Kinetic theory — pressure derivation, temperature as mean KE

1.7.9 · Physics › Thermodynamics

WHY kinetic theory ki zaroorat hai?

WHAT ye karta hai: ye connect karta hai microscopic world ko (molecules with mass $m$ aur speed $v$ ) macroscopic world se jo hum measure karte hain ( $P$ , $V$ , $T$ ).

WHY ye important hai: ideal gas law $P V = n R T$ experimentally discover ki gayi thi. Kinetic theory isse Newton's laws se derive karti hai — ye explain karti hai kyun ye law sach hai, sirf state karne ki bajaye.

Pressure SCRATCH SE derive karna

Socho ek cube hai jiske side $L$ hai, volume $V = L^{3}$ hai. EK molecule lo jiske velocity components $(v_{x}, v_{y}, v_{z})$ hain. Hum focus karte hain us wall par jo $x$ -axis ke perpendicular hai (right wall).

Step 1 — Ek bounce mein momentum change. Molecule right wall se takraata hai, elastically wapas bounce karta hai. Sirf $v_{x}$ sign palatta hai. $Δ p_{x} = m v_{x} - (- m v_{x}) = 2 m v_{x}$ Ye step kyun? Elastic collision ka matlab hai speed unchanged rehti hai lekin $x$ -direction reverse ho jaati hai, isliye momentum mein change incoming $x$ -momentum ka double hota hai.

Step 2 — Wo us wall se kitni baar takraata hai? Bounce karne ke baad, use far wall tak jaana aur wapas aana padta hai: doori $2 L$ speed $v_{x}$ par. $Δ t = \frac{2 L}{v _{x}}$ Ye step kyun? Hits ki frequency round-trip travel time par depend karti hai. Zyada $v_{x}$ ⇒ zyada frequent hits.

Step 3 — Ek molecule se Force (average). Force = rate of momentum transfer: $F_{1} = \frac{Δ p _{x}}{Δ t} = \frac{2 m v _{x}}{2 L / v _{x}} = \frac{m v _{x}^{2}}{L}$ Ye step kyun? Newton's 2nd law $F = \frac{Δ p}{Δ t}$ ke form mein. Bahut saari tiny kicks average out ho jaati hain = ek steady force.

Step 4 — Saare $N$ molecules par sum karo. $F = \frac{m}{L} (v_{x 1}^{2} + v_{x 2}^{2} + \dots + v_{x N}^{2}) = \frac{m}{L} N \overline{v_{x}^{2}}$ jahan $\overline{v_{x}^{2}}$ $v_{x}^{2}$ ka mean hai. Ye step kyun? Total force saari individual forces ka sum hai. Hum average use karke $N$ factor out karte hain.

Step 5 — Isotropy (randomness) use karo. Kyunki directions random hain, koi bhi axis special nahi hai: $\overline{v_{x}^{2}} = \overline{v_{y}^{2}} = \overline{v_{z}^{2}}$ Aur kyunki $v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$ : $\overline{v^{2}} = 3 \overline{v_{x}^{2}} \Rightarrow \overline{v_{x}^{2}} = \frac{1}{3} \overline{v^{2}}$ Ye step kyun? Ye crucial trick hai. Hum sirf $x$ track nahi karte; hum full speed $\overline{v^{2}}$ use karte hain, jo energy se relate karta hai.

Step 6 — Pressure = Force / Area. Wall ka area $= L^{2}$ hai. $P = \frac{F}{L ^{2}} = \frac{m N v _{x}^{2}}{L \cdot L ^{2}} = \frac{m N \frac{1}{3} v ^{2}}{L ^{3}} = \frac{1}{3} \frac{N m v ^{2}}{V}$

Figure — Kinetic theory — pressure derivation, temperature as mean KE

Temperature as mean kinetic energy

Ab pressure result ko rewrite karo taaki energy dikhaye. 2 se multiply aur divide karo: $P V = \frac{1}{3} N m \overline{v^{2}} = \frac{2}{3} N (\frac{1}{2} m \overline{v^{2}}) = \frac{2}{3} N \overline{E_{k}}$ jahan $\overline{E_{k}} = \frac{1}{2} m \overline{v^{2}}$ average translational KE per molecule hai.

Experimental ideal gas law $P V = N k_{B} T$ se compare karo (jahan $k_{B} = R / N_{A}$ Boltzmann constant hai): $\frac{2}{3} N \overline{E_{k}} = N k_{B} T$

Worked examples

Common mistakes ko steel-man karna

Active recall

Wall ke saath elastic collision mein (⊥ x-axis), ek molecule ka momentum change kya hota hai?

Δ p_{x} = 2 m v_{x}

(sirf

v_{x}

sign reverse karta hai)

Ek molecule ki same wall par successive hits ke beech ka time (cube side

L

Δ t = 2 L / v_{x}

(speed

v_{x}

par round trip)

Wall par ek molecule se average force?

F_{1} = m v_{x}^{2} / L

Kinetic-theory pressure formula?

P = \frac{1}{3} \frac{N m v ^{2}}{V} = \frac{1}{3} ρ v_{r m s}^{2}

\overline{v_{x}^{2}} = \frac{1}{3} \overline{v^{2}}

kyun?

Random directions (isotropy):

\overline{v_{x}^{2}} = \overline{v_{y}^{2}} = \overline{v_{z}^{2}}

aur ye sum hokar

\overline{v^{2}}

dete hain

Mean translational KE aur temperature mein kya relation hai?

\overline{E_{k}} = \frac{1}{2} m \overline{v^{2}} = \frac{3}{2} k_{B} T

T ke terms mein rms speed ka formula?

v_{r m s} = 3 k_{B} T / m = 3 R T / M

Same temperature par, H₂ ya O₂ mein se kiska mean KE zyada hai?

Same — mean KE sirf

T

par depend karta hai

Same T par, light ya heavy molecules tez chalte hain?

Light wale;

v_{r m s} \propto 1/ m

Translational degree of freedom per energy (equipartition)?

Har ek mein

\frac{1}{2} k_{B} T

, teeno milake

\frac{3}{2} k_{B} T

dete hain

Pressure mein average speed

\overset{v}{ˉ}

ki jagah

v_{r m s}

kyun use karte hain?

Pressure

\overline{v^{2}}

par depend karta hai;

v_{r m s} = \overline{v^{2}} \neq = \overset{v}{ˉ}

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho ek room bhar ke bouncy balls hain jo aadar-udhar ud rahi hain aur walls se super fast bang kar rahi hain. Har ball itni chhoti hai ki feel nahi hoti, lekin milke unka constant tap-tap-tap-tap wall ko push karta hai — whi push pressure hai. Agar tum room ko heat karo, balls tez ho jaati hain, zyada baar zyada hard hit karti hain, isliye zyada push karti hain. Toh "hot" ka matlab sirf balls ka zyada zoom karna hai. Temperature bas itna hi hai: ek fancy word jiske maane hain tiny balls kitna tezi se hilaati hain. Aur ek cool fact: heavy balls aur light balls, agar room same temperature par ho, toh average "oomph" of energy same carry karte hain — heavy wale sirf iske liye slower chalte hain.

Connections

Ideal Gas Law PV=nRT — kinetic theory ise derive karti hai; yahan ye ek experimental input hai jise hum match karte hain.
Boltzmann Constant and Equipartition — $\frac{1}{2} k_{B} T$ per degree of freedom ko generalise karta hai.
Maxwell-Boltzmann Speed Distribution — $\overset{v}{ˉ}$ , $v_{r m s}$ , $v_{m p}$ ke peechhe full shape deta hai.
Degrees of Freedom and Molar Heat Capacity — translation se aage rotational/vibrational modes.
Internal Energy of Ideal Gas — $U = \frac{3}{2} N k_{B} T$ monatomic gas ke liye directly follow karta hai.
Elastic Collisions and Momentum — derivation ka Newtonian backbone.
Pressure as Force per Area — macroscopic definition jise hum connect karte hain.