Heat and internal energy — microscopic vs macroscopic: Foundations

This page is the toolbox. The parent note fires off symbols like $v_{x}$ , $⟨ v^{2} ⟩$ , $k_{B}$ , $Δ U = Q - W$ as if you already own them. Here we earn every single one, from zero, in an order where each rests on the one before.

0. The picture we keep coming back to

Everything below lives inside one image: a cube of side $L$ with a handful of little balls (molecules) darting around, bouncing off the walls.

Figure — Heat and internal energy — microscopic vs macroscopic

Keep this cube in your head. Every symbol we define is either a property of one ball, a property of the crowd, or a property of the box.

1. Length, area, volume — the box itself

Why the topic needs it: pressure is a force spread over the wall area $A$ , and internal energy is shared among the molecules packed into the volume $V$ . You cannot talk about "how crowded" or "how hard they push" without these.

2. Counting the crowd — $N$ , $n$ , $R$ , $k_{B}$

There are two ways to count molecules, and two matching constants. This trips everyone up, so we go slow.

Micro-fact

N

counts molecules and pairs with

k_{B}

Macro-fact

n

counts moles and pairs with

R

3. Position and velocity of one ball — $r$ , $v$ , $v_{x}$

Now we describe the motion of a single molecule.

Why the topic needs it: when a ball hits the right wall, only its $x$ -motion matters (it's what carries it toward that wall). So the pressure derivation is built entirely on $v_{x}$ , not the full $v$ . Splitting into components is the tool that lets us handle "one wall at a time."

U

(state function).

Transfer-by-temperature symbol

Q

(heat).

Transfer-by-piston symbol

W

(work).

How the foundations feed the topic

Follow the arrows: geometry + counting + one ball's motion → averages → pressure → the ideal gas law → temperature → internal energy → the first law. Each node is a symbol you now own.

Where each foundation goes next

Counting & averaging power the whole Kinetic Theory of Gases.
"Energy per direction" becomes the Equipartition Theorem — $\frac{1}{2} k_{B} T$ per degree of freedom.
How many directions a molecule can store energy in is set by its Degrees of Freedom and Molecular Structure.
The stored-vs-transferred split powers Specific Heats Cv and Cp and Isothermal and Adiabatic Processes.

Equipment checklist

V

in terms of

L

V = L^{3}

; area of one wall is

A = L^{2}

Difference between

N

and

n

N

counts molecules;

n

counts moles;

N = n N_{A}

Which constant pairs with

N

, which with

n

k_{B}

with

N

(per molecule);

R

with

n

(per mole);

R = N_{A} k_{B}

What

v_{x}

the part of a molecule's velocity pointing along the

x

-axis (toward the right wall).

How speed relates to components

v^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}

Meaning of

\sum_{i}

add the quantity over every molecule

i = 1 \dots N

Meaning of

⟨ ⟩

the average — sum over all molecules divided by

N

Why

⟨ v_{x}^{2} ⟩ = \frac{1}{3} ⟨ v^{2} ⟩

the three directions share the speed-squared equally.

Momentum change when a ball bounces off a wall

Δ p = 2 m v_{x}

(velocity flips sign).

Force in terms of momentum

F = Δ p /Δ t

, momentum delivered per second.

Pressure in terms of force

P = F / A

, force per unit wall area.

Deep meaning of temperature

T

the average translational kinetic energy of one molecule.

What is stored vs what crosses the boundary

U

is stored (state function);

Q

and

W

only exist during transfer.

The first-law bookkeeping

Δ U = Q - W

Foundations — Heat and internal energy — microscopic vs macroscopic

0. The picture we keep coming back to

1. Length, area, volume — the box itself

2. Counting the crowd — $N$ , $n$ , $R$ , $k_{B}$

3. Position and velocity of one ball — $r$ , $v$ , $v_{x}$

4. The subscript $i$ and the sum $\sum$

5. The angle bracket $⟨ ⟩$ — averaging

6. Momentum $p$ and its change $Δ$

7. Force $F$ , pressure $P$ , temperature $T$ — the macroscopic dials

8. Energy — the star of the show ( $K E$ , $U$ , $Q$ , $W$ )

How the foundations feed the topic

Where each foundation goes next

Equipment checklist