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Kinetic Theory of Gases formulas

7 formulas and 12 common traps for MHT-CET Physics Kinetic Theory of Gases, grouped by subtopic.

Full notes with worked examples

The Gas Laws and the Ideal Gas Equation

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One Law at a Time

Combined gas law

P1V1T1=P2V2T2\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}

PV = nRT and Counting Molecules

Ideal gas equation

PV=nRT=NkT,ρ=PMRTPV = nRT = NkT, \qquad \rho = \frac{PM}{RT}

Common traps

Equal percentages in Boyle's law

PV is constant, not P + V. Reducing V by 5% multiplies P by 1/0.95 — an increase of 5.26%. The options include the plain 5%.

Using degrees Celsius in P ∝ T

27 °C to 37 °C is 300 K to 310 K, a 3.3% rise — not 37/27. Convert before taking the ratio.

Leaving T out of a molecule count

The number of molecules is PV/kT. Two jars at different temperatures cannot be compared by PV alone.

Pressure and R.M.S. Speed From Molecular Motion

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Pressure as Two Thirds of the Kinetic Energy Density

Kinetic pressure

P=13ρ⟨v2⟩=23EVP = \frac{1}{3}\rho\langle v^2\rangle = \frac{2}{3}\frac{E}{V}

R.M.S. Speed, Temperature and Molar Mass

R.M.S. speed

vrms=3RTM=3kTmv_{\text{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}

R.M.S. Speed Through an Adiabatic Expansion

Adiabatic cooling

V2V1=(T1T2)1/(γ−1),T1T2=(v1v2)2\frac{V_2}{V_1} = \left(\frac{T_1}{T_2}\right)^{1/(\gamma - 1)}, \qquad \frac{T_1}{T_2} = \left(\frac{v_1}{v_2}\right)^2

Common traps

Writing P = ⅓ of the energy density

P = ⅓ρv², but the kinetic energy density is ½ρv². So P is TWO thirds of the energy per unit volume.

Heating a gas at constant volume changes its mean free path

The mean free path depends on molecules per unit volume. In a rigid vessel that number does not change, so neither does the mean free path.

Scaling speed with temperature in °C

−68 °C is 205 K; doubling the speed needs 820 K = 547 °C. Multiplying −68 by 4 gives nonsense.

Scaling speed with T instead of √T

v_rms ∝ √T. Four times the temperature doubles the speed; the options offer 4x as well.

Using the speed ratio as the temperature ratio

Speed goes as √T, so halving the speed quarters the temperature. Feeding 2 instead of 4 into the adiabatic relation gives 4 instead of 16.

Average Kinetic Energy, Equipartition and Specific Heats

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Average Kinetic Energy Depends Only on Temperature

Average kinetic energy

Eˉ=32kT,each degree of freedom 12kT\bar{E} = \frac{3}{2}kT, \qquad \text{each degree of freedom } \tfrac{1}{2}kT

Degrees of Freedom, C_v, C_p and γ

Equipartition

Cv=f2R,Cp=Cv+R,γ=1+2fC_v = \frac{f}{2}R, \qquad C_p = C_v + R, \qquad \gamma = 1 + \frac{2}{f}

Common traps

Letting the heavier gas have more kinetic energy

At one temperature all gases have the same average kinetic energy. Molar mass changes the SPEED, not the energy.

Halving the Celsius temperature

E ∝ T in kelvin. E/2 at 399 °C (672 K) means 336 K, which is 63 °C — not 199.5 °C.

Averaging the specific heats of a mixture by gas, not by mole

Weight each C_v by its number of moles. Four moles of hydrogen count four times as much as one mole of water vapour.

Counting one vibrational degree of freedom

A vibrational mode stores both kinetic and potential energy, so it adds 2 to f. A vibrating diatomic has f = 7 and C_v = 7R/2.

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