Physics – Ch # 13 – Exercise Solutions -Thermal Physics – New Book 2026

Ye poora chapter Kinetic Theory of Gases (gases ka kinetic nazariya) par based hai. Bunyadi idea ye hai ke gas ke molecules hamesha random motion mein hote hain, container ki walls se takrate hain, aur isi takrane (collisions) se pressure paida hota hai. Chapter mein temperature, pressure, rms speed, aur energy ka aapas mein taluq samjhaya gaya hai, aur aakhir mein stars (sitaron) ki stability bhi isi pressure-gravity balance se explain ki gayi hai.

Section A — Multiple Choice Questions (MCQs)

Ye MCQs bunyadi formulas aur concepts test karte hain:

• Q1 (Ideal gas condition): Gas ideal behaviour tab dikhata hai jab high temperature aur low pressure ho — kyunke high T par molecules tezi se move karte hain aur low P par unke beech fasla zyada hota hai, isliye intermolecular forces negligible ho jaati hain.
• Q2, Q3, Q4 (vrms ka formula use karke calculation): In sab mein ek hi formula use hota hai:
  vrms∝TMv_{rms} \propto \sqrt{\frac{T}{M}}vrms​∝MT​​ Matlab rms speed temperature ke square root ke seedha proportional hai, aur molecular mass ke square root ke ulta proportional hai. Jab bhi T ya M change ho, is ratio se naya speed nikal lete hain.
• Q5 (Density se speed nikalna): Formula use hua: vrms = √(3P/ρ). Pressure aur density diye the, seedha substitute kar diya.
• Q6 (Kinetic theory ka galat statement pehchano): Yaad rakhna zaroori hai ke kinetic theory mein collisions hamesha elastic (bilkul energy loss ke bagair) hoti hain, inelastic nahi. Ye ek common trick question hai.
• Q7, Q8 (Stars ki stability): Star tab stable rehta hai jab hydrostatic equilibrium ho, matlab gravity (andar ki taraf khenchna) aur pressure (bahar ki taraf dhakka) barabar hon. Agar gravity zyada ho jaye to star contract (sikur) jata hai, aur core ka temperature barh jata hai.

Section B — Short Answer Questions

Ye sawalat concepts ko simple explanation ke saath poochte hain:

• Q13.1, 13.2 (Sab gases ki KE barabar kyun): Formula yaad rakho: ⟨K.E.⟩ = (3/2)kT. Is formula mein sirf temperature hai, koi mass ya gas ki type nahi. Isliye chahe O₂ ho ya N₂, agar temperature same hai to unki average kinetic energy bhi same hogi — chahe unki speeds alag hon (lighter gas tezi se move karta hai lekin KE phir bhi barabar rehti hai).
• Q13.3 (Density double hone se pressure): Formula P = (1/3)ρ⟨v²⟩ se pata chalta hai ke pressure density ke seedha proportional hai. Isliye density double karne se pressure bhi double ho jayega.
• Q13.4 (Heating se pressure kyun barhta hai): Simple logic — jab gas ko heat karte hain, molecules tez move karte hain, isliye walls se zyada bar aur zyada force se takrate hain, jisse pressure barh jata hai.
• Q13.5 (Molecules ki tadaad double hone ka asar): Ye thora tricky hai — rms speed sirf temperature aur mass par depend karti hai (formula: vrms=√(3kT/m)), molecules ki tadaad (N) is formula mein hai hi nahi. Isliye N double karne se rms speed par koi asar nahi parta (lekin total pressure zaroor barh jayega).
• Q13.6 (P aur V constant rakhtay hue T kyun nahi badal sakte): Ideal gas law se: T = PV/(nR). Agar P, V, aur n sab fix hain to T bhi automatically fix ho jata hai — koi azaad tabdeeli nahi ho sakti jab tak P ya V ko badalne na diya jaye.
• Q13.7 (Volume kam karne se pressure kyun barhta hai): Jab volume kam hota hai, molecules ek chhoti jagah mein confine ho jate hain, isliye wo walls se zyada bar takrate hain (collision frequency barh jaati hai). Yehi zyada collisions pressure barhate hain.
• Q13.8 (Stars mein gravity vs pressure): Jab internal pressure gravity se zyada ho to star phailta hai (red giant banta hai). Jab gravity pressure se zyada ho to star sikurta hai, aur core ka density aur temperature barh jata hai.

Section C — Constructed Response Questions

Ye thore detailed sawalat hain, kabhi calculation ke saath:

• Q13.1 (Absolute zero par KE): Formula ⟨K.E.⟩=(3/2)kT se seedha pata chalta hai ke jaise jaise T→0, KE bhi zero ki taraf jaati hai. Matlab absolute zero par molecular motion ruk jata hai (classical theory ke hisaab se).
• Q13.2 (Numerical — O₂ aur N₂ ki average KE): Simple substitution:
  • T=300K, k=1.38×10⁻²³ diya
  • Formula mein daal diya: ⟨K.E.⟩ = (3/2)(1.38×10⁻²³)(300) = 6.21×10⁻²¹ J
  • Dono gases ki KE same aayegi kyunke formula mein mass hai hi nahi
• Q13.3 (P aur V dono badalne se T kaisa change hota hai): Yahan trick ye hai ke PV=nRT se T∝PV hota hai (jab n constant ho). Bas naye P aur V ko multiply kar ke dekho purane PV se kitna guna hai:
  • (a) P aur V dono double → PV, 4 guna ho gaya → T bhi 4 guna
  • (b) V half, P triple → PV, 1.5 guna ho gaya → T bhi 1.5 guna
• Q13.4 (Bina heat kiye paani ubalna): Ye interesting concept hai — boiling tab hoti hai jab liquid ka vapour pressure bahar ke pressure ke barabar ho jaye. Agar hum temperature na barhayein balke bahar ka pressure kam kar den (vacuum pump se), to room temperature par bhi paani ubal sakta hai.
• Q13.5 (Ink drop ka phailna): Ye Brownian motion ki wajah se hota hai — paani ke molecules random tareeqe se move karte hue ink ke particles se takrate rehte hain, jisse ink dheere dheere poore paani mein phail jaati hai, bina hilaye bhi.

Section D — Comprehensive Questions (Derivations)

Ye sabse important section hai kyunke ismein derivations hain — inhe step-by-step samajhna zaroori hai:

Q13.1 — Gas Pressure Ki Derivation

Concept: Hum ek molecule ko lete hain jo box ki wall se takrata hai, aur uske collision se paida hone wali force nikalte hain, phir sab molecules par sum karte hain.

Steps:

1. Jab molecule wall se elastic collision karta hai to uski velocity ulti ho jaati hai, isliye momentum change = 2mvx
2. Wall tak wapis aane mein time lagta hai 2L/vx (L = box ka side)
3. Force = momentum change ÷ time = mvx²/L
4. Sab N molecules par sum kiya, aur 3D random motion ki wajah se ⟨vx²⟩ = ⟨v²⟩/3 use kiya
5. Aakhir mein Pressure = Force/Area nikal kar formula mila:

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

Ye formula chapter ki sabse important derivation hai — isay achi tarah yaad karo.

Q13.2 — rms Speed aur Most Probable Speed

Do formulas yaad karo:

• rms speed: vrms = √(3kT/m) = √(3RT/M)
• Most probable speed: vp = √(2kT/m) = √(2RT/M)

Farq sirf constant (3 vs 2) ka hai.

Q13.3 — Boltzmann Distribution Law

Concept: Kisi system mein particles alag alag energy levels par hote hain. Boltzmann law kehta hai ke kisi bhi energy level par particles ki tadaad us energy ke exponential se ulta proportional hoti hai (e^(−E/kT)). Matlab zyada energy wale states mein kam particles hote hain, aur temperature barhne se higher energy states mein particles ki tadaad barh jaati hai.

Q13.4 — Neutron Star Ki Pressure

Neutron star mein gravity ko neutron degeneracy pressure balance karta hai — 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). Jab tak ye pressure gravity ko balance karti hai, star stable rehta hai.

Q13.5 — Temperature aur KE Ka Taluq Sabit Karna

Do equations barabar ki gayi:

• Kinetic theory se: PV = (1/3)Nm⟨v²⟩
• Ideal gas law se: PV = NkT

In dono ko equate karke aur simplify karke final formula mili:

⟨K.E.⟩=12m⟨v2⟩=32kT\langle K.E.\rangle = \frac{1}{2}m\langle v^2\rangle = \frac{3}{2}kT⟨K.E.⟩=21​m⟨v2⟩=23​kT

Ye sabit karta hai ke temperature seedha average kinetic energy ka measure hai.

Q13.6 — Stellar Evolution (Sitaron Ka Irtiqa)

Ye ek conceptual essay-type answer hai:

• Star nebula (gas/dust cloud) se banta hai, gravity se sikurta hai, temperature barhti hai, aur nuclear fusion shuru hota hai jo gravity ko balance karta hai
• Jab fuel khatam ho jata hai, core sikurna shuru hota hai
• Chhote/medium stars (Sun jaise) → electron degeneracy pressure collapse rokti hai → white dwarf banta hai
• Bade stars → supernova hota hai → neutron degeneracy pressure collapse rokti hai → neutron star banta hai
• Bohot bade stars → koi pressure collapse nahi rok sakti → black hole banta hai

Yaad rakhne wala nuqta: Star ka final anjaam is baat par depend karta hai ke koi pressure (thermal, electron degeneracy, ya neutron degeneracy) gravity ko hamesha ke liye balance kar sakti hai ya nahi.

Section E — Numerical Problems

Ye numericals hain, jinme formulas mein values substitute kar ke answer nikalna hai. Har numerical mein tareeqa same hai: sahi formula pehchano, values daalo, calculate karo.

• Q13.1 (Hydrogen ki rms speed): Pehle mercury column se atmospheric pressure nikala (P=hρg), phir vrms=√(3P/ρ) formula use kiya → ≈1.84×10³ m/s
• Q13.2 (Oxygen ka pressure): Formula P=(1/3)ρv² use kiya, density aur rms speed diye the → ≈1.0×10⁵ N/m²
• Q13.3 (Temperature nikalna jab speed 3 guna ho): vrms∝√T ka concept use kiya: agar speed 3 guna honi hai to (3)²=9 guna temperature chahiye → T₂=9×273=2457K = 2184°C
• Q13.4 (Molecules ki tadaad nikalna): Ideal gas law PV=NkT se N=PV/(kT), values substitute ki → ≈1.02×10¹¹ molecules
• Q13.5 (Sun ke atmosphere mein electron ki rms speed): Formula vrms=√(3kT/m) use kiya, electron ka mass diya gaya tha → ≈9.5×10⁶ m/s. Note: yahan pressure ki value zaroorat nahi thi is calculation ke liye, sirf T aur m chahiye the.
• Q13.6 (Star ke andar gravitational force): Newton ka shell theorem use kiya: F/m = GM/r², values substitute karke field strength nikali → ≈2.7×10² N/kg

Numericals solve karne ka General Tareeqa:

1. Pehle pehchano konsi quantity maloom hai aur konsi nikalni hai
2. Sahi formula select karo
3. Units check karo (SI units mein convert karo agar zaroorat ho, jaise °C ko K mein)
4. Values substitute kar ke calculate karo

Sabse Zaroori Formulas Jo Poore Chapter Mein Baar Baar Aate Hain: