Work done in each process — derivation: Foundations

1.7.15 · D1 · Physics › Thermodynamics › Work done in each process — derivation

Is page par assume kiya gaya hai ki aap kuch nahi jaante. Parent note ka har letter — $P$ , $V$ , $T$ , $d V$ , $\int$ , $ln$ , $γ$ , $n$ , $R$ — yahan ek picture se build kiya gaya hai, pehle kabhi formula mein use karne se. Upar se neeche padho; har block agli cheez earn karta hai.

0. Physical scene (wo picture jis par sab kuch tika hai)

Kisi bhi symbol se pehle, wo machine dekho jo hum describe kar rahe hain.

Ek cylinder mein gas hai. Ek piston (sliding lid) move kar sakta hai. Gas ke molecules piston ko andar se baar baar thokate hain aur use bahar ki taraf push karte hain. Agar piston thoda bahar nikle, to gas ne work kiya hai — usne kisi cheez ko ek force ke against distance tak move kiya. Neeche sab kuch usi push ki careful bookkeeping hai.

1. Force $F$ aur distance $x$ — "work" ka matlab kya hai

Simple alfaz mein: zyada force, lambi distance = bahut kaam; thoda-sa nudge = chhota kaam.
Picture: ek arrow (force) jo uss direction mein point kare jis taraf piston slide karta hai, times kitna door slide karta hai.
Topic ko yeh kyun chahiye: work woh output hai jo hum calculate karna chahte hain. Gas ka poora kaam piston ko push karna hai, aur "kitni push × kitni door" wahi work hai.

$x$ sirf position hai — piston cylinder mein kitna door baitha hai. $d x$ (padho "dee-ex") matlab hai $x$ mein ek tiny change — piston bahut thoda sa bahar khisak jaata hai.

2. Pressure $P$ aur area $A$ — push ko pressure mein convert karna

Simple alfaz mein: gas ek point par push nahi karti — woh piston ki poori face par push karti hai. Pressure batata hai ki per unit surface kitna hard push hai; total force paane ke liye face area $A$ se multiply karo.
Picture: bahut saare chhote molecule-arrows piston face par hit kar rahe hain; unhe sabko add karo aur ek bada force arrow milta hai $F = P A$ .
Topic ko yeh kyun chahiye: hum piston ki force directly kabhi measure nahi karte. Hum pressure (gauge reading) aur piston area measure karte hain. Isliye $F = P A$ woh tarika hai jisse hum "jo hum jaante hain" ko "jo work ko chahiye" mein convert karte hain.

$A$ = piston face ka area, square metres mein. Yeh constant rehta hai (cylinder ek fixed width ka hota hai).

3. Volume $V$ aur swept-volume trick $A d x = d V$

Simple alfaz mein: jab piston thoda $d x$ bahar slide karta hai, gas ko naye space ka ek patla disc milta hai. Us disc ka volume = uska face area $A$ times uski thickness $d x$ .
Picture (red disc): ek coin-thin slab jo moving piston ne sweep kiya. Uska volume $A d x$ hai.
Topic ko yeh kyun chahiye — key swap: yeh humein distance ko volume se trade karne deta hai:

$d W = F d x = (P A) d x = P (A d x) = P d V$

Yahi poora master formula janam leta hai: tiny work pressure times tiny volume swept hai. Hum $d x$ ko $d V$ se swap karte hain kyunki pressure aur volume woh cheezein hain jo gas actually humein deti hai — aap piston ki position kabhi gauge se nahi padhte, lekin $P$ aur $V$ hamesha pata hota hai.

$d V$ = volume ka ek tiny sliver. $V_{1}$ = starting volume, $V_{2}$ = final volume.

4. Integral sign $\int$ — "saare slivers ko add karo"

Simple alfaz mein: lambaa patla "S" sirf ek fancy "Sum" hai. Chhote numbers $V_{1}$ (neeche) aur $V_{2}$ (upar) batate hain ki summing kahan se shuru hoti hai aur kahan khatam.
Picture: pressure curve ke neeche ka area, patli vertical rectangles mein kata hua. Har rectangle $P$ lamba aur $d V$ chauda hai, isliye uska area $P d V$ hai — exactly work ka ek tiny bit. Har rectangle add karo → total work → poora shaded area.
Yeh tool kyun aur plain multiplication kyun nahi? Kyunki $P$ badalta rehta hai jab gas expand karti hai (jaise gas phelti hai pressure drop hota hai). Agar pressure frozen hoti to hum sirf $P \times (V_{2} - V_{1})$ kar lete. Kyunki yeh nahi hai, humein rectangles itne patli chahiye ki har ek ke andar $P$ steady ho — exactly wahi ek integral karta hai.

5. Temperature $T$ , moles $n$ , gas constant $R$ — ideal gas law

$T$ = temperature kelvin (K) mein: yeh measure karta hai ki molecules kitni tezi se hil rahe hain. Zyada garam = zyada zyada thokna = zyada push.
$n$ = number of moles: gas particles ki ginti (ek mole ≈ $6 \times 1 0^{23}$ unka). Zyada gas = zyada pushers.
$R$ = universal gas constant $= 8.314 J K^{- 1} mol^{- 1}$ : ek fixed conversion number jo nature humein deti hai taaki pressure×volume ke units energy mein nikle.

Topic ko yeh kyun chahiye: master formula ko $P$ chahiye $V$ ke function ke roop mein. Gas law woh machine hai jo yeh deti hai. Rearrange karke, $P = \frac{n R T}{V}$ — aur wahi hum isothermal case mein integral mein substitute karte hain.

6. Logarithm $ln$ — " $1/ V$ add karne" ka jawab

Simple alfaz mein: jab pressure sliver ki height $\frac{n R T}{V}$ ho, to sum karne wali cheez $\frac{1}{V}$ hai har sliver par. $\frac{1}{V}$ ka running total koi normal power nahi hai — yeh logarithm hai.
Picture: $V_{1}$ aur $V_{2}$ ke beech curve $y = 1/ V$ ke neeche ka area; woh area $ln V_{2} - ln V_{1} = ln \frac{V _{2}}{V _{1}}$ ke barabar hai.
Topic ko yeh kyun chahiye: yeh wahi ek naya maths tool hai jo isothermal case demand karta hai. Iske bina aap $1/ V$ curve ke neeche ka area express nahi kar sakte.

$ln \frac{V _{2}}{V _{1}}$ positive hota hai jab $V_{2} > V_{1}$ (expansion → positive work), zero jab kuch nahi badlata, negative jab compress hota hai. Woh sign behaviour exactly wahi hai jo physics ko chahiye.

7. Gamma $γ$ aur $P V^{γ} = K$ — adiabatic curve

Simple alfaz mein: jab koi heat andar ya bahar nahi jaati (ek perfect blanket mein lapti gas), pressure aur volume gentle $P V = const$ curve nahi follow karte — woh zyada steep rule $P V^{γ} = K$ follow karte hain.
Picture: $P$ – $V$ diagram par adiabatic curve isothermal wale se zyada steeply dive karta hai, kyunki gas jaise expand karti hai woh thandi bhi hoti jaati hai, isliye pressure zyada tezi se girta hai.
Topic ko yeh kyun chahiye: $\int P d V$ ko adiabatically integrate karne ke liye humein phir se $P$ chahiye $V$ ke function ke roop mein. Yahan $P = K V^{- γ}$ hai — $V$ ki ek power — isliye integral ek power-rule integral hai, log se mushkil lekin phir bhi ho sakta hai.

Prerequisite map

Equipment checklist

Khud se test karo — sirf aloud jawab dene ke baad reveal karo.

Symbol

d x

ka plain words mein kya matlab hai

Ek tiny sliver of distance jitna piston move karta hai.

Force ko pressure aur area ke terms mein likho

F = P A

Hum

A d x

d V

se kyun swap kar sakte hain

Moving piston ek patla disc sweep karta hai jiska volume area × thickness

= A d x = d V

hai.

Integral sign

\int

actually kya karta hai

Path ko tiny bits mein kaata hai, har ek par

P d V

evaluate karta hai, aur unhe sabb add karta hai.

General mein hum sirf

P

(V_{2} - V_{1})

se kyun multiply nahi kar sakte

Kyunki

P

path ke saath badalta rehta hai, isliye har sliver ki alag height hoti hai aur use add karna padta hai, ek baar multiply nahi.

Ideal gas law state karo aur

P

ke liye rearrange karo

P V = n R T

, isliye

P = \frac{n R T}{V}

ln x

kis sawaal ka jawab deta hai

"e ko kis power par raise karne se

x

milega?" — aur yeh

1/ V

ko add karne ka total hai.

R

ki numerical value aur units

8.314 J K^{- 1} mol^{- 1}

γ

kya hai aur monatomic gas ke liye roughly uski value kya hai

Ratio

C_{p} / C_{v}

, lagbhag

1.667

(

= 5/3

P-V diagram par curve ke neeche ka area kya represent karta hai

Gas dwara kiya gaya work,

W = \int P d V

Connections

Parent: Work done in each process — yeh page har woh symbol build karta hai jo woh use karta hai
Hinglish version →
P-V Diagrams — jahan "area under the curve" rehta hai
Isothermal Process and Boyle's Law — jahan $ln$ tool use hota hai
Adiabatic Process and PV^gamma — $γ$ aur $P V^{γ} = K$ ka source
Internal Energy and Cv — $γ$ ke peeche $C_{v}$ define karta hai
First Law of Thermodynamics — jahan yeh $W$ values plug in hoti hain
Carnot Cycle — charon processes ko combine karta hai