Kinetic Theory: All Important Formulas Sheet

Lock these groups. GAS LAW: PV = nRT = N kB T, where n = number of moles, N = number of molecules, R = 8.314 J/mol/K, kB = 1.38e-23 J/K. PRESSURE: P = (1/3) rho v_rms^2 = (1/3)(N/V) m v_rms^2, where rho is gas density and m is mass of one molecule. SPEEDS: v_rms = sqrt(3RT/M) = sqrt(3 kB T/m); average speed v_avg = sqrt(8RT/(pi M)); most probable speed v_mp = sqrt(2RT/M); ratio v_mp : v_avg : v_rms = 1 : 1.13 : 1.22. ENERGY: average KE per molecule = (3/2) kB T; KE per mole = (3/2) RT; total internal energy of n moles of an ideal gas = (f/2) nRT where f is degrees of freedom. SPECIFIC HEATS: Cv = (f/2)R, Cp = (f/2 + 1)R, Cp - Cv = R (Mayer), gamma = Cp/Cv = 1 + 2/f. MEAN FREE PATH: lambda = 1/(sqrt(2) n pi d^2), where n is number density and d is molecular diameter.

They give the SAME answer, just different unit systems. Use v_rms = sqrt(3RT/M) when M is the molar mass in kg per mole and R = 8.314. Use v_rms = sqrt(3 kB T/m) when m is the mass of ONE molecule in kg and kB = 1.38e-23. The link between them is M = m x NA (Avogadro number) and R = kB x NA, so the two forms are identical. Trap: NEET often gives molar mass in g/mol - convert to kg/mol (divide by 1000) before using R = 8.314.

R is the universal gas constant, used PER MOLE (works with n and M). kB is the Boltzmann constant, used PER MOLECULE (works with N and m). They are connected by R = kB x NA, so R = 8.314 J/mol/K and kB = R/NA = 1.38e-23 J/K. Rule of thumb: if the formula counts moles use R; if it counts individual molecules use kB. That is why average KE per molecule is (3/2)kB T but per mole it is (3/2)RT.

Only find f, the degrees of freedom, then everything follows. Monatomic (He, Ar): f = 3, so Cv = (3/2)R, Cp = (5/2)R, gamma = 5/3 = 1.67. Diatomic (O2, N2, H2 at room temp): f = 5, so Cv = (5/2)R, Cp = (7/2)R, gamma = 7/5 = 1.4. Triatomic/polyatomic (linear f = 7, non-linear f = 6): gamma ranges about 1.28 to 1.33. Master formula: gamma = 1 + 2/f. So gamma tells you f: f = 2/(gamma - 1).

Step 1: molar mass of O2 = 32 g/mol = 0.032 kg/mol. Step 2: use v_rms = sqrt(3RT/M) = sqrt(3 x 8.314 x 300 / 0.032). Step 3: numerator = 3 x 8.314 x 300 = 7482.6. Step 4: divide by 0.032 = 233,831. Step 5: square root = about 483 m/s. So v_rms of oxygen at 300 K is roughly 483 m/s. Check: lighter gases move faster - hydrogen (M = 0.002) at the same T gives about 1934 m/s, exactly 4 times faster because v_rms scales as 1/sqrt(M).

Step 1: internal energy U = (f/2) nRT. Step 2: diatomic means f = 5. Step 3: substitute U = (5/2) x 2 x 8.314 x 400. Step 4: (5/2) x 2 = 5. Step 5: U = 5 x 8.314 x 400 = 16,628 J, about 16.6 kJ. Note: for a monatomic gas you would use f = 3 and get U = 3 x 8.314 x 400 = 9,977 J. The only thing that changes between gas types on this sheet is f.

Yes. Start from P = (1/3) rho v_rms^2. Since PV = N kB T, you can show (1/2) m v_rms^2 = (3/2) kB T. This single link joins pressure, RMS speed, molecular mass and temperature - it is the heart of Kinetic Theory.

Use R = 8.314 J/mol/K when working in SI units (pascals, cubic metres, kelvin) - this is the value for all speed, energy and specific-heat formulas. R = 0.0821 L atm/mol/K is only for gas-law volume problems in litres and atmospheres. For NEET Kinetic Theory numericals, 8.314 is almost always the right one.

No. v_rms = sqrt(3RT/M) depends only on temperature T and molar mass M. At the same temperature, changing pressure or volume does not change RMS speed. This is a very common NEET tricky point - RMS speed is set by T alone.

Remember only f (degrees of freedom): 3 for monatomic, 5 for diatomic. Then Cv = (f/2)R, Cp = Cv + R, and gamma = 1 + 2/f. That gives gamma = 1.67 monatomic and 1.4 diatomic without memorising separate numbers.

lambda = 1/(sqrt(2) n pi d^2) depends inversely on number density n and on the square of molecular diameter d. Higher pressure means more molecules per volume (larger n), so mean free path shrinks. At fixed T, lambda is inversely proportional to pressure.