3.5 Kinetic Molecular Theory

1. AThe atoms move faster after the compression, so each one strikes the walls with more force.
   A student who thinks a gas at higher pressure must have faster particles picks this. The average kinetic energy of the atoms depends only on the Kelvin temperature, which is held constant here; the pressure rises because the collisions with the walls are more frequent, not harder.
2. BThe atoms strike the walls more often, but their average kinetic energy has not changed. Correct
   The temperature is constant, so the average kinetic energy and average speed of the atoms do not change. The same number of atoms now moves in half the volume, so the atoms reach the walls more often: more collisions with the walls each second on each unit of area means a higher pressure.
3. CThe atoms are pushed closer together, so they repel each other more strongly than before.
   A student who thinks gas pressure comes from repulsion between the particles picks this. In the kinetic molecular theory the pressure results from collisions of the moving atoms with the walls; it does not depend on forces of repulsion between the atoms.
4. DThe atoms collide with each other more often, and such collisions are the gas pressure.
   A student who thinks gas pressure is caused by particles colliding with each other picks this. The atoms do collide with one another more often in the smaller volume, but the pressure on the container results from the collisions of the atoms with its walls.

CED 3.5.A.1 · Read this in Fix

1. ADiagram 1
   A student who thinks all the particles of a sample at one temperature move at the same speed picks this diagram, in which every arrow has the same length. The temperature fixes the average kinetic energy; the individual atoms have a range of speeds.
2. BDiagram 2
   A student who pictures the atoms of a gas moving together in one direction, like a wind, picks this diagram. In a gas that is not flowing, the atoms move randomly in all directions with a range of speeds.
3. CDiagram 3
   A student who thinks the particles of an undisturbed sample are at rest picks this diagram, which has no arrows. The atoms of a gas are in continuous motion even when the gas as a whole is still.
4. DDiagram 4 Correct
   The particles of a gas are in continuous, random motion: at any instant the atoms are moving in all directions, and at a given temperature they have a range of speeds. This diagram shows arrows of different lengths pointing in different directions.

CED 3.5.A.2 · Read this in Fix

1. ADiagram 1
   A student who thinks the particles of a substance expand when it is heated picks this diagram, with larger atoms. Heating does not change the size of an atom; it increases the atoms' average kinetic energy, which this diagram does not show.
2. BDiagram 2 Correct
   The Kelvin temperature is proportional to the average kinetic energy of the atoms, so at 600 K the atoms move faster on average: the arrows are longer on average. The atoms still have a range of speeds, they are the same size, and they are still spread through the whole container.
3. CDiagram 3
   A student who thinks heating pushes gas particles apart, so that they collect at the walls, picks this diagram. In a sealed, rigid container the atoms stay spread through the whole volume; what changes is their average speed.
4. DDiagram 4
   A student who thinks all the particles of a sample at one temperature move at the same speed picks this diagram, in which every arrow has the same length. The atoms do move faster on average at 600 K, but they still have a range of speeds.

CED 3.5.A.3 · Read this in Fix

1. AThe fastest molecules in the sample have speeds that are near 400 m/s.
   A student who thinks the peak of the curve marks the greatest speed in the sample picks this. The peak marks the most common speed; the curve continues to the right of the peak, so many molecules have speeds greater than 400 m/s.
2. BMore molecules have speeds near 400 m/s than near any other speed. Correct
   The height of the curve at a given speed shows the number of molecules with that speed. The curve is highest just above 400 m/s, so speeds near 400 m/s are the most common in the sample.
3. CMolecules with speeds near 400 m/s have the highest kinetic energy.
   A student who reads the height of the curve as the energy of the molecules picks this. The height shows how many molecules have each speed; the molecules with the highest kinetic energy are the fastest ones, at the far right of the curve.
4. DMolecules with speeds above 400 m/s are at a higher temperature than 300 K.
   A student who thinks all the molecules of a gas at one temperature move at the same speed picks this, taking a spread of speeds to mean a spread of temperatures. Temperature describes the sample as a whole: it is related to the average kinetic energy of the molecules, and a sample at 300 K contains molecules with a wide range of speeds.

CED 3.5.A.4 · Read this in Fix

1. AThe molecules have a higher average speed, so they hit the walls more often and with more force. Correct
   The Kelvin temperature is proportional to the average kinetic energy of the molecules, so at 450 K the molecules move faster on average. Faster molecules reach the walls more often and deliver more force in each collision, and both effects raise the pressure.
2. BThe molecules expand as they are heated, so they take up more of the space in the container.
   A student who thinks particles themselves expand when a substance is heated picks this. Heating does not change the size of an N₂ molecule; it increases the average kinetic energy of the molecules.
3. CThe molecules move farther apart as they are heated, so they press outward on the walls.
   A student who thinks heating always pushes gas particles farther apart picks this. The container is sealed and rigid, so the same number of molecules occupies the same volume and their average spacing cannot change; the pressure rises because the molecules move faster.
4. DThe molecules collide with one another more often, so these extra collisions add to the gas pressure.
   A student who thinks gas pressure is caused by particles colliding with each other picks this. Collisions between molecules do become more frequent, but the pressure on the container results from the collisions of the molecules with its walls, which become more frequent and more forceful.

CED 3.5.A.1 · Read this in Fix

1. AEqual numbers of molecules have kinetic energy greater than E at T₁ and T₂.
   A student who reasons that the two curves enclose the same total area, so nothing about the numbers can differ, picks this. The total area is the same because the sample is the same, but the part of the area to the right of E is larger for T₂.
2. BFewer molecules have kinetic energy greater than E at T₂ than at T₁.
   A student who reads the height of a curve as a measure of how energetic the sample is picks this, because the curve for T₂ has the lower peak. Height shows the number of molecules at each energy; to the right of E it is the T₂ curve that is higher.
3. CMore molecules have kinetic energy greater than E at T₂ than at T₁. Correct
   The number of molecules with kinetic energy greater than E is represented by the area under a curve to the right of the dotted line. To the right of E the curve for T₂ lies above the curve for T₁, so that area, and the number of molecules, is larger at T₂.
4. DNone of the molecules has kinetic energy greater than E at T₁ or at T₂.
   A student who thinks the peak of a curve marks the greatest kinetic energy in the sample picks this, because E lies to the right of both peaks. The peak marks the most common kinetic energy; both curves continue to the right of E, so some molecules have more kinetic energy than E at each temperature.

Working The number of molecules with kinetic energy greater than E is the area under a curve to the right of the dotted line. Beyond E the T₂ curve lies above the T₁ curve everywhere, so the area to the right of E is larger for T₂. (T₂ is the higher temperature: its curve is broader, with its peak at a higher kinetic energy.)

CED 3.5.A.1 · Read this in Fix

1. A7.0
   A student who thinks average speed is inversely proportional to mass picks this, using 28.02/4.00 = 7.0. Because the kinetic energy depends on v², it is the square of the speed that is inversely proportional to the mass, and the ratio of the speeds is √7.005 = 2.6.
2. B1.9
   A student who uses the atomic mass of nitrogen from the periodic table as the mass of the N₂ particle picks this: √(14.01/4.00) = 1.9. The moving particle is the N₂ molecule, with a molar mass of 28.02 g/mol.
3. C2.6 Correct
   Equal temperatures mean equal average kinetic energies, so ½mv² has the same average value for both gases and the ratio of the speeds is the square root of the inverse ratio of the masses: √(28.02/4.00) = √7.005 = 2.6.
4. D1.0
   A student who thinks all gases at the same temperature have the same average speed picks this. The gases have the same average kinetic energy; the lighter He atoms must move faster to have it.

Working Same temperature, so the average kinetic energies are equal: ½m(He)v(He)² = ½m(N₂)v(N₂)². v(He)/v(N₂) = √[m(N₂)/m(He)] = √(28.02/4.00) = √7.005 = 2.647 ≈ 2.6. Distractors: mass ratio without the square root, 28.02/4.00 = 7.0; atomic mass of N used, √(14.01/4.00) = 1.9; equal speeds assumed, 1.0.

CED 3.5.A.2 · Read this in Fix

1. AThe H₂ sample
   A student who thinks the gas with the lightest particles has the fastest particles whatever the temperature picks this. The H₂ sample is much colder than the He sample: T/M is 100/2.016 = 49.6 for H₂ but 300/4.00 = 75.0 for He.
2. BThe Ne sample
   A student who thinks temperature alone fixes the average speed picks the hottest sample. The highest temperature gives Ne the greatest average kinetic energy, but Ne atoms are much heavier than He atoms: T/M is 1200/20.18 = 59.5 for Ne and 75.0 for He.
3. CThe Ar sample
   A student who thinks the particles of a larger sample have more kinetic energy and move faster picks the sample with the most moles. The average speed does not depend on the amount of gas; for Ar, T/M = 800/39.95 = 20.0, the smallest of the four values.
4. DThe He sample Correct
   The average speed is greater where T/M (Kelvin temperature divided by molar mass) is greater, because ½mv² is proportional to T. T/M is 300/4.00 = 75.0 for He, compared with 49.6 for H₂, 59.5 for Ne and 20.0 for Ar, so the He atoms have the greatest average speed.

Working Average kinetic energy ∝ T and KE = ½mv², so v² ∝ T/M: the average speed is greatest where T/M is greatest. H₂: 100/2.016 = 49.6; He: 300/4.00 = 75.0; Ne: 1200/20.18 = 59.5; Ar: 800/39.95 = 20.0 (K·mol/g). The amount of gas does not affect the average speed. The He sample has the greatest value.

CED 3.5.A.2 · Read this in Fix

1. A3.0
   A student who thinks kinetic energy is directly proportional to speed picks this. The speed is squared in KE = ½mv², so 3 times the speed gives 3² = 9 times the kinetic energy.
2. B9.0 Correct
   KE = ½mv², and the two molecules have the same mass, so the kinetic energy is proportional to the square of the speed: 3² = 9.0.
3. C1.7
   A student who applies the square root in the wrong direction picks this, taking √3 = 1.7. A square root gives a ratio of speeds from a ratio of kinetic energies; here the ratio of the speeds is known, so it must be squared.
4. D6.0
   A student who reads the exponent in v² as multiplication by 2 picks this: 2 × 3 = 6.0. Squaring means multiplying the factor by itself: 3 × 3 = 9.

Working KE = ½mv². Both molecules are N₂, so m is the same and KE ∝ v². KE(fast)/KE(slow) = (3v)²/v² = 9.0. Distractors: KE ∝ v gives 3.0; a square root instead of a square gives √3 = 1.7; 'squared' read as 'times 2' gives 6.0.

CED 3.5.A.2 · Read this in Fix

1. ACloser to the HCl plug, because the lighter NH₃ molecules have the higher average speed Correct
   Both gases are at the same temperature, so their molecules have the same average kinetic energy. NH₃ molecules (17.03 g/mol) have less mass than HCl molecules (36.46 g/mol), so they have the higher average speed, travel farther along the tube in the same time, and meet the HCl nearer the HCl plug.
2. BAt the center of the tube, because gases at one temperature have equal average speeds
   A student who thinks all gases at the same temperature have the same average speed picks this. The two gases have the same average kinetic energy; the lighter NH₃ molecules move faster, so the gases meet nearer the HCl plug.
3. CCloser to the NH₃ plug, because heavier HCl molecules have the higher kinetic energy
   A student who thinks heavier particles have more kinetic energy at the same temperature, and so travel farther, picks this. At the same temperature both gases have the same average kinetic energy, and the heavier HCl molecules are the slower ones.
4. DCloser to the HCl plug, because lighter NH₃ molecules have a higher kinetic energy
   A student who thinks the faster, lighter particles have more kinetic energy picks this. The position is right, but the reasoning is not: both gases are at the same temperature and so have the same average kinetic energy. The NH₃ molecules travel farther because their smaller mass gives them a higher average speed.

CED 3.5.A.2 · Read this in Fix

1. AThe dye does not spread, as water molecules at temperatures near 0°C stop moving.
   A student who thinks particles stop moving at about 0°C picks this. 5°C is 278 K: the molecules have about 82% of the average kinetic energy they have at 65°C, so they are still moving and the dye still spreads, more slowly.
2. BThe dye spreads in less time, as the water molecules have shrunk and left wider gaps.
   A student who thinks particles shrink when a substance is cooled picks this. Cooling does not change the size of a water molecule; it lowers the average kinetic energy of the molecules, so the dye spreads more slowly.
3. CThe dye takes longer to spread out, as the water molecules have a lower average speed. Correct
   The particles of the water and the dye are in continuous, random motion, and this motion spreads the dye. At 5°C (278 K) the average kinetic energy of the molecules is lower than at 65°C (338 K), so the molecules move more slowly on average and the dye takes longer to spread.
4. DThe dye takes the same time to spread, as it spreads by sinking, not by molecular motion.
   A student who thinks the particles of still water are at rest, so that the dye must be moved by something else, picks this. The molecules of still water are in continuous, random motion, and their average speed, and so the rate of spreading, depends on the temperature.

CED 3.5.A.2 · Read this in Fix

1. AThe He sample
   A student who thinks the lightest, fastest particles have the greatest kinetic energy picks this. He atoms do have the greatest average speed here, but their small mass means that their average kinetic energy is set by the temperature, 350 K, which is lower than that of the Ar sample.
2. BThe Ne sample
   A student who thinks a gas at a higher pressure has more energetic particles picks the sample with the highest pressure. Pressure also depends on how many particles are in the container; the Ne sample has the lowest temperature, 300 K, and so the lowest average kinetic energy.
3. CThe Xe sample
   A student who thinks heavier particles have a greater average kinetic energy picks the gas with the greatest atomic mass. The average kinetic energy depends only on the Kelvin temperature, and the Xe sample (400 K) is cooler than the Ar sample (450 K).
4. DThe Ar sample Correct
   The Kelvin temperature of a sample is proportional to the average kinetic energy of its particles, whatever the gas and whatever its pressure. The Ar sample has the highest temperature, 450 K, so its atoms have the greatest average kinetic energy.

CED 3.5.A.3 · Read this in Fix

1. A1.04 Correct
   The average kinetic energy is proportional to the Kelvin temperature, and KE = ½mv², so the average speed is proportional to the square root of the Kelvin temperature: √(323.15 K/298.15 K) = √1.084 = 1.04.
2. B1.08
   A student who thinks average speed is directly proportional to the Kelvin temperature picks this: 323.15/298.15 = 1.08. It is the average kinetic energy that increases by this factor; the speed increases by its square root, 1.04.
3. C1.41
   A student who uses Celsius temperatures in the ratio picks this: √(50/25) = 1.41. Average kinetic energy is proportional to the Kelvin temperature, and from 298 K to 323 K the temperature rises by only about 8%.
4. D1.17
   A student who squares the temperature ratio instead of taking its square root picks this: (323.15/298.15)² = 1.17. Because temperature is proportional to v², the ratio of the speeds is the square root of the ratio of the Kelvin temperatures.

Working T₁ = 25 + 273.15 = 298.15 K; T₂ = 50 + 273.15 = 323.15 K. Average KE ∝ T and KE = ½mv², so v ∝ √T. v₂/v₁ = √(323.15/298.15) = √1.0838 = 1.04. Distractors: v ∝ T gives 1.08; Celsius temperatures give √(50/25) = 1.41; squaring the Kelvin ratio gives 1.17.

CED 3.5.A.3 · Read this in Fix

1. AIt is correct: the temperature of a sample measures the average speed of its particles, whatever the mass.
   A student who thinks temperature is a measure of particle speed, so that equal temperatures mean equal average speeds, picks this. Temperature is proportional to the average kinetic energy, ½mv², so at one temperature the lighter He atoms move faster.
2. BIt is correct: the particles in a sample at one temperature all move at one speed, whatever their mass.
   A student who thinks every particle in a sample at one temperature has the same speed picks this. The particles have a range of speeds at any temperature, and the average speed of the He atoms is higher than that of the heavier Ar atoms.
3. CIt is incorrect: the average speed of a particle is set by its mass, and does not depend on temperature.
   A student who thinks the mass of a particle alone sets its speed picks this. The claim is incorrect, but not for this reason: the average speed depends on both the mass and the Kelvin temperature, because the average of ½mv² is proportional to T.
4. DIt is incorrect: the two average kinetic energies are equal, and the He atoms have the higher average speed. Correct
   The Kelvin temperature is proportional to the average kinetic energy, so the He atoms and the Ar atoms have the same average kinetic energy. Because KE = ½mv², the He atoms, which have about one-tenth of the mass of the Ar atoms, must have the higher average speed.

CED 3.5.A.3 · Read this in Fix

1. AGraph 1
   A student who thinks heating moves the whole curve to higher speeds without changing its shape picks this graph. At the higher temperature some atoms still have very low speeds, and the range of speeds is wider, so the curve becomes broader and lower as its peak moves to the right.
2. BGraph 2
   A student who thinks the curve grows when the sample is heated, because the atoms have more energy, picks this graph. The peak does move to a higher speed, but the area under the curve represents the number of atoms, which has not changed, so the curve cannot become both broader and taller.
3. CGraph 3 Correct
   At the higher temperature the atoms have a greater average kinetic energy, so the peak moves to a higher speed and the range of speeds widens. The number of atoms is unchanged, so the area under the curve stays the same and the broader curve must have a lower peak.
4. DGraph 4
   A student who reads the height of the curve as a measure of energy or temperature picks this graph, with a taller peak at the same speed. Height shows the number of atoms at each speed; a higher temperature moves the peak to a higher speed and lowers it.

CED 3.5.A.4 · Read this in Fix

1. A800 K Correct
   The peaks are at 300 m/s (200 K) and 600 m/s (T₂), so the most common speed is twice as great at T₂. The Kelvin temperature is proportional to the average kinetic energy, which depends on the square of the speed, so T₂ = 200 K × 2² = 800 K.
2. B400 K
   A student who thinks molecular speed is directly proportional to the Kelvin temperature picks this, doubling the temperature because the peak speed doubles. Temperature is proportional to kinetic energy, and so to the square of the speed: T₂ = 200 K × 2² = 800 K.
3. C100 K
   A student who reads the height of a curve as a measure of temperature picks this, because the peak of the T₂ curve is half as high. Height shows the number of molecules at each speed; the T₂ curve lies at higher speeds, so T₂ is the higher temperature.
4. D283 K
   A student who takes the square root of the speed ratio instead of squaring it picks this: 200 K × √2 = 283 K. Temperature is proportional to v², so the factor of 2 in speed must be squared.

Working Read the peak positions: 300 m/s at 200 K and 600 m/s at T₂, a factor of 2 in speed. T ∝ average KE and KE = ½mv², so T ∝ v²: T₂ = 200 K × (600/300)² = 800 K. Distractors: T ∝ v gives 400 K; using the ratio of the peak heights (½) gives 100 K; taking √2 instead of 2² gives 283 K.

CED 3.5.A.4 · Read this in Fix