Formula sheet: Kinetic Theory

Kinetic Theory14 min readLesson 1 of 5

1.1 Pressure and temperature from molecular motion An ideal gas is a collection of point molecules in random motion, with no forces between collisions. The pressure on the wall comes from the momentum delivered by molecules hitting it, and the result links pressure, number density and the mean square speed. The kinetic energy of translation per molecule is proportional to the absolute temperature, which is what a thermometer really measures.

P = \frac{1}{3} \frac{m N}{V} v^{2}, P V = \frac{1}{3} m N v^{2}, \frac{1}{2} m v^{2} = \frac{3}{2} k_{B} T

Pressure of an ideal gas, the kinetic form of the gas equation and the mean translational kinetic energy per molecule

1.2 Molecular speeds and the gas laws The root mean square speed is the square root of the mean square speed, and it is the speed that enters the pressure formula. The most probable speed and the average speed differ from it by constant factors. From the kinetic form of the gas equation, the rms speed depends only on the temperature and the molecular mass, so lighter gases move faster at the same temperature. Boyle's law, Charles's law and Avogadro's law all follow from PV = Nk_BT.

v_{r m s} = \frac{3 k _{B} T}{m} = \frac{3 R T}{M}, v_{a v} = \frac{8 R T}{π M}, v_{m p} = \frac{2 R T}{M}

Root mean square, average and most probable speeds of gas molecules

1.3 Equipartition and specific heats Each independent way in which a molecule can store energy is a degree of freedom, and in thermal equilibrium each quadratic degree of freedom carries (1/2)k_BT of energy per molecule. A monatomic gas has three translational degrees of freedom, a diatomic gas adds two rotational ones at ordinary temperatures, and vibration appears only at high temperature. Counting degrees of freedom gives C_V and C_P and therefore gamma for each kind of gas.

U = \frac{f}{2} n R T, C_{V} = \frac{f}{2} R, C_{P} = C_{V} + R, γ = 1 + \frac{2}{f}

Internal energy from f degrees of freedom, the molar specific heats and gamma

1.4 Mean free path and real gases Molecules collide with each other, and the average distance travelled between two collisions is the mean free path. It falls as the number density rises and as the molecular diameter grows. Real gases follow the ideal equation only at low pressure and high temperature; at high pressure the molecules occupy a noticeable volume and attract one another, so the pressure and volume behave differently from the ideal prediction.

λ = \frac{1}{2 π d ^{2} n}, n = \frac{N}{V}

Mean free path in terms of molecular diameter and number density

Exam

Use the molar mass in kg mol^-1 when you use the form with R, and the molecular mass in kg when you use the form with k_B. Mixing the two units is the most common mistake in rms-speed questions.

in hinglish

Kinetic theory ka core hai molecules ki random motion. Deewar par pressure molecules ke collisions se banta hai, aur formula P = (1/3)(mN/V) v_rms^2 hai. Temperature ka molecular meaning yeh hai ki average translational KE per molecule (3/2) kT hoti hai. v_rms = sqrt(3RT/M) se speed nikalte hain, aur halki gas zyada tez chalti hai. Degrees of freedom se internal energy aur specific heats milte hain: monatomic ke liye f = 3, diatomic ke liye f = 5. Mean free path collisions ke beech ki average distance hai. Real gases ideal sirf low pressure aur high temperature par behave karti hain. Units me molar mass kg/mol me daalo. Iske saath hi pressure aur temperature ke relation ko graph par dekhna kabhi mat bhoolo. Formula sheet ko revise karte waqt ek baar har row ko apne shabdon me bol ke dekho, isse concept pakka hota hai.