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ThermodynamicsIntroStarter track

Concept module

Temperature and Internal Energy

Compare average particle motion with whole-sample energy, vary amount and heating, and see why a phase-change shelf breaks naive temperature-only reasoning on one compact thermal bench.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 20.0 sLivePause to inspect a specific moment, then step or scrub through it.

0.00 s20.0 s

Temperature and Internal Energy

One bounded particle box keeps temperature, internal energy, amount of substance, and a simple phase-change shelf on the same honest bench instead of splitting them into separate widgets.

Graphs

Switch graph views without breaking the live stage and time link.

Temperature over time

Follow how the sample temperature changes in time for the current setup. A flat region does not automatically mean no energy is entering.

time (s): 0 to 20temperature (arb): 0 to 4

Temperature

Hover or scrub to link the graph back to the stage.time (s) / temperature (arb)

Controls

Adjust the physical parameters and watch the motion respond.

Particle count

N

Changes how much sample is present while keeping the same bounded particle-box picture.

Heater power

P_{heater}

2.6 u/s

Raises or lowers the energy input rate without changing the sample amount.

Starting temperature

T_{0}

2.4 arb

Changes the starting average particle motion for the same sample.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

MisconceptionPrompt 1 of 1

A flat temperature line does not mean the heater turned off. If the shelf is still filling, internal energy is still climbing even while temperature is nearly constant.

Try this

Use the on-the-shelf preset, scrub through the flat part of the temperature line, and compare it with the energy-breakdown graph and the shelf-progress bar.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

N

Particle count

Changes how much substance is present. At the same temperature, more particles mean more total internal energy and a slower temperature rise under the same heater.

Graph: Internal-energy bookkeepingGraph: Internal energy vs particle countGraph: Temperature rise rate vs particle countOverlay: Particle and temperature chipsOverlay: Internal-energy split

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Use one prompt at a time. Each one is there to keep temperature, amount, and internal-energy bookkeeping on the same compact bench.

MisconceptionPrompt 1 of 1

Graph: Temperature over time

A flat temperature line does not mean the heater turned off. If the shelf is still filling, internal energy is still climbing even while temperature is nearly constant.

Try this

Use the on-the-shelf preset, scrub through the flat part of the temperature line, and compare it with the energy-breakdown graph and the shelf-progress bar.

Why it matters

This is the simplest honest preview of phase change you can keep on one beginner bench.

Control: Heater powerGraph: Temperature over timeGraph: Internal-energy bookkeepingOverlay: Phase shelfOverlay: Internal-energy splitEquation

P_{heater}

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

4 visible

Overlay focus

Motion vectors

Shows short motion traces on representative particles so average microscopic motion stays visible.

What to notice

Faster-looking average motion corresponds to higher temperature, but not necessarily to the largest total internal energy.

Why it matters

It keeps temperature tied to average microscopic motion instead of to a disconnected thermometer icon.

Control: Starting temperatureGraph: Temperature over timeEquation

T_{0}

Challenge mode

Use the same particle bench for amount and shelf targets. The checks read the live metrics, so average motion, total energy, and compare state all stay honest.

0/2 solved

MatchStretch

1 of 8 checks

Same temperature, bigger store

Start from the small warm sample, switch to compare mode, and edit only Setup B until it keeps about the same temperature as Setup A but clearly stores much more internal energy.

Compare modeGraph-linkedGuided start2 hints

Suggested start

Leave Setup A as the baseline and edit only Setup B.

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Open the amount scan.

Temperature over time

Matched

Keep the internal-energy split visible.

Pending

Keep Setup A near the small-sample amount.

Pending

Keep Setup A near the warm baseline temperature.

Pending

Raise Setup B to a much larger amount.

Pending

Keep Setup B near the same temperature.

Pending

Make Setup B's internal energy clearly larger.

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the sample contains 18 particles with temperature 2.4 arb. The average thermal kinetic energy is 1.8 u per particle, while the total internal energy is 48.6 u. In this single-phase stretch, temperature tracks the average thermal kinetic energy per particle directly.

Equation detailsDeeper interpretation, notes, and worked variable context.

Average kinetic energy and temperature

Temperature is represented here by the average thermal kinetic energy per particle.

K_{avg} = α T

The constant $\alpha$ only sets the display scale in this bounded model.

The key idea is average per-particle motion, not a new independent energy pool.

T_{0}

Starting temperature 2.4 arb

Total thermal kinetic energy

The whole-sample thermal kinetic energy depends on both temperature and particle count.

K_{thermal} = N K_{avg}

N

Particle count 18

T_{0}

Starting temperature 2.4 arb

Internal-energy bookkeeping

Internal energy includes the thermal motion and the other microscopic stores represented on the bench.

U = K_{thermal} + U_{bond}

N

Particle count 18

T_{0}

Starting temperature 2.4 arb

Energy added by heating

The heater adds energy at a rate. That input can raise temperature directly or feed other internal stores.

Δ U_{in} = P_{heater} t

P_{heater}

Heater power 2.6 u/s

Direct temperature rise away from the shelf

When the input is going mainly into average particle motion, larger samples warm more slowly under the same heater power.

Δ T = \frac{Δ U _{in}}{N α}

N

Particle count 18

P_{heater}

Heater power 2.6 u/s

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

Starter track

Step 1 of 40 / 4 complete

Thermodynamics and Kinetic Theory

Next after this: Ideal Gas Law and Kinetic Theory.

1. Temperature and Internal Energy2. Ideal Gas Law and Kinetic Theory3. Heat Transfer4. Specific Heat and Phase Change

This concept is the track start.

See next concept

Short explanation

What the system is doing

Temperature and internal energy are related, but they are not the same thing. In this compact particle bench, temperature follows the average kinetic energy per particle, while internal energy counts the whole thermal story across the entire sample.

That means amount matters. Two samples can have the same temperature even though the larger sample stores more total internal energy because more particles are sharing that same average microscopic motion.

This is the bookkeeping underneath later heat transfer, specific heat, and phase-change ideas. Energy can enter a sample while temperature behaves in a less naive way because some of that added energy can go into changing internal stores instead of only speeding particles up.

Key ideas

01Temperature tracks the average microscopic motion per particle in this bounded model, not the total energy of the whole sample.

02Internal energy depends on both the temperature-like average motion and how much substance is present.

03At the same heater power, a smaller sample warms faster because the same incoming energy is shared across fewer particles.

04During a phase-change shelf, internal energy can keep increasing even while temperature stays nearly flat.

Live worked example

Solve the exact state on screen.

Use the current sample directly. The same particle bench, energy bars, and graphs drive each worked example, so amount, heating, and shelf behavior stay tied to one state.

Live valuesFollowing current parameters

For the current sample with $N = 18$ particles and temperature $2.4$ , how much thermal kinetic energy and total internal energy does that setup represent?

N

Particle count

T

Temperature

2.4 arb

K_{avg}

Average kinetic energy per particle

1.8 u

U_{bond}

Bond and phase store

16.2 u

1. Read the per-particle thermal scale

In this model, the current temperature corresponds to an average kinetic energy of

K_{avg} = 1.8 u

per particle.

2. Turn that per-particle scale into a total thermal part

Multiply by the amount of substance:

K_{thermal} = N K_{avg} = 32.4 u

3. Add the bond and phase store

The current non-kinetic store is

U_{bond} = 16.2 u

, so the total internal energy is

U = 48.6 u

Thermal total and internal total

K_{thermal} = 32.4 u, U = 48.6 u

This smaller sample can still have the same temperature because temperature follows the average particle motion, not the total amount of energy stored across all particles.

Common misconception

If two samples have the same temperature, they must contain the same amount of internal energy.

Equal temperature only tells you that their average microscopic kinetic energy per particle is similar.

The larger sample can still have much more total internal energy because more particles and internal stores are contributing to the total.

Mini challenge

Two samples start at the same temperature. Sample A has three times as many particles as Sample B, and both receive the same heater power for the same short time. Which sample ends with more internal energy, and which one changes temperature faster?

Prediction prompt

Decide whether total stored energy and temperature change must always rank the samples the same way.

Check your reasoning

The larger sample ends with more total internal energy, but the smaller sample changes temperature faster.

Temperature follows the per-particle energy scale, so the same heater input is diluted across more particles in the larger sample. Internal energy totals still favor the larger sample because more particles and internal stores are counted in the whole-sample total.

Quick test

Compare cases

Question 1 of 5

Answer from the particle bench, not from a memorized slogan. The goal is to separate average-motion reasoning from whole-sample energy reasoning.

Two samples have the same temperature, but one contains three times as many particles. Which statement is best?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a bounded particle box on the left and a thermal-state card on the right. The particle box uses moving dots to show average microscopic motion, while chips and horizontal bars summarize particle count, temperature, internal energy, and any phase-shelf progress.

The same sample state drives the graphs below the bench. One graph tracks temperature in time, another tracks the internal-energy bookkeeping, and two response graphs sweep particle count to compare whole-sample internal energy and temperature rise rate.

The model is intentionally simplified. It does not simulate real intermolecular forces or real substances in detail. It is a compact educational bench for separating temperature, amount of substance, heating rate, and a simple phase-change shelf without pretending to be a full thermodynamics engine.

Graph summary

The temperature-history graph shows how the current sample warms in time. A flat segment can still correspond to ongoing energy input, so it should be read together with the energy-breakdown graph and the shelf overlay.

The energy-breakdown graph tracks total internal energy, thermal kinetic energy, and the other internal store together. The amount-response graphs sweep only particle count, which makes them the cleanest way to compare same-temperature samples or to see why larger samples warm more slowly under the same heater.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Gas pressure bridgeThermodynamics

Temperature and Internal Energy

See each variable before you move it.

Motion vectors

Same temperature, bigger store

Thermodynamics and Kinetic Theory

What the system is doing

Solve the exact state on screen.

For the current sample with N=18 particles and temperature 2.4, how much thermal kinetic energy and total internal energy does that setup represent?

Two samples have the same temperature, but one contains three times as many particles. Which statement is best?

Keep moving through the concept map.

Ideal Gas Law and Kinetic Theory

Heat Transfer

Specific Heat and Phase Change

For the current sample with $N = 18$ particles and temperature $2.4$ , how much thermal kinetic energy and total internal energy does that setup represent?