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.
If the molecular masses of gases are M1 and M2 respectively, then the mean square velocity of the gases are proportional to:
📅BP 2011
Q2.
Average translational kinetic energy of a molecule is given by:
📅BP 2011
Q3.
If all CO2 is removed, Earth's temperature will:
📅BP 2010
Q4.
Absolute zero is accurately:
📅BP 2010
Q5.
For material with P = aT2/V, work done when T changes from T0 to 2T0 at constant P is:
📅BP 2009
Q6.
For ideal gas with cp=525 J/kg°C and cv=315 J/kg°C, density at NTP is:
📅BP 2009
Q7.
Relation between average KE (E) per unit volume and pressure (P):
📅BP 2012
Q8.
Critical temperature for Van der Waals equation is:
📅BP 2013
Q9.
For gas with γ=1.4, which is true?
📅BP 2013
Q10.
At same T, P, and V for two gases, which quantity is constant?
📅IOM 2013
Q11.
150cc ideal gas at 27°C and 650mm pressure. Volume at 0°C (constant P)?
📅IOM 2014
Q12.
When absolute T increases 3×, RMS velocity becomes:
📅IOM 2011
Q13.
Single molecule in closed vessel. When T increases:
📅
Q14.
Average speed of gas molecules ∝
📅MOE/KU
Q15.
Mean square speed ∝
📅
Q16.
RMS velocity of gas molecules ∝
📅
Q17.
Which gas has maximum RMS speed at given T?
📅
Q18.
Which gas has maximum KE at given T?
📅
Q19.
Two gases A and B at same P,V,T are mixed. Final pressure?
📅
Q20.
At 0K, which property of gas becomes zero?
📅
Q21.
Two vessels: H2 at 1atm and He at 2atm (same V,T). Mean velocity ratio?
📅
Q22.
Heated monoatomic gas (300K→600K). Average KE change?
📅
Q23.
If gas molecule masses halved and speeds doubled, new pressure?
📅
Q24.
For gas mixture (N molecules mass m + 2N molecules mass 2m), vB/vA = ?
📅
Q25.
If P increases 0.4% when heated 1°C, initial T is:
📅
Q26.
Ratio of molecules in jar A (P,V,T) to jar B (2P,V/4,2T):
📅
Q27.
H2:O2 = 1:5 ratio. KE ratio of molecules?
📅
Q28.
6g O2 at 400K leaks until P/2 at 300K. Mass leaked?
📅
Q29.
Molar specific heat (Cv) of 1 mole mono + 1 mole diatomic gas mix:
📅
Q30.
If intermolecular forces vanish, volume of 4.5kg water at NTP:
📅
Q31.
Mean KE per mole per degree of freedom:
📅
Q32.
If vrms=200m/s at 27°C, new vrms at 127°C and 0.5×105 N/m²?
📅
Q33.
22g CO2 at 27°C + 16g O2 at 37°C. Mixture temp?
📅
Q34.
When P increases from 1 to 4 atm, vrms:
📅
Q35.
If vrms=v at NTP, when P becomes 4× at constant V:
📅
Q36.
Open vessel at 60°C heated until 1/4 air escapes. Final T?
📅
Q37.
Mean KE at 0°C is E. At 273°C, KE becomes:
📅
Q38.
If n → 2n molecules in box, pressure becomes:
📅
Q39.
vrms(H2)=1930m/s at 300K. vrms(O2) at 900K?
📅
Q40.
Pressure increase needed for 10% volume decrease at constant T:
📅
Q41.
Gas with 6 degrees of freedom at 300K has vrms=c. Speed of sound?
📅
Q42.
For diatomic gas with average KE/molecule=0.10eV, total KE/molecule?
📅
Q43.
Vessel with 1 mole O2 at T has pressure P. Identical vessel with 1 mole He at 2T has pressure?
📅KU 2015
Q44.
When diatomic gas is heated at constant P, fraction of heat increasing internal energy?
📅KU 2015