spherical, neutral and elastic molecules
  • •
    No intermolecular attraction between molecules
  • •
    Volume occupied by gas molecules is negligible compared to container volume
  • •
    Molecules move randomly in all directions with all possible speeds
  • •
    Molecular speeds follow Maxwell's distribution law
  • •
    Time of contact during collision is negligible compared to time between collisions
  • •
    Collisions between molecules and with container walls are perfectly elastic
  • •
    Pressure is due to elastic collision of molecules with container wall
  • •
    Average velocity and average momentum are zero at equilibrium
  • •
    Average speed and average kinetic energy depend on temperature and molecular mass
  • •
    Average kinetic energy is proportional to absolute temperature
▢ Pressure Exerted by Ideal Gas:

Table 1: Pressure Formulae

Form
Formula
Basic kinetic theory form
\(P=\frac{1}{3}\frac{mv*{rms}^2}{V}\)
Using density
\(P=\frac{1}{3}\rho v*{rms}^2\)
Using total kinetic energy
\(P=\frac{2}{3}\frac{E}{V}\)
Total kinetic energy
\(E=\frac{3}{2}PV\)
❖ Symbols:
  • •
    \(m\) = total mass of gas
  • •
    \(V\) = volume of container
  • •
    \(\rho\) = density of gas
  • •
    \(v*{rms}\) = root mean square speed
  • •
    \(E\) = total translational kinetic energy
▢ Speeds of Gas Molecules:

Table 1: Different Speeds of Gas Molecules

Speed
Definition
Formula
Root mean square speed
Square root of mean of squares of molecular speeds
\(v*{rms}=\sqrt{\frac{3RT}{M}}=\sqrt{\frac{3P}{\rho}}\)
Average speed
Mean speed of gas molecules
\(v*{av}=\sqrt{\frac{8RT}{\pi M}}=\sqrt{\frac{8P}{\pi\rho}}\)
Most probable speed
Speed possessed by maximum number of molecules
\(v*{mp}=\sqrt{\frac{2RT}{M}}=\sqrt{\frac{2P}{\rho}}\)
Speed of sound in gas
Propagation speed of sound in gas
\(v_s=\sqrt{\frac{\gamma RT}{M}}=\sqrt{\frac{\gamma P}{\rho}}\)
❖ Order: \(v*{rms}>v*{av}>v*{mp}\)
❖ Ratio: \(v*{mp}:v*{av}:v*{rms}=\sqrt2:\sqrt{\frac{8}{\pi}}:\sqrt3\)
❖ Symbols:
  • •
    \(M\) = molecular mass
  • •
    \(R\) = universal gas constant
  • •
    \(T\) = absolute temperature
  • •
    \(P\) = pressure
  • •
    \(\rho\) = density
  • •
    \(\gamma=\frac{C_p}{C_v}\)
▢ Kinetic Interpretation of Temperature:

Table 1: Temperature and Kinetic Energy

Quantity
Formula / Point
Total translational K.E.
\(E=\frac{3}{2}PV=\frac{3}{2}nRT=\frac{3}{2}NkT\)
Average K.E. per molecule
\(\bar E=\frac{3}{2}kT\)
Temperature
Measure of average translational K.E. of molecules
Depends on
Absolute temperature
Independent of
Nature of gas
At absolute zero
Translational K.E. becomes zero if gas remains gaseous
❖ Important Points:
  • •
    T.K.E. of gas molecules is independent of pressure for given temperature
  • •
    In closed vessel, T.K.E. is directly proportional to pressure because volume is constant
  • •
    Translatory motion of gas molecules determines temperature
▢ Degree of Freedom:
❖ Definition: Number of independent ways in which a molecule can possess energy
❖ Energy Types:

Table 1: Molecular Energy Types

Energy
Temperature condition
Translational K.E.
All temperatures except absolute zero
Rotational K.E.
Significant above \(70K\)
Vibrational K.E.
Significant at very high temperature \(\geq1000K\)
❖ Formulae:
  • •
    In Chemistry: \(f*{av}=f_T+f_R+f_V\)
  • •
    In Physics: \(f*{av}=f_T+f_R\)

Table 1: Degree of Freedom of Gases

Gas
Degree of freedom \(f\)
Monoatomic gas
3
Diatomic gas
5
Triatomic / polyatomic gas
6
▢ Relation Between Degree of Freedom, Gamma, Cp and Cv:

Table 1: Degree of Freedom Relations

Quantity
Formula
Ratio of specific heats
\(\gamma=1+\frac{2}{f}\)
Molar heat at constant volume
\(C_V=\frac{f}{2}R\)
Molar heat at constant pressure
\(C_P=\left(1+\frac{f}{2}\right)R\)
▢ Mixing of Gases:
❖ Based On: Conservation of energy
❖ Key Point: For ideal gases, potential energy is zero; total energy is kinetic

Table 1: Mixture of Gases

Quantity
Formula
Gamma of mixture
\(\frac{n_1+n_2}{\gamma*{mix}-1}=\frac{n_1}{\gamma_1-1}+\frac{n_2}{\gamma_2-1}\)
Molar heat at constant pressure
\((C_P)*{mix}=\frac{n_1C*{P1}+n_2C*{P2}}{n_1+n_2}\)
Molar heat at constant volume
\((C_V)*{mix}=\frac{n_1C*{V1}+n_2C*{V2}}{n_1+n_2}\)
Resulting temperature
\(T*{mix}=\frac{\mu_1T_1+\mu_2T_2}{\mu_1+\mu_2}\)
❖ Symbols:
  • •
    \(T_1,T_2\) = absolute temperatures
  • •
    \(\mu_1,\mu_2\) = number of moles
▢ Constants:

Table 1: Important Constants

Constant
Formula / Value
Boltzmann constant
\(k=\frac{R}{N_A}\)
Universal gas constant
\(R=8.314\ J\ mol^{-1}K^{-1}\)
Universal gas constant
\(R\approx0.0821\ L\ atm\ mol^{-1}K^{-1}\)
Universal gas constant
\(R\approx2\ cal\ mol^{-1}K^{-1}\)
Avogadro number
\(N_A\)
❖ Point: \(R\) is universal and independent of temperature, pressure and volume but its numerical value depends on units
▢ Read and Digest:

Table 1: Important Gas Theory Points

Fact
Point
Average velocity of gas molecules
Zero at equilibrium
Average momentum of gas molecules
Zero at equilibrium
Average speed
\(\propto\sqrt T\)
Temperature of gas
Measure of average K.E. of molecules
Gas in thermal equilibrium
Molecules have different energies but average remains constant
Internal energy of ideal gas
Depends only on temperature
Internal energy of real gas
Depends on temperature and volume
Gas state
Greatest potential energy
Pressure
\(P\propto N\)
Isothermal expansion
Pressure falls due to decreased collision frequency
Adiabatic expansion
Pressure falls due to decreased momentum per collision and decreased collision frequency
Absolute zero
Molecular motion of gases ceases
Evaporation
Temperature decreases
Ideal gas
Possesses only K.E., no P.E.
Ideal gas molecules
No size and no intermolecular force
Gas molecules
Behave as elastic rigid spheres
Order of K.E.
Gas > Liquid > Solid
Energy of ideal gas
State function; independent of path
Electric fan in closed room
Air becomes slightly heated
Real gases ideal behaviour
High temperature and low pressure
▢ High-Yield Recall:

Table 1: Gas Laws and Kinetic Theory One-Liners

Fact
Answer
Boyle's law
\(PV=constant\)
Gay-Lussac's law
\(\frac{P}{T}=constant\)
Charles law
\(\frac{V}{T}=constant\)
Combined gas equation
\(\frac{PV}{T}=constant\)
Avogadro's law
Equal volume at same \(P,T\) contains equal molecules
Ideal gas equation
\(PV=nRT\)
Molecular ideal gas equation
\(PV=NkT\)
Boltzmann constant
\(k=\frac{R}{N_A}\)
Specific gas constant
\(r=\frac{R}{M}\)
Van der Waals equation
\(\left(P+\frac{a}{V^2}\right)(V-b)=RT\)
Critical temperature
\(T_c=\frac{8a}{27Rb}\)
Critical pressure
\(P_c=\frac{a}{27b^2}\)
Critical volume
\(V_c=3b\)
Boyle's temperature
\(T_B=\frac{a}{Rb}=\frac{27}{8}T_c\)
Graham's law
\(r\propto\frac{1}{\sqrt\rho}\)
Dalton's law
\(P=P_1+P_2+...+P_n\)
Mean free path
\(\lambda=\frac{1}{\sqrt2\pi d^2n}\)
Pressure of ideal gas
\(P=\frac{1}{3}\rho v*{rms}^2\)
K.E.-pressure relation
\(P=\frac{2E}{3V}\)
RMS speed
\(v*{rms}=\sqrt{\frac{3RT}{M}}\)
Average speed
\(v*{av}=\sqrt{\frac{8RT}{\pi M}}\)
Most probable speed
\(v*{mp}=\sqrt{\frac{2RT}{M}}\)
Speed of sound
\(v_s=\sqrt{\frac{\gamma RT}{M}}\)
Speed order
\(v*{rms}>v*{av}>v*{mp}\)
Total T.K.E.
\(E=\frac{3}{2}nRT=\frac{3}{2}NkT\)
Average K.E. per molecule
\(\frac{3}{2}kT\)
Degree of freedom monoatomic
3
Degree of freedom diatomic
5
Degree of freedom triatomic
6
Gamma-degree relation
\(\gamma=1+\frac{2}{f}\)
\(C_V\)
\(\frac{f}{2}R\)
\(C_P\)
\(\left(1+\frac{f}{2}\right)R\)
Gas mixing temperature
\(T*{mix}=\frac{\mu_1T_1+\mu_2T_2}{\mu_1+\mu_2}\)
Ideal gas internal energy
Depends only on temperature
Real gas ideal condition
High temperature and low pressure
Q1.
For ideal gas, Joule Thomson effect is:
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Q2.
If volume of gas is suddenly reduced when temperature doubles, then ratio of final volume to initial volume will be:
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Q3.
At constant temperature, by how much should pressure be increased in order to decrease volume by 10%?
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Q4.
A vessel contains 6 g of oxygen at pressure \(P\) and temperature 400 K. A small hole is made so that oxygen leaks out. How much oxygen leaks out if final pressure is \(P/2\) and temperature is 300 K?
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Q5.
According to kinetic theory of gases, absolute zero temperature is the temperature at which:
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Q6.
Root mean square speed is:
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Q7.
If the pressure of a closed container is \(P\), the translational kinetic energy per unit volume (KE) is:
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Q8.
Velocity of sound is measured in hydrogen and oxygen gases at a given temperature. The ratio of two velocities will be \((v_H/v_O)\):
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Q9.
Two ideal gases of molecular masses \(M_1\) and \(M_2\) are at the same absolute temperature. What is the ratio of their mean-square speeds \(\langle v_1^2\rangle/\langle v_2^2\rangle\)?
📅BP 2011
Q10.
What is the mean translational kinetic energy per molecule of an ideal gas at absolute temperature \(T\)?
📅BP 2011
Q11.
If atmospheric \(\mathrm{CO_2}\) were removed while other conditions stayed approximately unchanged, Earth’s average surface temperature would:
📅BP 2010
Q12.
What is the exact value of absolute zero in the Celsius scale?
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Q13.
A system obeys \(PV=aT^2\), where \(a\) is constant. Find work done by the system during an isobaric expansion from \(T_0\) to \(2T_0\).
📅BP 2009
Q14.
An ideal gas has specific heat capacities \(c_p=525\,\mathrm{J/(kg\,K)}\) and \(c_v=315\,\mathrm{J/(kg\,K)}\). Find its approximate density at NTP (\(P=1.013\times10^5\,\mathrm{Pa}\), \(T=273\,\mathrm K\)).
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Q15.
If \(E\) is the mean translational kinetic energy per unit volume of an ideal gas, how is its pressure \(P\) related to \(E\)?
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Q16.
For one mole of a van der Waals gas, \((P+a/V_m^2)(V_m-b)=RT\). What is its critical temperature?
📅BP 2013
Q17.
For an ideal gas with \(\gamma=C_p/C_v=1.4\), which statement is correct? Assume no vibrational excitation.
📅BP 2013
Q18.
Two different ideal gases have the same absolute temperature. Which quantity is necessarily equal per molecule?
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Q19.
An ideal gas occupies \(150\,\mathrm{cm^3}\) at \(27^\circ\mathrm C\). Find its volume at \(0^\circ\mathrm C\) if pressure remains constant.
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Q20.
If the absolute temperature of an ideal gas triples, how does its RMS molecular speed change?
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Q21.
For a large sample of ideal gas molecules following a Maxwell–Boltzmann speed distribution, which ordering is correct?
Q22.
At fixed molecular mass, the mean speed of ideal-gas molecules is proportional to:
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Q23.
For a fixed gas, the mean-square molecular speed is proportional to:
Q24.
At fixed absolute temperature, RMS molecular speed is proportional to which power of molecular mass \(m\)?
Q25.
Which gas has the greatest RMS molecular speed at the same absolute temperature?
Q26.
At the same absolute temperature, which gas has the greatest mean translational kinetic energy per molecule?
Q27.
Two gases, each initially at pressure \(P\), volume \(V\) and temperature \(T\), are mixed in a single rigid vessel of volume \(V\) at the same temperature. What is the final total pressure?
Q28.
According to a classical ideal-gas extrapolation to \(T=0\,\mathrm K\), which molecular quantity tends to zero?
Q29.
In two vessels at equal temperature, \(\mathrm{H_2}\) is at \(1\,\mathrm{atm}\) and He at \(2\,\mathrm{atm}\). What is the ratio of their mean molecular speeds \(\bar v_{H_2}/\bar v_{He}\)?
Q30.
The absolute temperature of a monatomic ideal gas rises from \(300\,\mathrm K\) to \(600\,\mathrm K\). How does mean translational kinetic energy change?
Q31.
At fixed molecular number density, the mass of each gas molecule is halved and every molecular speed is doubled. How does pressure change?
Q32.
A gas mixture contains \(N\) molecules of species A (mass \(m\) each) and \(2N\) of species B (mass \(2m\) each), in thermal equilibrium. Find \(v_{\mathrm{rms},B}/v_{\mathrm{rms},A}\).
Q33.
For an ideal gas in a rigid sealed container, pressure increases by \(0.4\%\) for a \(1\,\mathrm K\) rise in temperature. Find the initial absolute temperature approximately.
Q34.
Jar A has ideal gas at \((P,V,T)\), and jar B at \((2P,V/4,2T)\). Find their molecule-number ratio \(N_A:N_B\).
Q35.
At the same absolute temperature, the numbers of \(\mathrm{H_2}\) and \(\mathrm{O_2}\) molecules are in the ratio \(1:5\). What is the ratio of their mean translational kinetic energies *per molecule*?
Q36.
A rigid vessel initially contains \(6\,\mathrm g\) of \(\mathrm{O_2}\) at \(400\,\mathrm K\). Some gas leaks out; its final pressure is half the initial pressure at \(300\,\mathrm K\). How much mass leaked?
Q37.
Equal amounts (one mole each) of a monatomic and a diatomic ideal gas are mixed. Find the molar heat capacity \(C_{V,\mathrm{mix}}\) of the mixture (vibrations neglected).
Q38.
If \(4.5\,\mathrm{kg}\) of water is completely vaporised and behaves as an ideal gas at STP (\(22.4\,\mathrm{L/mol}\)), what is its volume approximately?
Q39.
According to the equipartition theorem, what is the average energy per mole associated with one quadratic degree of freedom?
Q40.
For a fixed ideal gas, \(v_{\mathrm{rms}}=200\,\mathrm{m/s}\) at \(27^\circ\mathrm C\). Find its RMS speed at \(127^\circ\mathrm C\) (the given final pressure is irrelevant).
Q41.
A mixture is prepared from \(22\,\mathrm g\) of \(\mathrm{CO_2}\) at \(27^\circ\mathrm C\) and \(16\,\mathrm g\) of \(\mathrm{O_2}\) at \(37^\circ\mathrm C\). Assuming equal molar heat capacities and no heat loss, find the mixture temperature.
Q42.
For the same ideal gas, its pressure changes from 1 atm to 4 atm *at constant absolute temperature*. What happens to RMS molecular speed?
Q43.
An ideal gas initially has RMS speed \(v\). At constant volume and fixed gas amount, the pressure becomes four times its initial value. Find its new RMS speed.
Q44.
An open rigid vessel containing air at \(60^\circ\mathrm C\) is heated at constant atmospheric pressure until one-fourth of the air escapes. What is its final temperature?
Q45.
The mean translational kinetic energy of an ideal gas at \(0^\circ\mathrm C\) is \(E\). What is it approximately at \(273^\circ\mathrm C\)?
Q46.
In a rigid vessel at constant absolute temperature, the number of ideal-gas molecules doubles. How does pressure change?
Q47.
At \(300\,\mathrm K\), the RMS speed of \(\mathrm{H_2}\) is \(1930\,\mathrm{m/s}\). Find the RMS speed of \(\mathrm{O_2}\) at \(900\,\mathrm K\). Use molar masses \(2\) and \(32\,\mathrm{g/mol}\), respectively.
Q48.
What percentage pressure increase is required to decrease the volume of a fixed amount of ideal gas by 10% at constant temperature?
Q49.
An ideal gas has \(6\) active degrees of freedom, and its RMS molecular speed is \(c\). What is the speed of sound in that gas, neglecting vibrational effects beyond the stated degrees of freedom?
Q50.
For a diatomic ideal gas with mean *translational* kinetic energy \(0.10\,\mathrm{eV}\) per molecule, what is the total mean thermal energy per molecule if there are five active degrees of freedom?
Q51.
One mole of \(\mathrm{O_2}\) in a rigid vessel at temperature \(T\) produces pressure \(P\). In an identical vessel, one mole of He is at \(2T\). What is its pressure?
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Q52.
For a diatomic ideal gas without active vibrations, what fraction of the heat added at constant pressure increases its internal energy?
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