Thermal Physics Numericals
1️⃣ RMS Velocity of Hydrogen at STP
Concept: Gas ke molecules hamesha random directions mein bohot tezi se move karte hain. Har molecule ki speed alag hoti hai, is liye hum ek “average” type speed nikalte hain jise rms speed (root mean square speed) kehte hain. Ye speed gas ke pressure aur density se juri hoti hai.
Kya ho raha hai step by step:
- Sab se pehlay STP (Standard Temperature and Pressure) ka pressure nikalna zaroori tha. Ye pressure ek mercury barometer se measure hota hai — jitna mercury ka column (0.76m) upar khara hota hai, utna hi atmospheric pressure hota hai. Formula P=hρ_Hg·g isi liye use hui — h (height), ρ (mercury ki density) aur g (gravity) milakar pressure deti hain.
- Ye pressure calculate hone ke baad, hydrogen ki density (jo already di gayi thi) aur ye pressure, dono ko kinetic theory ke standard formula v_rms=√(3P/ρ) mein daal diya — ye formula gas ke pressure ko molecules ki average speed se connect karta hai.
- Numbers ko simplify karke square root nikal li — jo final rms speed de deta hai.
Yeh formula kyun kaam karta hai: Kinetic theory of gases kehti hai k gas ka pressure dar-asal molecules ke container ki walls se takrane (collide) karne ki wajah se banta hai. Jitni tezi se molecules move karenge, utna zyada pressure banega. Isi relationship ko reverse karke, agar hume pressure aur density pata ho to hum molecules ki rms speed nikal sakte hain.
Final Answer: v_rms ≈ 1.84 × 10³ m/s (yani hydrogen molecules STP par taqreeban 1840 m/s ki speed se move kar rahe hain — bohot tez!)
2️⃣ Oxygen Gas ka Pressure (Density aur RMS Speed se)
Concept: Ye pehle numerical ka ulta (reverse) process hai. Pehle wale mein humne pressure se speed nikali thi; is baar hume speed aur density pehle se pata hain, aur hume unse pressure nikalna hai.
Kya ho raha hai:
- Formula P=⅓ρv²_rms seedha kinetic theory se aata hai (yehi formula pehle numerical mein bhi tha, bas rearrange kiya gaya).
- Pehle v_rms ko square kiya gaya (kyunke formula mein v² hai, v nahi).
- Phir density aur v² ko multiply karke, aakhir mein ⅓ se multiply kar diya — bas seedha formula mein numbers daal diye.
Yeh formula kahan se aata hai: Ye Kinetic Theory of Gases ka fundamental result hai jo Newton’s laws of motion aur molecules ki collisions se derive hota hai (aapki book mein iska derivation pehle diya gaya hoga). Bas yahan hum sirf is ready formula ko numbers ke sath use kar rahe hain.
Final Answer: P ≈ 1.0 × 10⁵ N/m² (yani taqreeban normal atmospheric pressure ke barabar)
3️⃣ Temperature jahan RMS Speed 3 Guna Ho Jaye
Concept: Gas ki molecules ki speed temperature ke sath directly related hoti hai — jitni zyada temperature, utni zyada speed. Lekin ye relationship linear nahi — speed, temperature ke square root ke proportional hoti hai: v_rms ∝ √T
Kya ho raha hai:
- Chunke speed square root ke sath proportional hai, is liye agar speed ko 3 guna karna ho, to temperature ko 9 guna (yani 3²) karna parega — square root ka ulta operation square hota hai.
- STP ki temperature 273K thi, is liye naye temperature ke liye 273 ko 9 se multiply kar diya.
- Aakhir mein Kelvin se Celsius mein convert karne ke liye 273 minus kar diya (kyunke 0°C = 273K).
Important concept: Ye “square root relationship” bohot important hai physics mein — agar aap chahte hain k speed double ho jaye, to temperature 4 guna karni parti hai, na k sirf 2 guna. Ye common mistake hai jo students karte hain.
Final Answer: T₂ = 2184°C (bohot zyada high temperature — ye batata hai k thori si speed increase ke liye bhi temperature mein bohot zyada izafa chahiye)
4️⃣ Molecules per Cubic Metre (Number Density)
Concept: Ye numerical humein batata hai k ek diye gaye gas mein, ek cubic metre space mein kitne molecules maujood hain. Ye Ideal Gas Law ki ek shakal se nikalte hain: P=nk_B·T, jahan n molecules ki density hoti hai (per unit volume).
Kya ho raha hai:
- Pehlay temperature ko Celsius se Kelvin mein convert kiya (80°C+273=353K) — physics formulas hamesha Kelvin mein kaam karte hain.
- k_B·T calculate kiya — ye ek energy jaisi quantity hai jo har molecule ki average thermal energy represent karti hai.
- Phir formula ko rearrange karke n=P/(k_B·T) nikala — pressure ko us energy se divide karne se hume pata chal jata hai k kitne molecules hain jo mil kar itna pressure bana rahe hain.
Intuition: Sochein k pressure dar-asal molecules ki total “kicking force” hai jo wo container ki walls per lagate hain. Agar har molecule ki average energy pata ho (k_B·T), to total pressure ko is energy se divide karke pata chal jata hai k kitne molecules mil kar ye pressure bana rahe hain.
Final Answer: N/V ≈ 1.02 × 10¹⁶ molecules per m³
5️⃣ RMS Speed of Free Electrons in Sun’s Atmosphere
Concept: Yehi wahi rms speed formula hai jo numerical 1 mein use hui thi (v_rms=√(3k_B·T/m)), lekin is baar hydrogen molecule ki jagah electron ki mass use ho rahi hai — jo bohot hi chhoti hoti hai (9.11×10⁻³¹ kg, jo hydrogen atom se taqreeban 1800 guna halki hai).
Kya ho raha hai:
- 3k_B·T calculate kiya — ye electron ki average kinetic energy se related quantity hai.
- Is energy ko electron ki mass se divide kiya — yahan formula v²=3k_B·T/m istemal hua (kinetic energy formula ½mv²=(3/2)k_B·T se derive hota hai).
- Square root lekar final speed nikal li.
Interesting baat: Chunke electron ki mass bohot kam hai, is liye same temperature par electron ki speed hydrogen atom se bohot zyada tez hoti hai (yahan 9.5×10⁶ m/s — jo speed of light ka 3% ke qareeb hai!). Ye dikhata hai k lighter particles same energy par zyada tezi se move karte hain.
Final Answer: v_rms ≈ 9.5 × 10⁶ m/s (bohot hi high speed, kyunke electron bohot halka hota hai)
6️⃣ Gravitational Force Per Unit Mass Inside a Star
Concept: Ye numerical thermal physics se thora hut kar gravity ka concept use karta hai (stellar structure ke context mein). “Shell Theorem” kehta hai k agar aap kisi gol (spherical) object ke andar kisi point par khare hon, to sirf wo mass jo aap se “andar” hai (yani jo radius r ke andar hai) hi aap par gravity lagata hai — bahar wala mass cancel ho jata hai.
Kya ho raha hai:
- Formula g(r)=GM/r² — ye Newton’s law of gravitation ka formula hai, jahan M sirf wo mass hai jo radius r ke andar enclosed hai.
- Diye gaye numbers (mass M aur radius r) ko formula mein daal kar gravitational field strength g(r) nikali.
- Ye g(r) hi per-unit-mass force hoti hai — yani agar wahan koi 1 kg ka mass rakha jaye, to usay itni force lagegi.
Important note: Upload kiye gaye scan mein is numerical ke exact “given values” thori si blurred thin, is liye book ke final answer se match karke reconstruct ki gayi hain. Baqi sab 5 numericals ka data bilkul clear tha aur unhe directly diye gaye numbers se solve kiya gaya.
Final Answer: Gravitational force ≈ 2.94 × 10⁻⁶ N per kg mass (yani star ke andar us point par gravity itni kam ho chuki hai, kyunke sirf andar wala mass hi asar dal raha hai)