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EAPCET Physics · Intermediate first year

Kinetic Theory

Kinetic Theory explains gas behaviour from molecular motion: pressure, molecular speeds, the law of equipartition and specific heats.

Subject
Physics
Taught in
Intermediate first year
Physics in the paper
40 of 160 questions

Key formulas and results

  • PV=nRTPV = nRT
  • P=13ρvrms2P = \tfrac{1}{3}\rho v_{\text{rms}}^{2}
  • vrms=3RT/Mv_{\text{rms}} = \sqrt{3RT/M}, vavg=8RT/πMv_{\text{avg}} = \sqrt{8RT/\pi M}, vmp=2RT/Mv_{\text{mp}} = \sqrt{2RT/M}
  • Average kinetic energy per molecule =32kT= \tfrac{3}{2}kT
  • Each degree of freedom carries 12kT\tfrac{1}{2}kT
  • γ=5/3\gamma = 5/3 for monatomic and 7/57/5 for diatomic gases
  • Mean free path: λ=12 πd2n=kT2 πd2P\lambda = \frac{1}{\sqrt{2}\,\pi d^{2}n} = \frac{kT}{\sqrt{2}\,\pi d^{2}P}, where dd is the molecular diameter and nn the number density
  • Pressure in terms of total mass: PV=13Mvrms2PV = \tfrac{1}{3}Mv_{\text{rms}}^{2}
  • Internal energy of nn moles: U=f2nRTU = \frac{f}{2}nRT
  • Degrees of freedom: monatomic 3, diatomic 5, non-linear polyatomic 6 (Cv=3RC_{v} = 3R, Cp=4RC_{p} = 4R, γ=4/3\gamma = 4/3)
  • Graham’s law of effusion: rate ∝1ρ∝1M\propto \frac{1}{\sqrt{\rho}} \propto \frac{1}{\sqrt{M}}
  • Boltzmann constant and gas constant: kNA=RkN_{A} = R

What this chapter covers

  • Ideal gas equation
  • Pressure of an ideal gas
  • Molecular speeds
  • Law of equipartition of energy
  • Specific heats
  • Mean free path

How it is usually asked

  • Compare molecular speeds
  • Degrees of freedom and γ
  • γ for a gas mixture
  • Mean free path

Understanding Kinetic Theory

Kinetic theory pictures a gas as a huge number of tiny molecules moving randomly and colliding elastically with each other and the walls. Pressure is the result of momentum transferred to the walls in these collisions, giving P = ⅓ρv²ᵣₘₛ. Comparing this with PV = nRT shows that temperature is a measure of the average translational kinetic energy of the molecules, (3/2)kT per molecule.

The rms, average and most probable speeds all scale as √(T/M), with v_rms > v_avg > v_mp. The law of equipartition gives each degree of freedom an average energy of ½kT. A monatomic gas has 3 degrees of freedom and a rigid diatomic gas 5 at ordinary temperatures, which fixes Cᵥ, Cₚ and γ.

EAPCET usually asks for speed ratios between gases or temperatures, γ from degrees of freedom, and kinetic energy per molecule or per mole. Remember that at the same temperature all ideal gases have the same average kinetic energy per molecule, whatever their mass; lighter molecules simply move faster. That is why hydrogen molecules have four times the rms speed of oxygen molecules at the same temperature.

Worked example

EAPCET-style question

The rms speed of oxygen molecules at 27 °C is vv. At what temperature will their rms speed be 2v2v?

  1. (1)927 °C
  2. (2)108 °C
  3. (3)1200 °C
  4. (4)54 °C

Solution

  1. vrms=3RT/Mv_{\text{rms}} = \sqrt{3RT/M}, so vrms∝Tv_{\text{rms}} \propto \sqrt{T} for the same gas.
  2. Doubling the speed needs four times the absolute temperature.
  3. T1=27+273=300 KT_{1} = 27 + 273 = 300\ \text{K}, so T2=4×300=1200 KT_{2} = 4 \times 300 = 1200\ \text{K}.
  4. T2=1200−273=927 ∘CT_{2} = 1200 - 273 = 927\ ^\circ\text{C}.

Answer: 927 °C

Shortcuts and results to remember

  • vmp:vavg:vrms=2:8/π:3≈1.41:1.60:1.73v_{\text{mp}} : v_{\text{avg}} : v_{\text{rms}} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3} \approx 1.41 : 1.60 : 1.73, so vmp<vavg<vrmsv_{\text{mp}} < v_{\text{avg}} < v_{\text{rms}}.
  • Internal energy of an ideal gas depends only on temperature: if TT is constant, ΔU=0\Delta U = 0.
  • For P–V isotherms, the curve farther from the origin is at the higher temperature.
  • vrms∝Tv_{\text{rms}} \propto \sqrt{T}, so doubling vrmsv_{\text{rms}} needs four times the absolute temperature.

More practice questions

Practice question 1

The rms speed of a gas at 27 °C is doubled when its temperature becomes

  1. (1)54 °C
  2. (2)327 °C
  3. (3)927 °C
  4. (4)1200 °C

Solution

  1. vrms∝Tv_{\text{rms}} \propto \sqrt{T}, so doubling vrmsv_{\text{rms}} needs T2=4T1T_{2} = 4T_{1}
  2. T1=27+273=300 KT_{1} = 27 + 273 = 300\ \text{K}
  3. T2=1200 KT_{2} = 1200\ \text{K}
  4. In Celsius: 1200−273=927 ∘C1200 - 273 = 927\ ^\circ\text{C}

Answer: 927 °C

Practice question 2

One mole of a monatomic gas is mixed with one mole of a diatomic gas. The value of γ\gamma for the mixture is

  1. (1)1.40
  2. (2)1.50
  3. (3)1.53
  4. (4)1.67

Solution

  1. Cv,mix=1×3R/2+1×5R/22=2RC_{v,\text{mix}} = \frac{1 \times 3R/2 + 1 \times 5R/2}{2} = 2R
  2. Cp,mix=Cv+R=3RC_{p,\text{mix}} = C_{v} + R = 3R
  3. γ=CpCv=3R2R=1.5\gamma = \frac{C_{p}}{C_{v}} = \frac{3R}{2R} = 1.5

Answer: 1.50

Common mistakes to avoid

  • Scaling Celsius temperatures in speed or energy ratios — convert to kelvin first, then back if the options are in °C.
  • Using molar mass in g/mol in v_rms = √(3RT/M) — M must be in kg/mol (0.032 kg/mol for O₂).
  • Taking 6 degrees of freedom for a diatomic gas at room temperature — it is 5 (3 translational + 2 rotational), giving γ = 7/5.
  • Confusing kinetic energy per molecule, (3/2)kT, with kinetic energy per mole, (3/2)RT.

How to prepare it

Molecular speeds all scale as √(T/M) — ratio questions need only that.

Once the formulas feel familiar, test them the way the exam does: practise Kinetic Theory questions chapter-wise, then attempt a full previous year paper to see whether it holds up under a 3-hour timer.

Related Physics chapters

See every chapter in the EAPCET Physics syllabus.

Practise Kinetic Theory chapter-wise

Chapter-wise quizzes show exactly which topics in this chapter you still get wrong — before a full paper does.

Frequently asked questions

Is Kinetic Theory part of the EAPCET Physics syllabus?
Yes. Kinetic Theory is taught in Intermediate first year Physics, and the EAPCET Engineering syllabus is based on the Intermediate first- and second-year syllabus. Check the official notification each year for any changes.
How should I prepare Kinetic Theory for EAPCET?
Molecular speeds all scale as √(T/M) — ratio questions need only that.
How many Physics questions are there in EAPCET?
40 of the 160 questions in the Engineering paper are Physics, spread across 30 chapters.
What mistakes do students make in Kinetic Theory?
Scaling Celsius temperatures in speed or energy ratios — convert to kelvin first, then back if the options are in °C. Using molar mass in g/mol in v_rms = √(3RT/M) — M must be in kg/mol (0.032 kg/mol for O₂). Taking 6 degrees of freedom for a diatomic gas at room temperature — it is 5 (3 translational + 2 rotational), giving γ = 7/5. Confusing kinetic energy per molecule, (3/2)kT, with kinetic energy per mole, (3/2)RT.