1.7.11 · D5 · HinglishThermodynamics
Question bank — Mean free path, mean speed, RMS speed — derivations
1.7.11 · D5· Physics › Thermodynamics › Mean free path, mean speed, RMS speed — derivations
True or false — justify
Har statement ya toh subtly galat hai ya sahi-magar-non-obvious-reason se. Bolo kaun sa, aur kyun.
Ek hi temperature par, ek bhaari gas ki RMS speed ek halki gas se kam hoti hai
Sahi — mass par depend karta hai, aur same ka matlab same energy hai, toh bada chota maangta hai.
Ek hi temperature par, har gas molecule ka average kinetic energy mass se independent hota hai
Sahi — yeh equipartition hai: mein right side par koi nahi, isliye KE sirf par depend karta hai, molecule kaun sa hai is par nahi.
aur equal hote hain kyunki dono "average speed" hain
Galat — ko average karta hai, jabki pehle average karta hai phir square-root leta hai; squaring se fast molecules ka weight zyada ho jaata hai, isliye kisi bhi real speed distribution ke liye.
Agar har molecule ki speed double kar do, toh aur dono double ho jaayenge
Sahi — dono speeds mein degree one ke homogeneous hain ( linear, ek quadratic average ka root), isliye saari speeds ko 2 se scale karne par dono exactly 2 se scale hote hain.
Absolute temperature double karne se double ho jaata hai
Galat — , isliye double karne par sirf se multiply hota hai; speed double karne ke liye ko chaar guna karna padega.
Mean free path is baat par depend karta hai ki molecules kitni tez move karti hain
Galat — mein koi speed nahi hai; speed cancel ho jaati hai kyunki fly ki gayi distance aur collisions ki sankhya dono speed ke saath proportionally badhte hain.
Fixed volume par, gas ko heat karne se mean free path badhti hai
Galat — fixed par number density unchanged rehta hai, aur mein koi nahi, isliye waisa hi rehta hai; sirf collision rate badhti hai ( ke zariye).
Fixed pressure par, gas ko heat karne se mean free path badhti hai
Sahi — fixed par, ; badhane par molecules alag ho jaate hain (lower ), isliye har molecule hits ke beech zyada door tak udta hai.
mein collision cylinder ki geometry se aata hai
Galat — cylinder se milta hai; aata hai do moving molecules ke beech relative speed se jo average mein hoti hai, jo collision rate badhata hai.
Collision cross-section hai, nahi, kyunki molecules bade hote hain
Galat — yeh isliye hai kyunki collision tab hoti hai jab centres ek diameter ke andar aate hain (do radii ka sum), isliye effective target radius hai, molecular radius nahi.
Spot the error
Har item mein ek plausible-sounding claim ya step hai. Galti batao.
"Kyunki N₂ ke liye g/mol hai, ."
Wrong units — SI mein hai, isliye kg/mol mein hona chahiye (), nahi; grams use karne se speed zyada badi ban jaati hai.
"Pressure hai kyunki molecules -direction mein move karti hain."
Factor galat hai — correct relation poori speed use karta hai: , aur sirf teen directions ki isotropy ke zariye aata hai.
"Most probable speed hi average speed hai, kyunki wahan sabse zyada molecules hain."
ka peak mode hai, mean nahi; distribution mein ek lamba high-speed tail hai, isliye (tail average ko peak se upar kheench leta hai).
"Kyunki KE hai, mean speed wali molecule mean energy carry karti hai."
Nahi — mean energy mein aata hai, lekin ; energy-carrying speed hai, jo strictly se badi hai.
" nikalne ke liye total distance ko collisions se divide karo: , jo deta hai — ho gaya."
Incomplete — yeh targets ko stationary maanta hai; unki motion account karne par collision count mein ki jagah relative speed aata hai, jo missing add karta hai.
" odd hai, isliye yeh zero ho jaata hai."
Galat — par kuch bhi parity se vanish nahi hota; sirf par symmetric integrals odd integrands ko kill karte hain, aur waise bhi even hai. Integral hai.
"Kyunki mein koi pressure nahi hai, RMS speed aur gas type ke alawa sab se independent hai."
Jaise kaha gaya waise ke liye sahi hai, lekin trap yeh hai ki ke baare mein bhi yahi soch lo — zaroor par depend karta hai ( ke zariye), chahe nahi karta.
Why questions
Mechanism ke saath jawab do, formula se nahi.
hamesha kyun hota hai, kisi bhi speed distribution ke liye?
Average se pehle squaring fast molecules ko extra weight deta hai ( upar push hota hai), jabki lamba fast tail ko peak se upar kheenchta hai; equality tab hi hogi jab har molecule ki speed identical ho.
Speed mean free path se cancel kyun ho jaati hai?
Ek tez molecule zyada door bhi udta hai aur targets se zyada baar bhi milta hai, same proportion mein, isliye distance-per-collision — yeh ratio — speed se unaffected rehta hai.
Collision rates aur compute karte waqt hum (na ki ) kyun use karte hain?
Collision frequency encounters per second count karta hai, jo ordinary average travel rate se govern hota hai; collision per energy ek alag sawaal hai jo se handle hota hai.
kyun hota hai?
Koi direction special nahi hai (isotropy), isliye ; kyunki inka sum hai, har ek uska ek-third ke barabar hai.
Wall par momentum change kyun hota hai, kyun nahi?
Wall -velocity ko se tak reverse kar deta hai; momentum ek vector hai, isliye magnitude mein change ka difference hai.
Collision cross-section radius ki jagah diameter se kyun banta hai?
Do spheres tab touch karte hain jab unke centres radii ke sum se separated hon; moving molecule ka centre har target centre ko se zyada miss karna chahiye, isliye "danger disc" ka radius hai.
Do molecules ki relative speed average mein kyun hoti hai?
Relative velocity do independent equal mean speed wali velocities ka vector difference hai; equal-variance independent vectors ke difference ka mean square ek variance ka do guna hota hai, isliye typical relative speed se scale hoti hai.
Gas squeeze karne par (fixed par badhane se) mean free path kyun chhoti ho jaati hai?
Fixed par zyada ka matlab zyada number density hai; zyada targets packed hone se ek molecule jaldi collide kar leti hai, isliye chhota ho jaata hai.
humein invisible molecular motion ko measurable pressure se kyun connect karne deta hai?
Pressure molecular impacts ki momentum delivery se aata hai, jo par depend karta hai; ko se match karne par (dekho Ideal gas law) macroscopic data se , hence , nikalta hai.
Edge cases
Boundary aur degenerate scenarios — jo ek memorised formula ko tod dete hain.
ka kya hoga jab pressure zero ke paas jaaye (near-vacuum)?
jab — molecules almost kabhi nahi milte, aur eventually container size se bada ho jaata hai isliye "ek doosre se collide karo" wali picture toot jaati hai aur wall collisions dominate karne lagte hain.
par ka kya hoga?
; classically saari motion band ho jaati hai, lekin real matter pehle liquify/solidify ho chuka hota aur quantum zero-point motion (is model se bahar) baaki rehti.
Agar gas ek box mein single molecule hai, toh kya "" ya "" meaningful bhi hai?
Ensemble averages ke taur par nahi — poora Maxwell–Boltzmann picture bade ki zaroorat rakhta hai; ek molecule ke saath yeh us molecule ki apni speed aur uske square-root-of-square ban jaate hain, jo coincide karte hain, isliye ordering collapse ho jaati hai.
Ek hypothetical gas ke liye jahan har molecule ki exactly same speed ho, kya hoga?
Exactly 1 — zero spread ke saath, , isliye strict inequality equality ban jaati hai; inke beech ka gap speed spread ka ek pure measure hai.
Jab container size ke comparable ho jaaye, tab kya mean-free-path formula ab bhi apply hota hai?
Nahi — derivation assume karti hai ki wall se takraane se pehle bahut saari molecule–molecule collisions hon; jab box size, transport ballistic ban jaata hai (effusion-jaisa) aur collisional formula flights describe nahi karta.
Bahut zyada density par (liquid-like packing), kya ab bhi hold karta hai?
Nahi — yeh assume karta hai ki molecules mostly empty space mein point-sized targets hain; jab packing tight ho, molecules "reach" mein overlap karte hain, multiple simultaneous encounters hote hain, aur dilute-gas assumption fail ho jaati hai.
Agar do directions mein somehow alag mean-square velocities hon (anisotropic gas), toh kya survive karta hai?
Nahi — yeh identity poori tarah isotropy par ruki hai; symmetry todo (jaise directional flow ya field se) aur har direction ko alag handle karna padega.
Recall Traps ka ek-line summary
Mass speed kam karta hai energy nahi; spread se; speed ignore karta hai density nahi; relative motion hai, aur cross-section diameter use karta hai, radius nahi.