Question bank — Kinetic theory — pressure derivation, temperature as mean KE

True. $\overline{E_k}=\frac{3}{2}{k}_{B}T$ sirf $T$ par depend karta hai, mass par nahi; same $T$ har ideal gas ke liye same average translational KE force karta hai.

False. Equal KE ke saath unequal mass ka matlab hai $v_{rms}=\sqrt{3k_{B}T/m}$ alag hoga — halka hydrogen bahut zyada fast hoga, kyunki sirf product $\frac{1}{2}m\overline{v^2}$ fixed hai.

False. $v_{rms}\propto \sqrt{T}$, toh $T$ double karne par rms speed $\sqrt{2}\approx 1.41$ se multiply hoti hai; speed double karne ke liye tumhe $T$ chaar guna karna hoga.

False. $v_{rms}=\sqrt{\overline{v^2}}$ hamesha $\bar{v}$ se zyada hoti hai kyunki squaring fast molecules ko extra weight deta hai; ye dono genuinely alag averages hain (dekho Maxwell-Boltzmann Speed Distribution).

True. $PV=N{k}_{B}T$ mein $N,V$ fixed hain, isliye $P\propto T$. Dhyan raho $T$ kelvin mein hona chahiye — °C double karne se $P$ double nahi hota.

False. Pressure hai $P=N{k}_{B}T/V$ — isme koi mass nahi aata. Bhaari molecules zyada hard hit karte hain lekin proportionally slower chalte hain, aur ye dono effects exactly cancel ho jaate hain.

False. Ye spatial dimensions ki sankhya hai: isotropy se $\overline{v_x^2}=\frac{1}{3}\overline{v^2}$ milta hai. Ek 2D gas mein factor $\frac{1}{2}$ hoga; ye pure geometry hai.

False. Wall rigid aur stationary hai aur collision elastic hai, isliye sirf $v_x$ ki direction flip hoti hai — speed, aur isliye KE, unchanged rehti hai (dekho Elastic Collisions and Momentum).

Partly true aur harmless. Intermolecular collisions sirf velocities ko molecules ke beech redistribute karti hain; kyunki ye total momentum aur energy conserve karti hain, population averages $\overline{v^2}$ aur $\overline{v_x^2}$ unaffected rehte hain, toh derived pressure same hota hai.

True is model ke andar. $\overline{E_k}=\frac{3}{2}{k}_{B}T\to 0$ jab $T\to 0$, toh classically sab motion ruk jaata hai — real gases bahut pehle liquefy ho jaate hain ya quantum effects dikhate hain, lekin ideal law yahi limit predict karta hai.

False. Temperature poori population ke average KE ki ek statistical property hai. Ek single molecule ki speed aur KE hoti hai, lekin "temperature" sirf ensemble ke liye meaningful hai.

Error: ye $\overline{v^2}$ ki jagah $\bar{v}$ use karta hai. Momentum transfer per unit time $v_x^2$ ke saath scale karta hai, isliye pressure mean of the squares par depend karta hai, jo $v_{rms}$ deta hai, $\bar{v}$ nahi.

Error: ye rebound bhool jaata hai. Elastic bounce ke baad momentum $-m{v}_x$ hota hai, isliye change hai $m{v}_x-(-m{v}_x)=2m{v}_x$ — factor of 2 essential hai.

Error: ye ek-taraf ka trip use karta hai. Molecule ko opposite wall tak jaana aur wapas aana hoga usi wall se phir hit karne se pehle, isliye $\Delta t=2L/v_x$.

Error: ye sum drop kar deta hai. Kyunki $v^2=v_x^2+v_y^2+v_z^2$, teeno equal parts add hote hain: $\overline{v^2}=3\overline{v_x^2}$, isliye $\overline{v_x^2}=\frac{1}{3}\overline{v^2}$.

Error: har translational degree of freedom $k_{B}T$ nahi balki $\frac{1}{2}{k}_{B}T$ carry karta hai. Teeno milke $\frac{3}{2}{k}_{B}T$ dete hain (dekho Boltzmann Constant and Equipartition).

Error: per-molecule aur per-mole quantities ko mix kar raha hai. $k_B$ ke saath molecular mass $m$ use karo; molar mass $M$ ke saath $R$ use karo: $v_{rms}=\sqrt{3RT/M}$.

Dono teen spatial dimensions se aate hain. $\overline{v_x^2}=\frac{1}{3}\overline{v^2}$ speed ko $x,y,z$ mein baanta hai; equipartition phir un teeno directions mein se har ek ko $\frac{1}{2}{k}_{B}T$ deta hai, jo $\frac{3}{2}{k}_{B}T$ deta hai.

Kyunki squared components ka sum, mean ki definition se $N$ guna average ke barabar hota hai: $\sum v_{xi}^2=N\overline{v_x^2}$. Ye sirf repackaging hai, koi approximation nahi.

$\sim 10^{23}$ molecules ke saath jo per second billions baar hit karte hain, discrete kicks ek constant average force mein blur ho jaate hain — everyday scales par fluctuations bilkul negligible hain.

Kyunki $PV=\frac{2}{3}N\overline{E_k}$ aur $\overline{E_k}=\frac{3}{2}{k}_{B}T$ sirf temperature par depend karta hai. Mass cancel ho jaata hai kyunki bhaari molecules sirf same KE rakhne ke liye slower chalte hain.

Kyunki physics — wall par momentum flux — $v_x^2$ se bani hai, isliye jo average emerge hota hai woh $\overline{v^2}$ hai, jiska square root construction se $v_{rms}$ hai. Baaki averages alag questions ka jawab dete hain (dekho Maxwell-Boltzmann Speed Distribution).

$x$-wall par pressure mein $\overline{v_x^2}$ involve hai, lekin energy mein poora $\overline{v^2}$. Isotropy hume $\overline{v_x^2}=\frac{1}{3}\overline{v^2}$ ke zariye ek ko doosre se swap karne deti hai, wall force ko molecular energy se connect karti hai.

Pressure derivation sirf centre-of-mass motion ($v_x,v_y,v_z$) track karta hai. Rotational ya vibrational energy exist karta hai lekin kisi wall ko push nahi karta, isliye ye kabhi $P$ mein enter nahi hota — lekin ye heat capacity ko zaroor affect karta hai (dekho Degrees of Freedom and Molar Heat Capacity).

$P$ double ho jaata hai, kyunki $P=N{k}_{B}T/V$ aur $P\propto 1/V$. Physically, molecules aadhe time mein box cross karte hain, hit frequency double kar dete hain.

Kuch nahi — $\overline{E_k}=\frac{3}{2}{k}_{B}T$ sirf $T$ par depend karta hai. Molecules walls ko zyada baar hit karte hain (higher $P$) lekin har ek abhi bhi same average KE carry karta hai.

Sirf do directions ke saath, $\overline{v_x^2}=\frac{1}{2}\overline{v^2}$, isliye pressure $\frac{1}{2}$ carry karta hai, aur equipartition deta hai $\overline{E_k}=\frac{2}{2}{k}_{B}T=k_{B}T$ (har dimension ke liye ek $\frac{1}{2}{k}_{B}T$).

Directly nahi — $v_x=0$ ke saath ye kabhi $x$-wall tak nahi pahunchti. Lekin ye $10^{23}$ mein se ek molecule hai; population average $\overline{v_x^2}$ hi pressure set karta hai, aur woh $\frac{1}{3}\overline{v^2}$ rehta hai.

$v_{rms}\to \infty$, toh vanishingly light molecules fixed $T$ par arbitrarily fast chalenge — isi liye sabse halke gases (hydrogen, helium) sabse fast hote hain aur planetary atmospheres se sabse aasani se escape karte hain.

Nahi. Ye relation absolute temperature demand karta hai; $0{}^{\circ }\text{C}$ par KE zero nahi hoti (woh $273\text{K}$ ki motion hai). Sirf Kelvin $T=0$ par $\overline{E_k}$ ko vanish karta hai.