Answer key · Unit 5

States of matter and gases.

Each lesson’s practice questions, their correct answers and the mistakes they spot. Answers read off the Scene use the same setting the learner sees.

Misconception catalogue

Answer key 5

Lesson 1 of 5 · About 20 minutes

The particle model

All matter is made of particles that never stop moving: how their arrangement and motion make a solid, a liquid and a gas, and why a gas takes a thousand times more room.

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The 5 steps and their answers

Step 1 of 5
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Three states
Practice · Choose
Question
Which state of matter has a fixed volume but no fixed shape: it takes the shape of whatever holds it?

Correct answer
- Correct Liquid
Setup, choices and feedback
The Scene at this step
Temperature
253.2 K
Zoom to the molecules
Surface
Choices
Shown in a random order.
- Correct Liquid
- Wrong Solid
  Feedback
  A solid has a fixed shape too: an ice cube stays a cube in any glass.
- Wrong Gas
  Feedback
  A gas has no fixed volume either: it spreads to fill any container, and it can be squeezed.
Feedback for a right answer
In a liquid the particles are packed close together, as in a solid, so it can hardly be squeezed. But they are not tied to fixed places: they slide past each other, so the liquid flows and takes the shape of its container.

Hint
Think of water poured from a narrow glass into a wide bowl.

Worked solution
1. Solid: the particles are close together and fixed in place, so it has a fixed shape and a fixed volume.
2. Liquid: the particles are close together but slide past each other, so the volume is fixed and the shape is not.
3. Gas: the particles are far apart and move freely, so it has neither a fixed shape nor a fixed volume.
The answer: a liquid.
Step 2 of 5
Open this step
Inside the ice
Practice · Predict, then look
Question
The ice in the Scene is at −20 °C. What are the water molecules inside it doing? Choose, and the Scene will zoom in on them.

Correct answer
- Correct Vibrating in place, each about a fixed position
Setup, choices and feedback
The Scene at this step
Temperature
253.2 Kfixed
Zoom to the molecules
Surfacefixed
Choices
- Correct Vibrating in place, each about a fixed position
- Wrong Standing still, not moving at all
  Misconception: Thinks the particles of a solid stand completely still.
  Feedback from the catalogue
  The particles of a solid do not stand still: each one vibrates about its fixed place in the lattice. What sets a solid apart from a liquid is that its particles do not leave their places, not that they do not move. Only at absolute zero, 0 K, would the motion almost stop, and nothing is ever there.
- Wrong Moving freely all through the cube
  Feedback
  If the molecules roamed freely, the ice would flow like a liquid. In ice each molecule is held to its place in the lattice.
Once the prediction is locked in, the Scene changes
Zoom to the molecules
Moleculesfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Each molecule vibrates about its place in the lattice, and none leaves it. The warmer the ice, the stronger the vibration, until at the melting point it shakes the molecules loose from the lattice.

Worked solution
1. Any substance above absolute zero has kinetic energy, so its particles move.
2. In a solid the attraction between the particles holds each one in a fixed place.
3. So the only motion it can have is a vibration about that place.
If you chose “standing still”, you took a solid to be matter without motion. At −20 °C, which is 253 K, there is still plenty of energy.
Step 3 of 5
Open this step
Kelvin and Celsius
Practice · Calculate
Question
The Scene shows the temperature in kelvin: it is now 293 K. How much is that in degrees Celsius?

Correct answer
20 °C ± 0.5 °C
Setup, choices and feedback
The Scene at this step
Temperature
293.1 Kfixed
Zoom to the molecules
Surface
Accepted within
± 0.5 °C
Computed from the Scene at this step’s setting
Temperature/nodes/sample/props/temperature
Any unit of the same quantity is accepted; it is converted before checking.
Feedback for a right answer
$293 - 273 = 20$ ⁠, that is 20 °C: room temperature. One kelvin is the same step as one degree Celsius; only the zero differs. The kelvin scale’s zero is absolute zero, −273 °C.

Hint
The two scales differ by a constant: $T = t + 273$ ⁠.

Worked solution
1. The link: $T\,[\mathrm{K}] = t\,[^\circ\mathrm{C}] + 273$ ⁠.
2. So $t = T - 273$ ⁠.
3. Substitute: $t = 293 - 273 = 20$ ⁠.
4. The answer: 20 °C.
If you got 566 °C, you added 273 instead of subtracting it: going from kelvin to Celsius the number goes down.
Step 4 of 5
Open this step
Melt it
Practice · Reach the target
Question
The ice is frozen at 253 K. Change Temperature until the whole sample is liquid: no ice and no vapour. Then press Check.

Goal
- 1 ± 0.001
  At the start of the step: 0
Setup, choices and feedback
The Scene at this step
Temperature
253.2 K
Zoom to the molecules
Molecules
Read from the Scene at
Share of the sample liquid, 0..1/nodes/sample/outputs/liquid
One setting that reaches it
Temperature
293.1 K
Feedback for a right answer
Any temperature from 293 K up to the boiling point, 373 K, gives liquid water only. In the Scene melting is spread over about 20 degrees so you can watch it; real ice melts entirely at exactly 0 °C, as long as heat keeps coming.

Feedback for a wrong answer
Not yet. If there is ice left, warm it; if there is already vapour, cool it.

Hint
Water is liquid between its melting point, 273 K, and its boiling point, 373 K. Keep a little away from both.

Worked solution
1. Water melts at 273 K (0 °C) and boils at 373 K (100 °C).
2. Between them it is a liquid.
3. Set Temperature to about 320 K, in the middle of the range.
Step 5 of 5
Open this step
Far apart
Practice · Choose
Question
Steam takes up about 1700 times the volume of the water it came from. Why?

Correct answer
- Correct The molecules are much further apart, with nothing between them
Setup, choices and feedback
The Scene at this step
Temperature
393.1 K
Zoom to the molecules
Molecules
Choices
Shown in a random order.
- Correct The molecules are much further apart, with nothing between them
- Wrong Each molecule has swollen with the heat
  Misconception: Thinks a substance expands, melts or boils because its particles themselves grow or melt.
  Feedback from the catalogue
  The particles themselves do not grow, soften or melt. A water molecule in ice, in water and in steam is the same molecule, $\mathrm{H_2O}$ ⁠, the same size. What changes is how fast the particles move and how far apart they are: in steam they are about ten times further apart.
- Wrong Air has got in between the molecules
  Misconception: Thinks the space between the particles of a gas is filled with air or some other substance.
  Feedback from the catalogue
  Between the particles of a gas there is nothing: empty space. Air is itself a mixture of particles, mostly $\mathrm{N_2}$ ⁠ and $\mathrm{O_2}$ ⁠, and between them there is nothing. That is why a gas can be squeezed: you squeeze the empty space between the particles, not the particles.
- Wrong There are more molecules
  Feedback
  The mass does not change when water boils, so neither does the number of molecules: the same molecules, further apart.
Feedback for a right answer
In a gas the molecules are about ten times further apart than in a liquid, so the volume is about a thousand times larger ( $10^3$ ⁠). The molecules themselves have not changed, and between them is empty space.

Hint
In a change of state a molecule stays the same molecule. What else can change?

Worked solution
1. When water boils, $\mathrm{H_2O}$ ⁠ molecules go from liquid to gas and stay $\mathrm{H_2O}$ ⁠.
2. The mass and the number of molecules do not change.
3. What changes is the distance between them: about 12 times larger, and $12^3 \approx 1700$ ⁠.
There is no air between the molecules: air is itself a gas of other molecules.

Lesson 2 of 5 · About 25 minutes

Changes of state

Melting, boiling, evaporation, condensation and sublimation: what happens to the particles in each change, where the energy goes, and why the temperature stands still while water boils.

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The 5 steps and their answers

Step 1 of 5
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Six changes
Practice · Choose
Question
Dry ice, solid carbon dioxide, turns straight into gas at room temperature without passing through a liquid. What is this change called?

Correct answer
- Correct Sublimation
Setup, choices and feedback
The Scene at this step
Temperature
283.1 K
Zoom to the molecules
Surface
Choices
Shown in a random order.
- Correct Sublimation
- Wrong Evaporation
  Feedback
  Evaporation is from liquid to gas. Dry ice has no liquid stage at one atmosphere.
- Wrong Melting
  Feedback
  Melting is from solid to liquid.
- Wrong Condensation
  Feedback
  Condensation goes the other way: from gas to liquid, like drops on a bathroom mirror.
Feedback for a right answer
Sublimation: from solid straight to gas. The reverse, from gas straight to solid, is deposition, which is how frost forms on a window on a cold night.

Worked solution
1. Solid → liquid: melting. Liquid → solid: freezing.
2. Liquid → gas: evaporation (and boiling). Gas → liquid: condensation.
3. Solid → gas directly: sublimation. Gas → solid: deposition.
Dry ice goes from solid to gas, so this is sublimation.
Step 2 of 5
Open this step
Melting takes energy
Practice · Calculate
Question
Melting 1 kg of ice at 0 °C takes 334 kJ, and after melting the water is still at 0 °C. Warming 1 kg of water by one degree takes 4.18 kJ. By how many degrees would that same energy warm 1 kg of liquid water? Answer in degrees.

Correct answer
79.9 ± 1
Setup, choices and feedback
The Scene at this step
Temperature
273.1 K
Zoom to the molecules
Molecules
Accepted within
± 1
A fixed value in the question
Wrong answers it recognises
- 0 ± 0.5
  Misconception: Thinks the temperature keeps rising while a substance melts or boils as long as it is heated.
  Feedback
  Zero is what happens during melting. Here the question is how far the same energy would warm water that has already melted.
Feedback for a right answer
$\dfrac{334}{4.18} \approx 80$ ⁠ degrees. Melting takes nearly all the energy it takes to warm the water from 0 °C to 80 °C, and yet the thermometer does not move. That is why ice cools a drink so well.

Hint
Divide the melting energy by the energy for one degree.

Worked solution
1. Energy to melt 1 kg of ice: 334 kJ.
2. Energy per degree for 1 kg of water: 4.18 kJ.
3. Degrees: $\dfrac{334}{4.18} \approx 79.9$ ⁠.
4. The answer: about 80 degrees.
Step 3 of 5
Open this step
Boiling
Practice · Choose
Question
What are the big bubbles rising from a pan of boiling water made of?

Correct answer
- Correct Water vapour
Setup, choices and feedback
The Scene at this step
Temperature
373.1 K
Zoom to the molecules
Surface
Choices
Shown in a random order.
- Correct Water vapour
- Wrong Air that was dissolved in the water
  Misconception: Thinks the bubbles in boiling water are air, or hydrogen and oxygen split from the water.
  Feedback from the catalogue
  The big bubbles rising through boiling water are water vapour: water turned to gas inside the liquid. Boiling is a change of state, not a chemical reaction, so the molecules stay $\mathrm{H_2O}$ ⁠ and do not split into hydrogen and oxygen. The air dissolved in the water comes out earlier, as small bubbles on the sides, well before it boils.
- Wrong Hydrogen and oxygen released from the water
  Misconception: Thinks the bubbles in boiling water are air, or hydrogen and oxygen split from the water.
  Feedback from the catalogue
  The big bubbles rising through boiling water are water vapour: water turned to gas inside the liquid. Boiling is a change of state, not a chemical reaction, so the molecules stay $\mathrm{H_2O}$ ⁠ and do not split into hydrogen and oxygen. The air dissolved in the water comes out earlier, as small bubbles on the sides, well before it boils.
- Wrong Nothing: empty holes in the water
  Feedback
  An empty bubble would be crushed at once by the pressure of the water and air above it. It is full of a gas whose pressure holds it open.
Feedback for a right answer
Boiling is evaporation from all through the liquid: at the boiling point water molecules turn to gas inside the water too, forming bubbles of vapour. The molecules stay $\mathrm{H_2O}$ ⁠.

Hint
Boiling is a change of state. What turns from liquid into gas?

Worked solution
1. When water boils it goes from liquid to gas: a change of state, not a reaction.
2. In a change of state the molecules do not break apart, so there is no hydrogen or oxygen here.
3. The gas in the bubbles is therefore water in gas form: water vapour.
Small air bubbles appear on the sides of the pan while it heats, and they run out long before it boils.
Step 4 of 5
Open this step
The temperature stands still
Practice · Choose
Question
A pan of water is boiling over a flame at sea level. You turn the flame up and wait five more minutes, and there is still water in the pan. What is the water’s temperature?

Correct answer
- Correct 100 °C
Setup, choices and feedback
The Scene at this step
Temperature
373.1 K
Zoom to the molecules
Surface
Choices
Shown in a random order.
- Correct 100 °C
- Wrong More than 100 °C, because it was heated more
  Misconception: Thinks the temperature keeps rising while a substance melts or boils as long as it is heated.
  Feedback from the catalogue
  As long as there is water boiling, its temperature stays at 100 °C (at one atmosphere), even over a fierce flame. The energy from the flame goes into breaking the attraction between the molecules, not into speeding them up. A stronger flame only boils the water away faster. The same goes for melting: ice and water together stay at 0 °C.
- Wrong Less than 100 °C, because the hot water has boiled off
  Feedback
  Water left in the pan is still boiling, so it is at its boiling point. It does not cool while the flame is on.
Feedback for a right answer
As long as there is water boiling, it stays at 100 °C. The extra energy from the flame goes into pulling the molecules apart, and the water just boils away faster. What raises a boiling point is pressure, as in a pressure cooker, not heat.

Worked solution
1. At sea level water boils at 100 °C.
2. While it boils, the energy coming in goes into breaking the attraction between the molecules, not into speeding them up.
3. Temperature measures the average speed of that motion, so it does not change.
4. The answer: 100 °C, until the last of the water boils off.
Step 5 of 5
Open this step
Evaporation without boiling
Practice · Choose
Question
A puddle dries up on a 25 °C day, far below the boiling point. How does the water leave?

Correct answer
- Correct The fastest molecules near the surface escape into the air
Setup, choices and feedback
The Scene at this step
Temperature
323.1 K
Zoom to the molecules
Molecules
Choices
Shown in a random order.
- Correct The fastest molecules near the surface escape into the air
- Wrong Water does not evaporate below 100 °C; it only soaks into the ground
  Misconception: Thinks a liquid can only turn into gas at its boiling point.
  Feedback from the catalogue
  A liquid evaporates at any temperature, from its surface. At any moment some molecules are faster than average, and those near the surface can break free. That is how a puddle dries on a 25 °C day. Boiling is something else: evaporation from all through the liquid, in bubbles, which starts only at the boiling point.
- Wrong The water splits into hydrogen and oxygen in the sun
  Misconception: Thinks the bubbles in boiling water are air, or hydrogen and oxygen split from the water.
  Feedback from the catalogue
  The big bubbles rising through boiling water are water vapour: water turned to gas inside the liquid. Boiling is a change of state, not a chemical reaction, so the molecules stay $\mathrm{H_2O}$ ⁠ and do not split into hydrogen and oxygen. The air dissolved in the water comes out earlier, as small bubbles on the sides, well before it boils.
Feedback for a right answer
At any temperature some molecules are much faster than average. Those near the surface and moving upwards can break free of their neighbours’ pull: that is evaporation. The warmer and windier it is, the faster it goes.

Worked solution
1. Temperature sets the average speed of the molecules; some are faster and some slower.
2. A fast enough molecule at the surface can leave for the air.
3. That happens at any temperature, slowly: evaporation. In boiling it happens inside the liquid too.

Lesson 3 of 5 · About 30 minutes

Pressure and the gas laws

Where a gas’s pressure comes from, and three laws that follow from the particles: half the volume gives twice the pressure, twice the kelvin temperature gives twice the pressure, and twice the gas gives twice the pressure.

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The 6 steps and their answers

Step 1 of 6
Open this step
What pressure is
Practice · Choose
Question
Where does the pressure a gas puts on the walls of its box come from?

Correct answer
- Correct From the hits of many particles on the walls, every moment
Setup, choices and feedback
The Scene at this step
Temperature
300 K
Volume
2 L
Amount of gas
200 particles
Show the speed curve
Particles
Choices
Shown in a random order.
- Correct From the hits of many particles on the walls, every moment
- Wrong From the weight of the gas pressing down
  Feedback
  The weight of the gas in a box is tiny, and the pressure pushes just as hard on the top wall as on the bottom and the sides.
- Wrong From the particles swelling and pushing the walls
  Misconception: Thinks a substance expands, melts or boils because its particles themselves grow or melt.
  Feedback from the catalogue
  The particles themselves do not grow, soften or melt. A water molecule in ice, in water and in steam is the same molecule, $\mathrm{H_2O}$ ⁠, the same size. What changes is how fast the particles move and how far apart they are: in steam they are about ten times further apart.
- Wrong From the air between the particles
  Misconception: Thinks the space between the particles of a gas is filled with air or some other substance.
  Feedback from the catalogue
  Between the particles of a gas there is nothing: empty space. Air is itself a mixture of particles, mostly $\mathrm{N_2}$ ⁠ and $\mathrm{O_2}$ ⁠, and between them there is nothing. That is why a gas can be squeezed: you squeeze the empty space between the particles, not the particles.
Feedback for a right answer
Each particle that hits the wall and bounces off gives it a brief push. Billions of hits a second add up to a steady force, and the force on each unit of area is the pressure. So the pressure rises when there are more hits or when each hit is harder.

Worked solution
1. In a gas the particles move fast and in every direction.
2. When a particle bounces off the wall its momentum reverses, and the wall gets a push.
3. All the pushes in a second, per unit of area, add up to the pressure: $p = \dfrac{F}{A}$ ⁠.
Step 2 of 6
Open this step
Squeeze
Practice · Predict, then look
Question
The box holds gas at one atmosphere, in 2 L. Temperature and amount of gas are fixed. What will the pressure be when you push the wall in to 1 L?

Correct answer
2 atm ± 0.05 atm
A number prediction counts as right within ± 0.05 atm
Setup, choices and feedback
The Scene at this step
Temperature
300 Kfixed
Volume
2 L
Amount of gas
200 particlesfixed
Show the speed curve
Particles
Once the prediction is locked in, the Scene changes
Volume
1 L
The Scene then shows: 2 atm · Ideal-gas pressure p = N k T / V of the molecules the particles stand for/nodes/gas/outputs/pressure
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Two atmospheres: half the volume, twice the pressure. That is Boyle’s law: at constant temperature $pV$ ⁠ stays the same, $1 \times 2 = 2 \times 1$ ⁠.

Hint
In a room half the size, each particle reaches the wall twice as often.

Worked solution
1. At constant temperature and amount: $p_1 V_1 = p_2 V_2$ ⁠ (Boyle’s law).
2. The givens: $p_1 = 1\ \mathrm{atm}$ ⁠, $V_1 = 2\ \mathrm{L}$ ⁠, $V_2 = 1\ \mathrm{L}$ ⁠.
3. $p_2 = \dfrac{p_1 V_1}{V_2} = \dfrac{1 \times 2}{1} = 2$ ⁠.
4. The answer: 2 atm.
If you guessed half an atmosphere, you turned the ratio upside down: pressure and volume are inversely related.
Step 3 of 6
Open this step
Boyle’s law
Practice · Calculate
Question
At 2 L the pressure was one atmosphere. Now the wall is at 1.6 L, at the same temperature and with the same gas. What is the pressure?

Correct answer
1.25 atm ± 0.01 atm
Setup, choices and feedback
The Scene at this step
Temperature
300 Kfixed
Volume
1.6 Lfixed
Amount of gas
200 particlesfixed
Show the speed curve
Particles
Accepted within
± 0.01 atm
Computed from the Scene at this step’s setting
Ideal-gas pressure p = N k T / V of the molecules the particles stand for/nodes/gas/outputs/pressure
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 0.8 atm ± 0.01 atm
  Misconception: Thinks squeezing a gas into less room lowers its pressure in proportion, as if pressure followed volume.
  Feedback from the catalogue
  At constant temperature pressure and volume are inversely related (Boyle’s law): $pV$ ⁠ stays the same. When the volume halves, the same particles hit each square centimetre of wall twice as often, and the pressure doubles. So $p_2 = p_1 \dfrac{V_1}{V_2}$ ⁠, not $p_1 \dfrac{V_2}{V_1}$ ⁠.
Feedback for a right answer
$p_2 = \dfrac{1 \times 2}{1.6} = 1.25$ ⁠ atmospheres. A volume 20% smaller raised the pressure by 25%: the ratio inverts, the difference does not stay equal.

Hint
$p_1 V_1 = p_2 V_2$ ⁠.

Worked solution
1. Boyle’s law: $p_1 V_1 = p_2 V_2$ ⁠.
2. The givens: $p_1 = 1\ \mathrm{atm}$ ⁠, $V_1 = 2\ \mathrm{L}$ ⁠, $V_2 = 1.6\ \mathrm{L}$ ⁠.
3. $p_2 = \dfrac{1 \times 2}{1.6} = 1.25$ ⁠.
4. The answer: 1.25 atm.
If you got 0.8, you multiplied by $\dfrac{V_2}{V_1}$ ⁠: a smaller volume does not lower the pressure, it raises it.
Step 4 of 6
Open this step
Heat it
Practice · Reach the target
Question
The box is at 300 K and one atmosphere. Volume and Amount of gas are fixed. Change Temperature until the pressure is exactly 2 atm, then press Check.

Goal
- 2 atm ± 0.005 atm
  At the start of the step: 1 atm
Setup, choices and feedback
The Scene at this step
Temperature
300 K
Volume
2 Lfixed
Amount of gas
200 particlesfixed
Show the speed curve
Particles
Read from the Scene at
Ideal-gas pressure p = N k T / V of the molecules the particles stand for/nodes/gas/outputs/pressure
One setting that reaches it
Temperature
599 K
Feedback for a right answer
At 600 K: twice the absolute temperature, twice the pressure (Gay-Lussac’s law). In Celsius that is from 27 °C to 327 °C, far more than “twice”.

Feedback for a wrong answer
Not yet. Work it out: $\dfrac{p_2}{p_1} = \dfrac{T_2}{T_1}$ ⁠, with $T$ ⁠ in kelvin.

Hint
At constant volume the pressure is proportional to the temperature in kelvin.

Worked solution
1. At constant volume and amount: $\dfrac{p_1}{T_1} = \dfrac{p_2}{T_2}$ ⁠.
2. $T_2 = T_1 \dfrac{p_2}{p_1} = 300 \times \dfrac{2}{1} = 600$ ⁠.
3. Set Temperature to 600 K.
Step 5 of 6
Open this step
Why kelvin
Practice · Calculate
Question
A sealed, rigid can holds gas at 27 °C and one atmosphere. It is heated to 127 °C. What is the pressure now?

Correct answer
1.333 atm ± 0.01 atm
Setup, choices and feedback
The Scene at this step
Temperature
300 K
Volume
2 Lfixed
Amount of gas
200 particlesfixed
Show the speed curve
Particles
Accepted within
± 0.01 atm
A fixed value in the question
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 4.704 atm ± 0.02 atm
  Misconception: Uses degrees Celsius instead of kelvin in a gas-law ratio.
  Feedback from the catalogue
  In the gas laws pressure and volume are proportional to the absolute temperature, in kelvin, measured from absolute zero. At 0 °C a gas still has a pressure, so you cannot divide by degrees Celsius. Convert first: $T = t + 273$ ⁠. From 27 °C to 327 °C the temperature goes from 300 K to 600 K: twice as much, not twelve times.
Feedback for a right answer
$300\ \mathrm{K} \to 400\ \mathrm{K}$ ⁠, so $p_2 = 1 \times \dfrac{400}{300} \approx 1.33$ ⁠ atmospheres. You can check it in the Scene: 400 K at 2 L.

Hint
Convert both temperatures to kelvin before you divide.

Worked solution
1. Constant volume: $\dfrac{p_2}{p_1} = \dfrac{T_2}{T_1}$ ⁠, with $T$ ⁠ in kelvin.
2. $T_1 = 27 + 273 = 300\ \mathrm{K}$ ⁠, $T_2 = 127 + 273 = 400\ \mathrm{K}$ ⁠.
3. $p_2 = 1 \times \dfrac{400}{300} \approx 1.33$ ⁠.
4. The answer: 1.33 atm.
If you got 4.7, you divided $\dfrac{127}{27}$ ⁠ in degrees Celsius. A gas at 0 °C still has a pressure, so the Celsius zero does not work in a ratio.
Step 6 of 6
Open this step
More gas
Practice · Predict, then look
Question
The box holds 200 particles at 300 K in 2 L: one atmosphere. What happens to the pressure if you double Amount of gas to 400 particles, without changing temperature or volume?

Correct answer
- Correct It doubles
Setup, choices and feedback
The Scene at this step
Temperature
300 Kfixed
Volume
2 Lfixed
Amount of gas
200 particles
Show the speed curve
Particles
Choices
- Correct It doubles
- Wrong It stays the same, because the temperature has not changed
  Feedback
  Temperature sets how hard each hit is. But twice the particles make twice the hits.
- Wrong It goes up four times
  Feedback
  Each particle hits as often and as hard as before; only the number of particles has doubled.
Once the prediction is locked in, the Scene changes
Amount of gas
400 particles
The Scene then shows: 2 atm · Ideal-gas pressure p = N k T / V of the molecules the particles stand for/nodes/gas/outputs/pressure
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Twice the particles, twice the hits a second, twice the pressure: 2 atm. That is how a pump inflates a tyre: it adds gas, and the pressure rises.

Worked solution
1. At constant temperature and volume, the pressure is proportional to the number of particles: $p \propto n$ ⁠.
2. The number of particles doubles.
3. So the pressure doubles: from 1 atm to 2 atm.

Lesson 4 of 5 · About 25 minutes

Molar volume

How many moles the box holds, how much room one mole of gas takes, and why a mole of hydrogen and a mole of carbon dioxide fill the same balloon.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
Counting moles
Practice · Calculate
Question
The box is at 300 K, 2 L and one atmosphere, 101 325 Pa. How many moles of gas does it hold? Use $pV = nRT$ ⁠ with $R = 8.314\ \mathrm{J/(mol \cdot K)}$ ⁠.

Correct answer
0.08124 mol ± 2%
Setup, choices and feedback
The Scene at this step
Temperature
300 Kfixed
Volume
2 Lfixed
Amount of gas
200 particlesfixed
Show the speed curve
Particles
Accepted within
± 2%
Computed from the Scene at this step’s setting
Amount of gas in moles: the molecules the particles stand for, over Avogadro’s number/nodes/gas/outputs/moles
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 81.24 mol ± 2%
  Misconception: Puts litres into pV = nRT where SI wants cubic metres.
  Feedback from the catalogue
  With $R = 8.314\ \mathrm{J/(mol \cdot K)}$ ⁠ everything is in SI units: pressure in pascals and volume in cubic metres. A litre is a thousandth of a cubic metre, 1 L = 0.001 m³, so putting in litres gives an answer a thousand times too big.
Feedback for a right answer
$n = \dfrac{pV}{RT} = \dfrac{101\,325 \times 0.002}{8.314 \times 300} \approx 0.081$ ⁠ mol. So each of the Scene’s 200 particles stands for about $2.4 \times 10^{20}$ ⁠ real molecules.

Hint
In SI the volume is in cubic metres: 1 L = 0.001 m³.

Worked solution
1. The ideal gas equation: $pV = nRT$ ⁠, so $n = \dfrac{pV}{RT}$ ⁠.
2. In SI units: $p = 101\,325\ \mathrm{Pa}$ ⁠, $V = 2\ \mathrm{L} = 0.002\ \mathrm{m^3}$ ⁠, $T = 300\ \mathrm{K}$ ⁠.
3. $n = \dfrac{101\,325 \times 0.002}{8.314 \times 300} = \dfrac{202.65}{2494.2} \approx 0.0812$ ⁠.
4. The answer: about 0.081 mol.
If you got 81, you put in litres instead of cubic metres, and the answer came out a thousand times too big.
Step 2 of 5
Open this step
One mole
Practice · Calculate
Question
In the same box, 0.0812 mol fills 2 L at 300 K and one atmosphere. What volume would one mole of gas take up under these conditions?

Correct answer
24.62 L/mol ± 1%
Setup, choices and feedback
The Scene at this step
Temperature
300 Kfixed
Volume
2 Lfixed
Amount of gas
200 particlesfixed
Show the speed curve
Particles
Accepted within
± 1%
Computed from the Scene at this step’s setting
Volume the box holds per mole of gas, V / n/nodes/gas/outputs/molarVolume
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 22.4 L/mol ± 0.1 L/mol
  Misconception: Uses 22.4 L/mol for the molar volume whatever the temperature.
  Feedback from the catalogue
  22.4 L/mol is the volume of a mole of gas at 0 °C and one atmosphere. At room temperature the gas is warmer and takes more room: about 24 L/mol at 20–25 °C. The molar volume depends on temperature and pressure, $V_m = \dfrac{RT}{p}$ ⁠, so check which conditions you are in.
Feedback for a right answer
$V_m = \dfrac{2}{0.0812} \approx 24.6$ ⁠ litres per mole. That is the molar volume at 27 °C: every gas, of any kind, takes the same volume per mole at the same temperature and pressure.

Hint
$V_m = \dfrac{V}{n}$ ⁠.

Worked solution
1. The molar volume is the volume over the number of moles: $V_m = \dfrac{V}{n}$ ⁠.
2. $V_m = \dfrac{2\ \mathrm{L}}{0.0812\ \mathrm{mol}} \approx 24.6\ \mathrm{L/mol}$ ⁠.
3. Check: $V_m = \dfrac{RT}{p} = \dfrac{8.314 \times 300}{101\,325} = 0.0246\ \mathrm{m^3}$ ⁠, that is 24.6 L.
If you answered 22.4, that is the molar volume at 0 °C. Here the gas is at 27 °C, and takes more room.
Step 3 of 5
Open this step
Standard conditions
Practice · Reach the target
Question
The box has been cooled to 273 K, 0 °C, at 2 L, and the pressure has dropped below one atmosphere. Temperature and Volume are fixed. Add gas with Amount of gas until the pressure is back to one atmosphere, then press Check.

Goal
- 1 atm ± 0.005 atm
  At the start of the step: 0.91 atm
Setup, choices and feedback
The Scene at this step
Temperature
273 Kfixed
Volume
2 Lfixed
Amount of gas
200 particles
Show the speed curve
Particles
Read from the Scene at
Ideal-gas pressure p = N k T / V of the molecules the particles stand for/nodes/gas/outputs/pressure
One setting that reaches it
Amount of gas
220 particles
Feedback for a right answer
At 220 particles, that is 0.0894 mol. Now $V_m = \dfrac{2}{0.0894} \approx 22.4$ ⁠ litres per mole: the molar volume at standard conditions, 0 °C and one atmosphere.

Feedback for a wrong answer
Not yet. $n \propto \dfrac{p}{T}$ ⁠: at a lower temperature it takes more gas for the same pressure.

Hint
In colder gas each hit is weaker, so it takes more particles for the same pressure: $\dfrac{300}{273}$ ⁠ times as many.

Worked solution
1. At 300 K, 200 particles gave one atmosphere.
2. At constant volume and pressure $nT$ ⁠ stays the same, so $n_2 = n_1 \dfrac{T_1}{T_2} = 200 \times \dfrac{300}{273} \approx 220$ ⁠.
3. Set Amount of gas to 220 particles.
Step 4 of 5
Open this step
Avogadro’s law
Practice · Choose
Question
One balloon holds a mole of hydrogen, $\mathrm{H_2}$ ⁠ (2 g), and another a mole of carbon dioxide, $\mathrm{CO_2}$ ⁠ (44 g), at the same temperature and pressure. Which balloon is bigger?

Correct answer
- Correct They are the same size
Setup, choices and feedback
The Scene at this step
Temperature
300 K
Volume
2 L
Amount of gas
200 particles
Show the speed curve
Speed curve
Choices
Shown in a random order.
- Correct They are the same size
- Wrong The $\mathrm{CO_2}$ ⁠ balloon, because its molecules are bigger and heavier
  Misconception: Thinks a mole of a heavier gas takes up more room than a mole of a lighter one.
  Feedback from the catalogue
  The volume of a gas depends on the number of particles, not on their size or mass (Avogadro’s law). The particles are tiny compared with the gaps between them, and a heavy molecule moves more slowly but hits harder, so the pressure comes out the same. A mole of hydrogen and a mole of carbon dioxide, at the same temperature and pressure, take up the same volume.
- Wrong The hydrogen balloon, because its molecules are faster
  Misconception: Thinks all gases at the same temperature move at the same speed.
  Feedback
  The hydrogen molecules are indeed faster, but they are also 22 times lighter. Each of their hits is weaker, and the pressure comes out the same.
Feedback for a right answer
Avogadro’s law: equal volumes of gases, at the same temperature and pressure, hold equal numbers of particles. A mole of any gas takes the same molar volume, about 24 L at room temperature.

Worked solution
1. $V = \dfrac{nRT}{p}$ ⁠: the formula has no mass and no molecule size in it.
2. Both balloons have the same $n = 1$ ⁠, $T$ ⁠ and $p$ ⁠.
3. So $V$ ⁠ is the same too.
The molecules are tiny compared with the gaps between them, so their size hardly matters.
Step 5 of 5
Open this step
From grams to litres
Practice · Calculate
Question
A bottle of fizzy drink releases 11 g of $\mathrm{CO_2}$ ⁠ (molar mass 44 g/mol) at room temperature, where the molar volume is 24 L/mol. What volume of gas came out?

Correct answer
6 L ± 0.05 L
Setup, choices and feedback
The Scene at this step
Temperature
300 Kdefault
Volume
2 Ldefault
Amount of gas
200 particlesdefault
Show the speed curve
Particlesdefault
Accepted within
± 0.05 L
A fixed value in the question
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 5.6 L ± 0.05 L
  Misconception: Uses 22.4 L/mol for the molar volume whatever the temperature.
  Feedback from the catalogue
  22.4 L/mol is the volume of a mole of gas at 0 °C and one atmosphere. At room temperature the gas is warmer and takes more room: about 24 L/mol at 20–25 °C. The molar volume depends on temperature and pressure, $V_m = \dfrac{RT}{p}$ ⁠, so check which conditions you are in.
Feedback for a right answer
$n = \dfrac{11}{44} = 0.25$ ⁠ mol, and $V = 0.25 \times 24 = 6$ ⁠ litres: six one-litre bottles full of gas, from one bottle of drink.

Hint
Moles first, $n = \dfrac{m}{M}$ ⁠; then volume, $V = n V_m$ ⁠.

Worked solution
1. Moles: $n = \dfrac{m}{M} = \dfrac{11}{44} = 0.25\ \mathrm{mol}$ ⁠.
2. Volume: $V = n V_m = 0.25 \times 24 = 6$ ⁠.
3. The answer: 6 L.
If you got 5.6, you used 22.4 L/mol, the molar volume at 0 °C.

Lesson 5 of 5 · About 20 minutes

Hydrogen, oxygen and nitrogen

Three colourless, odourless gases, and one test that tells them apart: a burning splint. And inside the jar: why hydrogen molecules move four times as fast as oxygen’s.

Open the lesson

The 5 steps and their answers

Step 1 of 5
Open this step
The splint test
Practice · Predict, then look
Question
The jar holds air now, and the burning splint keeps burning quietly in it. What will the splint do in a jar of pure Oxygen?

Correct answer
- Correct The flame will flare up and burn brightly
Setup, choices and feedback
The Scene at this step
Gas in the jar
Airfixed
Temperature
293 K
Zoom to the molecules
Splint test
Choices
- Correct The flame will flare up and burn brightly
- Wrong The flame will go out
  Feedback
  The flame goes out in nitrogen, which does not support burning. Oxygen is the gas burning uses up.
- Wrong There will be a small pop
  Feedback
  The pop is the test for hydrogen: a fuel gas that burns in a small explosion. Oxygen does not burn itself; it lets other things burn.
Once the prediction is locked in, the Scene changes
Gas in the jar
Oxygenfixed
A prediction is never marked as a failure: the learner sees the outcome and the explanation either way.
Feedback for a right answer
Air is only about a fifth oxygen; a jar of pure oxygen has five times as much, and the wood burns fast and bright. In the classic test a glowing splint, with no flame, bursts back into flame: that is how oxygen is identified.

Worked solution
1. Burning is a reaction with oxygen.
2. Air is about 21% oxygen, and the rest is mostly nitrogen, which does not react.
3. In pure oxygen every molecule that hits the hot wood can react, so the burning is much faster and brighter.
Step 2 of 5
Open this step
The pop
Practice · Choose
Question
At a jar of hydrogen the splint gives a pop, and a thin mist is left on the glass. What is it?

Correct answer
- Correct Water: the hydrogen combined with oxygen from the air
Setup, choices and feedback
The Scene at this step
Gas in the jar
Hydrogen
Temperature
293 K
Zoom to the molecules
Splint test
Choices
Shown in a random order.
- Correct Water: the hydrogen combined with oxygen from the air
- Wrong Smoke from the splint
  Feedback
  Smoke is soot and ash, grey and gritty. The mist on the glass is clear, and it dries off as the glass warms.
- Wrong Hydrogen that has condensed
  Feedback
  Hydrogen turns liquid only at −253 °C. On glass at room temperature it stays a gas.
Feedback for a right answer
The pop is hydrogen burning fast with oxygen from the air: $2\mathrm{H_2} + \mathrm{O_2} \rightarrow 2\mathrm{H_2O}$ ⁠. The water forms as hot vapour and condenses on the cold glass.

Worked solution
1. Hydrogen is a fuel: it reacts with oxygen.
2. The product of hydrogen and oxygen is water: $2\mathrm{H_2} + \mathrm{O_2} \rightarrow 2\mathrm{H_2O}$ ⁠.
3. The hot vapour condenses on the cold glass and leaves a mist.
Step 3 of 5
Open this step
Nitrogen
No practice question in this step.
The Scene at this step
Gas in the jar
Nitrogen
Temperature
293 K
Zoom to the molecules
Splint test
Step 4 of 5
Open this step
Inside the jar
Practice · Calculate
Question
The jar holds hydrogen at 293 K. The average speed of gas molecules is $\bar v = \sqrt{\dfrac{8RT}{\pi M}}$ ⁠, with $M$ ⁠ in kilograms per mole. For hydrogen $M = 0.002016\ \mathrm{kg/mol}$ ⁠. What is the average speed?

Correct answer
1754 m/s ± 2%
Setup, choices and feedback
The Scene at this step
Gas in the jar
Hydrogenfixed
Temperature
293 Kfixed
Zoom to the molecules
Molecules
Accepted within
± 2%
Computed from the Scene at this step’s setting
Mean molecular speed of the gas in the jar/nodes/jar/outputs/averageSpeed
Any unit of the same quantity is accepted; it is converted before checking.
Wrong answers it recognises
- 440 m/s ± 2%
  Misconception: Thinks all gases at the same temperature move at the same speed.
  Feedback from the catalogue
  At the same temperature every gas has the same average kinetic energy per molecule, $\tfrac32 kT$ ⁠, not the same speed. With $\tfrac12 mv^2$ ⁠ equal, a light molecule moves faster: speed goes as $\dfrac{1}{\sqrt M}$ ⁠. Hydrogen is 16 times lighter than oxygen, so about 4 times faster.
Feedback for a right answer
About 1750 m/s: more than five times the speed of sound in air. Oxygen molecules in the same jar, at the same temperature, move at only about 440 m/s.

Hint
$R = 8.314\ \mathrm{J/(mol \cdot K)}$ ⁠.

Worked solution
1. The givens: $R = 8.314$ ⁠, $T = 293\ \mathrm{K}$ ⁠, $M = 0.002016\ \mathrm{kg/mol}$ ⁠.
2. $\bar v = \sqrt{\dfrac{8 \times 8.314 \times 293}{\pi \times 0.002016}} = \sqrt{\dfrac{19\,488}{0.006333}} \approx \sqrt{3.08 \times 10^6}$ ⁠.
3. $\bar v \approx 1754$ ⁠.
4. The answer: about 1750 m/s.
If you answered 440 m/s, that is oxygen’s speed. At the same temperature the average kinetic energy is the same, not the speed.
Step 5 of 5
Open this step
Light and fast
Practice · Calculate
Question
The molar mass of oxygen, $\mathrm{O_2}$ ⁠, is 32 g/mol, and of hydrogen, $\mathrm{H_2}$ ⁠, 2 g/mol. At the same temperature, how many times faster are the hydrogen molecules?

Correct answer
4 ± 0.05
Setup, choices and feedback
The Scene at this step
Gas in the jar
Oxygen
Temperature
293 K
Zoom to the molecules
Molecules
Accepted within
± 0.05
A fixed value in the question
Wrong answers it recognises
- 1 ± 0.05
  Misconception: Thinks all gases at the same temperature move at the same speed.
  Feedback from the catalogue
  At the same temperature every gas has the same average kinetic energy per molecule, $\tfrac32 kT$ ⁠, not the same speed. With $\tfrac12 mv^2$ ⁠ equal, a light molecule moves faster: speed goes as $\dfrac{1}{\sqrt M}$ ⁠. Hydrogen is 16 times lighter than oxygen, so about 4 times faster.
Feedback for a right answer
$\sqrt{\dfrac{32}{2}} = \sqrt{16} = 4$ ⁠. That is why hydrogen leaks from a balloon so fast, and why it rises in air: a mole of it weighs 2 g, and a mole of air about 29 g. In the Scene, 1754 over 440 is almost exactly 4.

Hint
$\bar v \propto \dfrac{1}{\sqrt M}$ ⁠.

Worked solution
1. At the same temperature $\tfrac12 m v^2$ ⁠ is the same for both gases, so $v \propto \dfrac{1}{\sqrt M}$ ⁠.
2. $\dfrac{v_{\mathrm{H_2}}}{v_{\mathrm{O_2}}} = \sqrt{\dfrac{M_{\mathrm{O_2}}}{M_{\mathrm{H_2}}}} = \sqrt{\dfrac{32}{2}} = 4$ ⁠.
3. The answer: 4 times.
If you answered 16, you left out the square root: energy goes as the square of the speed.

Across the unit · 12 misconceptions

Misconception catalogue

Each common mistake this unit recognises, in a few words. Open one for what the learner reads when an answer matches it, and where it is asked.

Index

The particle model

Three states

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Inside the ice

Question

Correct answer

The Scene at this step

Choices

Once the prediction is locked in, the Scene changes

Worked solution

Kelvin and Celsius

Question

Correct answer

The Scene at this step

Hint

Worked solution

Melt it

Question

Goal

The Scene at this step

One setting that reaches it

Hint

Worked solution

Far apart

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

Changes of state

Six changes

Question

Correct answer

The Scene at this step

Choices

Worked solution

Melting takes energy

Question

Correct answer

The Scene at this step

Wrong answers it recognises

Hint

Worked solution

Boiling

Question

Correct answer

The Scene at this step

Choices

Hint

Worked solution

The temperature stands still

Question

Correct answer

The Scene at this step

Choices

Worked solution

Evaporation without boiling

Question

Correct answer

The Scene at this step

Choices

Worked solution

Pressure and the gas laws

What pressure is

Question

Correct answer

The Scene at this step

Choices

Worked solution

Squeeze

Question

Correct answer

The Scene at this step

Once the prediction is locked in, the Scene changes