Physics – Ch # 13 – Long Questions -Thermal Physics – New Book 2026

Overview

Ye Long Questions section pichli exercise ke Comprehensive Questions se milta julta hai, lekin yahan har jawab ko exam-style, thora zyada detailed tareeqe se likha gaya hai. Ismein 3 bari derivations, kinetic theory ki assumptions, aur stars/Brownian motion se related conceptual essays shamil hain. Ye sab exam mein zyada marks wale sawalat hote hain, isliye inhe step-by-step samajhna zaroori hai.

Q1 — Gas Ki Pressure Ki Derivation (PV = ⅓Nm⟨v²⟩)

Ye chapter ki sabse important derivation hai, jo pehle bhi cover ho chuki thi lekin yahan exam-style poore steps ke saath di gayi hai.

Concept: Hum sochte hain ke ek molecule cubical box (side L) ke andar move kar raha hai, aur uski ek wall se takrane ki wajah se jo force paida hoti hai, wo nikaalte hain. Phir sab molecules ke liye sum karte hain.

Step-by-step tareeqa:

1. Momentum change: Jab molecule wall se elastic collision karta hai, uski velocity (vx) ulti ho jaati hai. Isliye momentum change = mvx − (−mvx) = 2mvx
2. Time between collisions: Molecule ko wapis usi wall tak pohanchne ke liye 2L distance tay karna parta hai, isliye time = 2L/vx
3. Ek molecule ki force: Force = momentum change ÷ time = 2mvx ÷ (2L/vx) = mvx²/L
4. Sab N molecules ki total force: Sab molecules ko add kiya: F = (m/L) × N × ⟨vx²⟩
5. 3D random motion ka concept: Chunke molecules teeno directions (x,y,z) mein equally random move karte hain, isliye ⟨vx²⟩ = ⟨v²⟩/3
6. Pressure nikalna: Pressure = Force/Area, aur V=L³ use kar ke final formula mili:

PV=13Nm⟨v2⟩PV = \frac{1}{3}Nm\langle v^2\ranglePV=31​Nm⟨v2⟩

Yaad rakhne ka tareeqa: Is derivation ke 6 steps ko yaad karo: (1) momentum change, (2) time between collisions, (3) force by one molecule, (4) total force, (5) 3D random motion ka trick (÷3), (6) pressure = force/area.

Q2 — Temperature Ka KE Se Taluq Sabit Karna

Ye derivation do formulas ko barabar karke nikali jaati hai:

Step-by-step:

1. Kinetic theory se mili formula: PV = (1/3)Nm⟨v²⟩ … (i)
2. Ideal gas law se: PV = NkT … (ii)
3. Dono equations (i) aur (ii) ko barabar kiya (kyunke dono PV ke barabar hain):
   13Nm⟨v2⟩=NkT\frac{1}{3}Nm\langle v^2\rangle = NkT31​Nm⟨v2⟩=NkT
4. N cancel ho gaya, aur simplify karte hue:
   12m⟨v2⟩=32kT\frac{1}{2}m\langle v^2\rangle = \frac{3}{2}kT21​m⟨v2⟩=23​kT
5. Left side asal mein average kinetic energy hi hai (½mv²), isliye:
   ⟨K.E.⟩=32kT\langle K.E.\rangle = \frac{3}{2}kT⟨K.E.⟩=23​kT

Nateeja: Chunke k ek constant hai, iska matlab hai ke average KE seedha T ke proportional hai. Yani T barhne se KE barhti hai, T ghatne se KE ghatti hai. Isi wajah se kehte hain ke temperature, molecules ki average kinetic energy ka direct measure hai.

Q3 — Kinetic Theory Ki Bunyadi Assumptions (Postulates)

Ye ek list-type sawal hai, jisme kinetic theory ki 5 bunyadi assumptions (mafrozat) yaad karni hain:

1. Gas mein bohot zyada chhote, gol molecules hote hain jo apne size ke muqablay mein bohot door door hote hain — isliye unka apna volume container ke volume ke muqablay mein negligible hota hai
2. Molecules hamesha random motion mein hote hain, har direction mein alag alag speeds ke saath
3. Molecules ke aapas mein aur walls ke saath collisions hamesha perfectly elastic hote hain — koi kinetic energy zaya nahi hoti
4. Molecules ke darmiyan koi attraction ya repulsion force nahi hoti, sirf collision ke waqt thori si force hoti hai
5. Molecules ki average kinetic energy sirf absolute temperature par depend karti hai

Yaad rakhne ka tareeqa: “Size negligible, Random motion, Elastic collisions, No intermolecular force, Temperature-dependent KE” — pehle letters se yaad kar sakte hain (SRENT jaisa kuch banaya ja sakta hai apni marzi se).

Q4 — rms Speed Aur Most Probable Speed

Ye do formulas hain jo Maxwell-Boltzmann speed distribution se related hain:

• rms speed: Ye sab molecules ki speeds ke squares ka average lekar phir uska square root nikalne se milti hai:
  vrms=⟨v2⟩=3kTm=3RTMv_{rms} = \sqrt{\langle v^2\rangle} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}vrms​=⟨v2⟩​=m3kT​​=M3RT​​
• Most probable speed: Ye wo speed hai jo sabse zyada molecules ke paas hoti hai — yani speed distribution graph ka peak point:
  vp=2kTm=2RTMv_p = \sqrt{\frac{2kT}{m}} = \sqrt{\frac{2RT}{M}}vp​=m2kT​​=M2RT​​

Farq samjho: Dono formulas mein sirf constant (3 vs 2) ka farq hai. Dono hi temperature barhne se barhti hain aur mass barhne se ghatti hain, lekin vrms hamesha vp se thori zyada hoti hai.

Q5 — Boltzmann Distribution Law

Concept: Kisi system mein bohot saare particles alag alag energy levels (E₁, E₂, E₃…) mein hote hain. Boltzmann law kehta hai ke kisi energy state mein particles ki tadaad us energy ke exponential factor e^(−E/kT) ke proportional hoti hai.

Do energy levels ke liye formula:

N2N1=e−ΔE/kT\frac{N_2}{N_1} = e^{-\Delta E/kT}N1​N2​​=e−ΔE/kT

Iska matlab: Jitni zyada energy hogi, utni kam population hogi us state mein — yani hamesha lower energy states mein zyada particles hote hain. Lekin jab temperature barhta hai, higher energy states ki population bhi barhne lagti hai (kyunke particles ko zyada thermal energy milti hai).

Q6 — Neutron Star Ke Andar Pressure Ka Role

Concept: Neutron star ek bohot dense star hota hai jo massive star ke supernova explosion ke baad bachta hai.

• Iski gravity star ko andar ki taraf khenchti rehti hai
• Isko balance karti hai neutron degeneracy pressure — ye ek quantum mechanical pressure hai jo Pauli Exclusion Principle ki wajah se paida hoti hai (neutrons ek hi quantum state share nahi kar sakte, isliye unhe alag alag higher energy states mein jana parta hai jab wo sikurte hain)
• Jab tak ye pressure gravity ko balance karti rehti hai (yani star ka mass Tolman–Oppenheimer–Volkoff limit — taqreeban 2-3 solar masses — se kam rahe), star stable rehta hai
• Agar mass is limit se zyada ho jaye to gravity is pressure ko bhi maat de deti hai, aur star black hole mein collapse ho jata hai

Q7 — Stellar Evolution (Sitaron Ka Poora Safar)

Ye is chapter ka sabse lamba conceptual answer hai. Poori kahani teen hisso mein samjho:

1. Star Kaise Banta Hai:
Gas-dust ka cloud (nebula) apni gravity ki wajah se sikurta hai, temperature barhti hai, aur aakhir mein nuclear fusion shuru ho jata hai (hydrogen, helium mein badalta hai). Ye fusion se paida hone wali thermal pressure gravity ko exactly balance karti hai — isay hydrostatic equilibrium kehte hain, aur star apni “main sequence” (stable) phase mein aa jata hai.

Is balance ko ek equation se bhi represent kiya jata hai:

ΔPΔr=−GMrρrr2\frac{\Delta P}{\Delta r} = -\frac{GM_r\rho_r}{r^2}ΔrΔP​=−r2GMr​ρr​​

2. Jab Fuel Khatam Ho Jata Hai:
Fusion ruk jata hai, thermal pressure kam ho jaati hai, aur gravity hawi ho jaati hai. Core sikurna shuru ho jata hai.

3. Star Ka Anjaam (Mass Par Depend Karta Hai):

Sabse Zaroori Nuqta: Star ki poori zindagi aur uska anjaam sirf ek cheez se decide hota hai — gravity (andar ki taraf) aur pressure (bahar ki taraf) ke darmiyan mukabla.

Q8 — Brownian Motion

Definition: Brownian motion wo musalsal, random, zig-zag harkat hai jo chhote particles (jaise pollen grains ya dust) fluid (liquid ya gas) mein dikhate hain. Ye pehli dafa Robert Brown ne 1827 mein observe ki thi.

Wajah (Cause): Fluid ke tez move karne wale molecules suspended particle ko har taraf se randomly takra rahe hote hain. Chunke ye collisions barabar nahi hoti (kabhi ek taraf se zyada, kabhi doosri taraf se), isliye particle idhar udhar dhakela jata hai, jisse ek unpredictable (zig-zag) raasta banta hai.

Kuch Zaroori Points:

1. Chhote (lighter) particles collisions se zyada speed paate hain
2. Particle ki speed fluid ki viscosity ke ulta proportional hoti hai — kam viscosity mein Brownian motion tez hoti hai
   3 Brownian motion particles ko hamesha motion mein rakhti hai

Ahmiyat (Significance): Brownian motion ne pehli dafa experimental proof diya ke atoms aur molecules waqai mojood hain aur wo hamesha random motion mein rehte hain — jo kinetic theory ki bunyadi assumption hai. Einstein ne 1905 mein isay mathematically explain kiya, aur Jean Perrin ke 1908 ke experiments ne is theory ko confirm kiya, jis par unhe 1926 ka Nobel Prize mila.

Sabse Zaroori Baat (Exam Ke Liye):

Long questions mein sabse zyada marks derivations (Q1, Q2) ke hote hain, isliye unke har step ko crmwise yaad karna zaroori hai — sirf final formula yaad karna kaafi nahi hota, examiner steps bhi check karta hai. Baaki conceptual answers (Q3, Q5, Q6, Q7, Q8) mein key terms (jaise “hydrostatic equilibrium”, “degeneracy pressure”, “Pauli Exclusion Principle”) zaroor use karo, ye examiner ko dikhata hai ke concept clear hai.