Noesis
1.7.3Thermodynamics

Heat and internal energy — microscopic vs macroscopic

1,913 words9 min readdifficulty · medium6 backlinks
🇮🇳 Yeh note Hinglish mein padho →

WHAT are these quantities?

WHY two descriptions? (Microscopic vs Macroscopic)

Figure — Heat and internal energy — microscopic vs macroscopic

HOW: Deriving UU for an ideal gas from first principles

We want to connect microscopic motion to the macroscopic UU.

Step 1 — Pressure from molecular collisions. Consider NN molecules of mass mm in a cube of side LL (volume V=L3V=L^3). Take one molecule with xx-velocity vxv_x. When it bounces elastically off the right wall, its momentum changes by Δp=2mvx\Delta p = 2 m v_x.

Why this step? Pressure is force/area, and force is rate of momentum delivery to a wall — so we start from momentum transfer per bounce.

Step 2 — Time between hits on that wall. It travels 2L2L between successive hits on the same wall, so time between hits =2Lvx=\dfrac{2L}{v_x}.

Step 3 — Force from one molecule. F1=ΔpΔt=2mvx2L/vx=mvx2LF_1 = \frac{\Delta p}{\Delta t} = \frac{2 m v_x}{2L/v_x} = \frac{m v_x^2}{L}

Why this step? Average force = total momentum delivered ÷ time.

Step 4 — Sum over all molecules, use averages. P=FtotalL2=mL3∑ivx,i2=NmV⟨vx2⟩P = \frac{F_{\text{total}}}{L^2} = \frac{m}{L^3}\sum_i v_{x,i}^2 = \frac{N m}{V}\langle v_x^2\rangle By symmetry ⟨vx2⟩=⟨vy2⟩=⟨vz2⟩=13⟨v2⟩\langle v_x^2\rangle = \langle v_y^2\rangle=\langle v_z^2\rangle = \tfrac13\langle v^2\rangle, so PV=13Nm⟨v2⟩PV = \tfrac13 N m \langle v^2\rangle

Step 5 — Compare with ideal gas law PV=NkBTPV = N k_B T. 13Nm⟨v2⟩=NkBT  ⇒  12m⟨v2⟩=32kBT\tfrac13 N m\langle v^2\rangle = N k_B T \;\Rightarrow\; \tfrac12 m\langle v^2\rangle = \tfrac32 k_B T

Why this step? This is the bridge: it defines temperature as average translational KE.

Step 6 — Total internal energy. For a monatomic ideal gas (only translation, no intermolecular PE): U=N(12m⟨v2⟩)=32NkBT=32nRTU = N\left(\tfrac12 m\langle v^2\rangle\right) = \tfrac32 N k_B T = \tfrac32 n R T

Flashcards

Internal energy UU is (process / state) function?
State function — depends only on current state, not path.
Heat is energy transferred because of a
temperature difference across a boundary.
Why is "heat stored in a body" wrong?
Bodies store internal energy; heat only exists during transfer due to ΔT\Delta T.
Average translational KE per molecule equals
32kBT\tfrac32 k_B T.
Energy per degree of freedom (equipartition)
12kBT\tfrac12 k_B T.
UU for a monatomic ideal gas
U=32nRTU=\tfrac32 nRT (depends only on TT).
Why does ideal-gas UU depend only on TT?
No intermolecular forces ⇒ no potential-energy term ⇒ only KE, which is set by TT.
First law of thermodynamics
ΔU=Q−W\Delta U = Q - W (WW = work done by gas).
In isothermal ideal-gas expansion, ΔU=?\Delta U=? and Q=?Q=?
ΔU=0\Delta U=0; Q=WQ=W.
Pressure from kinetic theory
PV=13Nm⟨v2⟩PV=\tfrac13 Nm\langle v^2\rangle.
What links micro KE to macro TT?
12m⟨v2⟩=32kBT\tfrac12 m\langle v^2\rangle=\tfrac32 k_B T.
Recall Feynman: explain to a 12-year-old

Imagine a box full of tiny bouncing balls (molecules). Internal energy is the total bounciness — how fast and how much they're all moving. Temperature is just the average speed of one ball. Heat is what happens when you put your warm box next to a cold box: the fast balls bump the slow ones until both boxes are equally bouncy — energy flows across. That flowing energy is heat; the bounciness inside is internal energy. You'd never say "the box is full of flowing" — flowing is something that happens, not something stored. Same with heat!

Connections

Concept Map

described two ways

described two ways

tracks

measures

are averages of

summed give

is a

stored inside

crosses boundary via

crosses boundary via

process quantity changes

process quantity changes

wrongly says Q stored in body

Gas in a box

Microscopic view

Macroscopic view

Molecular KE and PE

T, P, V and U

Internal energy U

State function

Heat Q

Temperature difference

Work W

Ordered macroscopic force

Caloric misconception

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, yahan do alag cheezein hain jinko log aksar mix kar dete hain: internal energy (U) aur heat (Q). Internal energy matlab box ke andar saare molecules ki total energy — unki speed (kinetic energy) plus aapas ki potential energy. Ye ek "state function" hai, yaani sirf current state pe depend karti hai. Heat alag cheez hai — heat tab hoti hai jab temperature difference ki wajah se energy boundary cross karti hai. Body mein "heat stored" nahi hoti, body mein internal energy stored hoti hai. Heat to bas ek transfer ka naam hai, jaise paani ka behna.

Microscopic vs macroscopic ka matlab: micro level pe har molecule ki velocity track karni padegi (10^23 molecules!), jo impossible hai. Isliye hum macro language use karte hain — TT, PP, VV, UU. Kinetic theory derive karke nikalta hai ki 12m⟨v2⟩=32kBT\frac12 m\langle v^2\rangle = \frac32 k_B T, yaani temperature basically average molecular kinetic energy hai. Aur ideal gas ke liye U=32nRTU = \frac32 nRT — yani UU sirf TT pe depend karta hai, VV ya PP pe nahi (kyunki ideal gas mein intermolecular force nahi, to potential energy term zero).

Sabse important formula: First Law, ΔU=Q−W\Delta U = Q - W. Isko "bank" wala mnemonic se yaad rakho — UU bank balance hai, QQ (heat in) deposit, WW (work gas karta hai) withdrawal. Jaise isothermal expansion mein TT same rehta hai to ΔU=0\Delta U = 0, isliye Q=WQ = W — jitni heat andar aati hai utni hi work ban ke bahar nikal jaati hai. Yehi proof hai ki "heat in" ka matlab hamesha "internal energy badhi" nahi hota. Bas yeh teen baatein pakad lo aur poora thermodynamics khul jayega.

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Connections

First law of thermodynamics — dU = dQ − dW, sign conventionsPhysics · 1.7.13Kinetic theory — pressure derivation, temperature as mean KEPhysics · 1.7.9Equipartition theorem — ½k_BT per quadratic degree of freedomPhysics · 2.4.14γ = Cp - Cv — for monatomic, diatomic, polyatomicPhysics · 1.7.17Zeroth law — transitivity of thermal equilibriumPhysics · 1.7.2First law - ΔU = q + w (chemist sign convention)Chemistry · 2.5.3Heat capacities Cp, Cv; relationship Cp − Cv = nR (ideal gas)Chemistry · 2.5.6EfficiencyPhysics · 1.3.10Temperature — thermal equilibrium, thermometers, scalesPhysics · 1.7.1Specific heat capacity — calorimetryPhysics · 1.7.4Ideal gas law PV = nRT — derivation from kinetic theoryPhysics · 1.7.8