The Rubber Band — A Spring Made of Counting

The Rubber Band

Teach me

Speed

Two-state chain — every arrangement has equal energy

segment → right segment ← left

Guided Inquiry

1 · Forty links, one energy

Let’s meet the chain: forty segments, each pointing right or left along x. No segment stores energy — every arrangement has exactly the same energy. Press ▶ Play and watch: the chain jiggles, and the counter reports how many arrangements share its present length.

2 · Hanging the weight

Now for your first prediction. Suppose you hang a 4 pN weight and let the chain straighten. Remember, energy is identical for every arrangement. What do you think the arrangement counter does?

3 · Pull

Let’s pull. Drag the force slider from 0 up to 4 pN. The weight tautens the cord, and you’ll see the extension tile is now open. Watch the counter as the skyline straightens and the jiggling end settles about the ⟨x⟩ tick. What did the counter do?

4 · The count is the resistance

Now compare the counter with the coiled reference pinned beneath it: orders of magnitude gone, while ⟨x⟩ has climbed. Straightening only cost arrangements. No energy moved — so what, exactly, pulled back? Have a think about that one.

5 · Warm the loaded chain

Here’s a real puzzle. The 4 pN weight hangs at 300 K. Suppose you warmed the chain to 450 K with a hairdryer, weight untouched. What’s your call — does the hanging weight sag, stay, or lift?

6 · The hairdryer

Time to switch on the hairdryer. Drag T from 300 K to 450 K, weight untouched, watching the weight’s height, ⟨x⟩ and the counter. The force–extension plot is now open too, with the 300 K isotherm pinned. What did the weight do?

7 · Tension is proportional to T

Isn’t that surprising? Contraction lifted the weight while the counter climbed — the chain retreats to where the arrangements are. On the plot the live 450 K curve sits exactly ×1.50 (= T/300) above the pinned 300 K reference, and the hotter curve’s k_BT/ℓ marker sits higher; read both at r = 0.60 in the isotherm panel. So why do you think hotter jiggling favors the coil?

8 · The endpoint is a limit

One last experiment. Set f = 20 pN and T = 150 K: the counter reads 1 and S = 0, yet ⟨x⟩ stays strictly inside the contour — watch 1 − r stay finite in the extension tile. Ω = 1 needs f → ∞. And that’s the big idea: entropy is the spring.

Listen

Temperature T300 K

150 K600 K

k_BT = 4.14 pN·nm (1 pN·nm = 10⁻²¹ J)

Pulling force f (the hanging weight)0.0 pN

0 pN20 pN

The weight in the scene IS this force; at f = 0 the cord goes slack.

Segments N40

10100

Segment length ℓ = 1 nm is fixed, never a slider. Contour = Nℓ.

Arrangements

▾

Ω of the chain now—

—

Extension

▾

⟨x⟩ closed form—

x̄ running average—

—

Isotherm check r = 0.60

▾

live T2.871 pN

300 K pinned2.871 pN

ratio 1.000 = T / 300 K, exactly, at every r

Fixed-force (Gibbs) ensemble, N segments of length ℓ each pointing ±x, all arrangements of equal energy. At f = 0 the mean extension is exactly 0.

Microstate = which segments point right; macrostate = how many (n_R), which fixes the length. Ω is taken at the nearest-integer macrostate of the mean; S = 0 when Ω = 1 — the third law in Boltzmann's language.

The isotherm: temperature enters only as the prefactor k_BT/2ℓ, so the live curve is the 300 K reference scaled by exactly T/300 at every r. It diverges as r → 1.

Jiggle amplitude of the length: √N ℓ at zero force (6.32 nm, 15.8% of the contour at N = 40), collapsing under strong pull. The running average x̄ converges onto ⟨x⟩.

The Physics

The rubber band · what this sim shows, the physics behind it, and every control

At a glance

A stretched rubber band pulls back, yet this chain has no bonds to stretch: every arrangement of its segments has the same energy. The sim hangs a weight on it, counts its arrangements and warms it, so you can find where the pull comes from.

What you'll learn

How the length of the chain follows from how many segments point right.
How to count the arrangements Ω that share one length.
How the force needed to hold a given length depends on the temperature.
How the jiggling chain differs from its average length ⟨x⟩.

N segments of length ℓ = 1 nm. A weight pulls the free end over a pulley. (Drawn with 12 segments; the sim has 10 to 100.)

What's on screen

Segment → right and segment ← left (slate): the chain as a skyline, one step per segment. A flash marks the segment that just flipped.
Pulley and weight: the pulling force f. At f = 0 the cord goes slack.
Ruler: marks the contour Nℓ. The ⟨x⟩ tick (average length) and the x now arrowhead (length this instant) appear with the Extension panel.
Sidebar panels and force–extension plot: Arrangements, Extension and Isotherm check open as the Guided Inquiry reaches them.

See itRun the chain with the weight off, at 300 K. Watch for: segments flashing as they flip, and the free end wandering to and fro.

The controls

Press ⌖ on a row to see where that control sits in the sim.

Control	What it changes	Range
Temperature T	The temperature of the surroundings. The note under the slider gives k_BT in pN·nm.	150–600 K opens at 300 K
Pulling force f	The hanging weight. The weight in the scene is this force.	0–20 pN opens at 0 pN (4 pN in free explore)
Segments N	How many segments the chain has. Each is 1 nm long, so the contour is N nm.	10–100 opens at 40
Arrangements	Ω for the length on screen now, and n right of N. Later it adds S = k_B ln Ω and the count for the coil.	panel
Extension	⟨x⟩ from the formula, and x, the running average of the chain on screen.	panel
Isotherm check	The force the live-T curve and the 300 K curve need at r = 0.60, and their ratio.	panel
▶ Play / ⏸ Pause	Starts or freezes the flipping.	top bar
Speed	How fast the flips run on screen. It changes how soon x settles, not where.	0.25×–4×
↻ Reset	Back to 300 K, 4 pN, N = 40, with the chain restarted.	top bar

Fixed: ℓ = 1 nm. Forces are in piconewtons; k_BT = 4.14 pN·nm at 300 K. Any change to f, T or N restarts the run.

Many arrangements share one length

The microstate is which segments point right. The macrostate is only how many, n_R, and that alone fixes the length: x = (2n_R − N)ℓ. Segments are distinguishable, each has its own place along the chain, so the number of arrangements is a binomial coefficient:

S is the Boltzmann entropy of the macrostate on screen.

Two arrangements the chain visits at N = 40: a coil, and a stretched chain.

Ω peaks at n_R = N/2, the coil, x = 0: 1.38 × 10¹¹ arrangements. At x = 30 nm (n_R = 35) only 658 008 remain. Nothing separates their energies, so straightening costs arrangements and no energy.

Key idea. The length is set by n_R. Longer does not mean more arrangements: the count is largest at the coil.

See itTake the weight off, then hang it back on. Watch for: Ω dropping by orders of magnitude when the weight returns.

Entropy supplies the pull

At fixed temperature, equilibrium minimises F = U − TS. Here U is the same for every arrangement, so ∂U/∂x = 0 and only the entropy term is left. The force needed to hold the chain at length x is its slope:

Every piconewton of the force comes from the entropy term.

S is largest at x = 0. For x > 0 its slope is negative, so the pull points back toward the coil.
The slope steepens without limit toward x = Nℓ, because the last arrangements are the costly ones. The force diverges as r → 1; there is no yield point.
T converts the entropy slope into a force. No stiffness constant is hiding anywhere.

S/k_B = ln C(40, n_R). The tangent marks ⟨x⟩ = 29.87 nm at 4 pN and 300 K.

See itStep the force up in stages. Watch for: S = k_B ln Ω falling with every step up.

Force sets the average length

The sim fixes the pull f and lets x fluctuate. On average the chain settles where its entropic tension equals f. The segments are independent, so this holds exactly for every N:

f T N

These sliders move the sim too.

Everything depends on one group, fℓ/k_BT. For fℓ ≪ k_BT the chain is a Hookean spring, but its stiffness is proportional to T:

At 300 K and N = 40, k = 0.104 pN/nm; a longer chain is softer. At 4 pN, fℓ/k_BT = 0.97 is beyond the Hookean range: ⟨x⟩ = 29.87 nm, short of the 38.6 nm a spring would give.

See itRestart the chain at your settings. Watch for: x climbing onto the ⟨x⟩ value.

Heating shortens a loaded chain

T enters only through the prefactor, so the live curve is the 300 K curve scaled by T/300 at every r.

At fixed f, raising T raises the tension the chain supplies at every r. The weight can no longer hold the chain out as far, so it retreats toward the coil. Equivalently, r = tanh(fℓ/k_BT): warming acts like a weaker pull.

The 4 pN working point moves from r = 0.747 at 300 K to r = 0.567 at 450 K.

⟨x⟩ falls from 29.87 nm to 22.70 nm: the chain contracts 7.17 nm, 18% of its contour, and the weight rises.
Ω climbs from 6.58 × 10⁵ to 2.73 × 10⁸. The chain retreats to where the arrangements are.

This is the Gough–Joule effect. A heated metal rod lengthens instead: its restoring force comes from bond energy, not entropy.

See itHeat the loaded chain by half again. Watch for: ⟨x⟩ falling and Ω rising as the weight lifts.

One chain jiggles, a band averages

⟨x⟩ is an ensemble average. The chain on screen never sits at it: x now wanders by about σ_x while the ⟨x⟩ tick stays put. The running average x is a time average over this run; it settles onto ⟨x⟩ only after many flips, and any slider change restarts it.

The width of the length distribution. At f = 0 it is √N ℓ.

At N = 40 and f = 0, σ_x = 6.32 nm, 16% of the contour.
Relative to the contour the wobble falls as 1/√N: 32% at N = 10, 10% at N = 100.
A real band holds a vast number of such chains, so its length is the average, not one chain's wobble.

See itRun 10 segments, then 100, with the weight off. Watch for: the chain’s free end wandering across the dashed ⟨x⟩ line, by a larger share of the contour at N = 10.

Where it matters

Rubber. A cross-linked network of long polymer chains. Its pull is mostly entropic, which is why a loaded band shortens when heated. A small energetic part survives.
Stretch and warm. Pull a band quickly and touch it to your lip: it is warm. Straightening the chains lowers their entropy, and the band gives out heat.
Single molecules. Optical tweezers pull DNA and other polymers with piconewton forces. At low force their response is entropic, and the force scale k_BT/ℓ, 4.14 pN for this chain, marks where it stops behaving as a spring.

Go deeper: three dimensions and a paramagnet

In three dimensions a freely jointed chain replaces tanh by the Langevin function, coth u − 1/u, with u = fℓ/k_BT. At low force it is again Hookean, with k = 3k_BT/Nℓ².
The one-dimensional chain is the same mathematics as N spin-½ moments in a field: segments are spins, f plays the field, and ⟨x⟩ ∝ tanh is the magnetisation.

Summary

In one line. With equal energy for every arrangement, the chain's pull is entropic: the force is −T ∂S/∂x, so it grows in proportion to T.

Key points

Length is x = (2n_R − N)ℓ; Ω = C(N, n_R) is largest at the coil.
⟨x⟩ = Nℓ tanh(fℓ/k_BT), exact for any N.
The stiffness k = k_BT/Nℓ² grows with T, so a loaded chain contracts when heated.
One chain wobbles about ⟨x⟩ by σ_x, near 1/√N of its contour at f = 0.

Common mix-ups

"Heating expands things, and a stretched band stores energy in its bonds." Here all arrangements share one energy, and heating contracts the loaded chain.
"A longer chain has more arrangements." Ω is largest at n_R = N/2. Pushing n_R toward N collapses it.
"Pulled hard means fully stretched, so Ω = 1." Ω = 1 needs f → ∞. At 20 pN and 150 K the counter reads 1, yet 1 − r is still 8 × 10⁻⁹.

Things to try on your own

At 300 K and N = 40, find the force that gives ⟨x⟩ = 20 nm (the formula says 2.28 pN), then check it on the Extension panel.
Hold f = 4 pN and compare ⟨x⟩ at 150, 300, 450 and 600 K with the formula.
Set f = 0 and compare how far x now wanders at N = 10 and N = 100.

Predict before reveal	The temperature plot and isotherm panel stay closed until after the sag/stay/lift prediction is committed.
Name the faulty intuition	Each distractor is a real student model (thermal expansion; "no energy so no effect"; instant full stretch) and its feedback names it.
Invariant made visible	Energy is invariant across every arrangement; only the count changes — so the counter, not a force law, carries the explanation.
Operational numbers	Ω, S = k_B ln Ω, closed-form ⟨x⟩ and the live time-average x̄ are all read off the scene as it runs.
Honest limits	Ω = 1 is shown as an f → ∞ limit: 1 − r is displayed, and ⟨x⟩ never touches the contour mark.
Formal layer optional	The four governing equations live behind the ⚛︎ The Physics button, inside a full revision sheet; the sim works before the formalism is taught.