Ideal gas equation | PhysicsRank

A sealed bicycle pump gets harder to compress, while a balloon can expand when warmed. The same gas equation can describe both, but only after we decide what is held fixed. A formula with three changing quantities needs a physical story to close it.

This chapter also follows a molecule between collisions. Mean free path will give a second kind of scale, one that helps decide when an ordinary continuous-fluid description is appropriate.

What you will learn to do

Use absolute pressure, Kelvin temperature and consistent mole/mass units.
Distinguish mass density, number density and partial pressure.
Derive the pressure relation from molecular momentum transfer.
Compare molecular speeds and translational energies at a common temperature.
Distinguish vector, mean-speed and rms averages.
Explain the collision cross-section and mean-free-path model.
Predict collision changes under fixed-pressure and fixed-volume constraints.

Useful before this chapterViscosity, terminal motion and flow regimes →Impulse, momentum and collisions →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

Why is a statement such as “pressure increases with temperature” incomplete for a gas?

Compare your reasoning

You must specify constraints, such as fixed amount and fixed volume. If volume also changes, pressure need not rise in the same way. Use absolute temperature in the ideal-gas relation.

Revisit the explanation →

01 / Gases: state variables, molecular speeds and collisions

An equation of state connects equilibrium properties

A gas sample can be described macroscopically by pressure, volume, temperature and amount. An equation of state relates these properties when the sample is adequately close to an equilibrium state. For a dilute classical ideal gas, pV = nRT, where n is amount in moles and T is absolute temperature.

The pressure in this equation is absolute pressure, measured from vacuum. A gauge often measures pressure above the surrounding atmosphere. Add atmospheric pressure before using the gas law, and subtract it again only if the requested answer is a gauge pressure.

Temperature must be in kelvins. Celsius temperature differs by an offset, so ratios of Celsius readings do not represent ratios of thermal energy or ideal-gas pressure. A temperature change has the same numerical size in kelvins and Celsius degrees, but a temperature itself does not.

The ideal model neglects molecular volume and interaction energy in the equation of state. It is often a good approximation at sufficiently low density and away from condensation. It is not a universal description of dense gases, phase changes or every temperature range.

pV=nRT=Nk_{\mathrm B}T,\qquad n=\frac{N}{N_{\mathrm A}},\qquad R=N_{\mathrm A}k_{\mathrm B}

N is the number of molecules or atoms, while n is the amount in moles. Keep particle mass and molar mass distinct in later formulas.

Explain this equation

Read it in words

Pressure times volume is proportional to the amount of gas and its absolute temperature. Mole counting and molecule counting describe the same state.

Symbols and units

$p$: Absolute gas pressure, including atmospheric pressure when converting a gauge reading.Pascals (Pa).
$V$: Volume occupied by the gas.Cubic metres (m³).
$n$: Amount of gas counted in moles.Moles (mol).
$N$: Number of gas particles.Dimensionless count.
$T$: Absolute temperature.Kelvin (K).
$R$: Molar gas constant.J/(mol·K).
$k_{\mathrm B}$: Boltzmann constant, the energy–temperature constant per particle.J/K.
$N_{\mathrm A}$: Avogadro constant: number of particles per mole.mol⁻¹.

Signs and operations

Products mean ordinary multiplication. Uppercase N counts particles while lowercase n counts moles; choose the matching constant. Kelvin and absolute pressure are required because their zero points matter.

When it applies

This is an ideal-gas equilibrium model. It works best for sufficiently dilute gases away from condensation; interactions and molecular volume can cause departures. It relates a state, not the heat or work of an entire process.

Make it concrete

At 300 K, 1 mol in 0.025 m³ has p ≈ 99.8 kPa using R ≈ 8.314 J/(mol·K). At fixed volume and amount, doubling the kelvin temperature doubles pressure; doubling the Celsius number does not.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Equation (LaTeX): pV=nRT=Nk_{\mathrm B}T,\qquad n=\frac{N}{N_{\mathrm A}},\qquad R=N_{\mathrm A}k_{\mathrm B}
Read it as: Pressure times volume is proportional to the amount of gas and its absolute temperature. Mole counting and molecule counting describe the same state.
Symbols:
p: Absolute gas pressure, including atmospheric pressure when converting a gauge reading.; Pascals (Pa).
V: Volume occupied by the gas.; Cubic metres (m³).
n: Amount of gas counted in moles.; Moles (mol).
N: Number of gas particles.; Dimensionless count.
T: Absolute temperature.; Kelvin (K).
R: Molar gas constant.; J/(mol·K).
k_{\mathrm B}: Boltzmann constant, the energy–temperature constant per particle.; J/K.
N_{\mathrm A}: Avogadro constant: number of particles per mole.; mol⁻¹.
Signs and operations: Products mean ordinary multiplication. Uppercase N counts particles while lowercase n counts moles; choose the matching constant. Kelvin and absolute pressure are required because their zero points matter.
Conditions: This is an ideal-gas equilibrium model. It works best for sufficiently dilute gases away from condensation; interactions and molecular volume can cause departures. It relates a state, not the heat or work of an entire process.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

WORKED EXAMPLE

Convert the pressure reference before taking a ratio

A sealed rigid vessel contains ideal gas at 300 K. Its gauge pressure is 100 kPa while atmospheric pressure is 100 kPa. It is heated to 450 K with no leakage. Find the final absolute and gauge pressures, assuming the atmospheric pressure stays fixed.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: An equation of state connects equilibrium properties
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
pV=nRT=Nk_{\mathrm B}T,\qquad n=\frac{N}{N_{\mathrm A}},\qquad R=N_{\mathrm A}k_{\mathrm B}
Meaning: N is the number of molecules or atoms, while n is the amount in moles. Keep particle mass and molar mass distinct in later formulas.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Gases: state variables, molecular speeds and collisions

Keep amount, particle count and mass density separate

Molar mass M is mass per mole, expressed in kg/mol when using SI values of R. If a sample has total mass m_gas, then n = m_gas/M. The mass of one molecule is m = M/N_A. Substituting a molar mass given in grams directly into an SI formula can introduce a factor of a thousand.

Mass density is ρ = m_gas/V. Combining it with the gas law gives p = ρRT/M. Number density is n_N = N/V, so p = n_Nk_BT. These densities count different things: one counts kilograms per volume and the other counts particles per volume.

At the same pressure and temperature, ideal gases have the same number density, regardless of molecular mass. Their mass densities differ in proportion to molar mass. Equal volumes then contain equal numbers of particles, but not generally equal masses.

For an ideal mixture at one temperature, each species contributes a partial pressure p_i = N_i k_BT/V and the total is their sum. Mole fraction determines pressure fraction. Mass fraction generally does not, because different species can have different molecular masses.

p=\frac{\rho RT}{M}=n_Nk_{\mathrm B}T,\qquad p_{\mathrm{total}}=\sum_i p_i

Use M in kg/mol with R in J/(mol·K). Partial pressures use the same total mixture volume.

WORKED EXAMPLE

Infer mass density from an equilibrium state

An ideal gas has molar mass 0.030 kg/mol, temperature 300 K and absolute pressure 83.1 kPa. Use R = 8.31 J/(mol·K). Find its mass density.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: Keep amount, particle count and mass density separate
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
p=\frac{\rho RT}{M}=n_Nk_{\mathrm B}T,\qquad p_{\mathrm{total}}=\sum_i p_i
Meaning: Use M in kg/mol with R in J/(mol·K). Partial pressures use the same total mixture volume.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Gases: state variables, molecular speeds and collisions

A state relation is not a complete process description

For a fixed amount of ideal gas, pV/T is constant between equilibrium states. But different physical constraints give different relationships: at fixed volume p ∝ T, at fixed temperature p ∝ 1/V, and at fixed pressure V ∝ T.

Read the apparatus or process description before selecting a ratio. A sealed rigid vessel fixes amount and volume. A freely moving piston under a constant external load may approximately fix pressure during slow motion. A leak means the amount can change, invalidating a fixed-n shortcut.

The gas law does not by itself say how much heat entered or how much work was done. Different paths can connect the same initial and final states. Energy transfer requires the first law and information about the path, which will be developed in the next chapter.

A useful check is to predict the direction of change before calculating. If a sealed gas doubles its volume at fixed temperature, pressure should decrease. A result with doubled pressure would signal that a constraint or ratio was reversed.

WORKED EXAMPLE

Account for two simultaneous changes

A fixed amount of ideal gas changes from p₁, V₁ and T₁ to a state with V₂ = 1.5V₁ and T₂ = 2T₁. Find p₂/p₁.

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STOP & THINK

A vessel is rigid but leaks gas while being heated. May p/T be assumed constant?

Yes, rigid always means p/T is constant
No; the particle amount also changes
Yes, if the temperature is in kelvins

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: A state relation is not a complete process description
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

04 / Gases: state variables, molecular speeds and collisions

Pressure arises from momentum transfer to walls

Consider a molecule moving between opposite walls of a box. In an elastic collision with a wall normal to x, its x momentum reverses, transferring 2m|v_x| to that wall. The return time between successive hits on the same wall is 2L/|v_x| in the elementary box argument.

The average force from that molecule is therefore m v_x²/L. Adding over molecules and dividing by wall area yields p = (Nm/V)⟨v_x²⟩. The square explains why molecules moving in opposite directions contribute positive pressure rather than canceling.

For an equilibrium gas with isotropic random motion and no bulk flow, the three component mean squares are equal. Since v² = v_x² + v_y² + v_z², each component contributes one third of ⟨v²⟩. Thus p = ρ⟨v²⟩/3.

The derivation assumes elastic collisions and a dilute gas with well-defined molecular motion between collisions. Molecular collisions redistribute momenta but do not change the equilibrium pressure relation in the ideal model. A gas with bulk flow requires separating that ordered motion from thermal random velocities.

p=\frac13\rho\langle v^2\rangle=\frac13\rho v_{\mathrm{rms}}^2

The velocities here are thermal velocities in the gas rest frame. Isotropy supplies the factor one third.

WORKED EXAMPLE

Infer rms speed from pressure and mass density

An equilibrium ideal gas has absolute pressure 120 kPa and mass density 2.0 kg/m³. Find its rms molecular speed.

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: Pressure arises from momentum transfer to walls
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
p=\frac13\rho\langle v^2\rangle=\frac13\rho v_{\mathrm{rms}}^2
Meaning: The velocities here are thermal velocities in the gas rest frame. Isotropy supplies the factor one third.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

05 / Gases: state variables, molecular speeds and collisions

Temperature fixes mean translational energy, not one common speed

Equate the kinetic pressure relation to pV = Nk_BT. The result is ½m⟨v²⟩ = 3k_BT/2, so v_rms = √(3k_BT/m) = √(3RT/M). Temperature is linked to the mean translational kinetic energy per particle in this classical model.

Molecules do not all move at the rms speed. Collisions continually change individual velocities, while an equilibrium distribution of speeds is maintained. The rms value is a statistical summary chosen because kinetic energy contains v².

At the same temperature, different ideal-gas species have the same mean translational kinetic energy per particle. Lighter molecules have larger rms speeds so that m⟨v²⟩ stays the same. Equal kinetic energies do not imply equal speeds or equal momenta.

Rms speed scales as √T and as 1/√M. Doubling the absolute temperature increases it by √2; doubling a Celsius reading does not establish that temperature ratio. For molecular gases, rotational and vibrational energy may also contribute to internal energy, so translational energy alone is not always the entire internal energy.

\frac12m v_{\mathrm{rms}}^2=\frac32k_{\mathrm B}T,\qquad v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}

Use absolute temperature and molar mass in kg/mol. This is a nonrelativistic classical result.

WORKED EXAMPLE

Compare species without calculating either absolute speed

Two ideal-gas species are in equilibrium at the same temperature. Species B has four times the molecular mass of species A. Compare their rms speeds and mean translational kinetic energies.

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STOP & THINK

By what factor must the absolute temperature of the same ideal gas increase to make its rms speed three times as large?

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: Temperature fixes mean translational energy, not one common speed
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
\frac12m v_{\mathrm{rms}}^2=\frac32k_{\mathrm B}T,\qquad v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}
Meaning: Use absolute temperature and molar mass in kg/mol. This is a nonrelativistic classical result.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / Gases: state variables, molecular speeds and collisions

Average velocity, average speed and rms speed answer different questions

Average velocity is a vector average. In an equilibrium gas at rest, opposite directions cancel, so the mean thermal velocity is zero. Average speed is the mean of nonnegative speed magnitudes and is not zero. Rms speed first averages the squared magnitudes and then takes a square root.

These averaging operations do not commute. For speeds 100, 200 and 300 m/s, the mean speed is 200 m/s while rms speed is √[(100² + 200² + 300²)/3], about 216 m/s. Squaring the mean would underestimate the mean squared speed and the kinetic energy.

For the classical Maxwell speed distribution, the most probable, mean and rms speeds are respectively √(2k_BT/m), √(8k_BT/(πm)) and √(3k_BT/m). They are ordered most probable < mean < rms. These formulas describe an equilibrium distribution, not every arbitrary collection of three particles.

The sound speed of an ideal gas is also related to molecular motion, but it is the speed of a collective pressure disturbance. In the adiabatic ideal-gas model it is √(γRT/M), which differs from v_rms = √(3RT/M). A disturbance does not travel simply by one molecule flying unchanged from source to receiver.

WORKED EXAMPLE

Check which average belongs in an energy calculation

A toy collection has three equal-mass particles with speeds 0, 3 and 6 m/s. Find the mean speed and rms speed. Which determines its mean kinetic energy?

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: Average velocity, average speed and rms speed answer different questions
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

07 / Gases: state variables, molecular speeds and collisions

A distribution predicts a fraction of molecules, not one speed

Imagine taking a snapshot of a gas and sorting molecules into speed bins. Even at one fixed temperature, the bins contain different numbers of molecules. The curve through many such bins describes an ensemble; it is not a speed–time graph for a particular molecule. Collisions keep replacing the occupants of each bin.

The probability density F(v) is defined so that F(v) dv is the fraction in a narrow speed interval of width dv. F itself is not a fraction: it has units inverse to speed. The area over all nonnegative speeds is one. For a finite interval, integrate the curve; height times width is only an approximation when the curve changes little inside the bin.

In a dilute classical gas at thermal equilibrium, isotropic velocity components lead to the Maxwell speed distribution. Its v² factor counts the growing number of velocity directions available at a given speed, while the exponential suppresses very high kinetic energies. These competing factors make the peak occur above zero.

Heating the same gas spreads the normalized curve to higher speeds. It becomes wider and its maximum falls because the total area must stay one. This says nothing by itself about the total number of molecules: doubling that number doubles a count histogram, but leaves a normalized probability density unchanged.

F(v)=4\pi\left(\frac{m}{2\pi k_{\mathrm B}T}\right)^{3/2}v^2e^{-mv^2/(2k_{\mathrm B}T)}

Integrate F over a speed interval to obtain a probability.

Explain this equation

Read it in words

The probability density at speed v is a normalization factor times the number-of-directions factor v squared and a decaying energy factor.

Symbols and units

$F(v)$: Probability per unit speed at vs/m
$v$: Nonnegative molecular speedm/s
$m$: Mass of one molecule, not molar masskg
$T$: Absolute gas temperatureK
$k_{\mathrm B}$: Boltzmann constantJ/K

Signs and operations

The negative exponential penalizes high kinetic energy. The integral from zero to infinity is one; the curve height is not a probability.

When it applies

Dilute, nonrelativistic classical gas in equilibrium; velocities are measured relative to the gas bulk motion.

Make it concrete

If F is nearly 0.0015 s/m throughout a 40 m/s bin, its probability is approximately 0.060, or 6%.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Equation (LaTeX): F(v)=4\pi\left(\frac{m}{2\pi k_{\mathrm B}T}\right)^{3/2}v^2e^{-mv^2/(2k_{\mathrm B}T)}
Read it as: The probability density at speed v is a normalization factor times the number-of-directions factor v squared and a decaying energy factor.
Symbols:
F(v): Probability per unit speed at v; s/m
v: Nonnegative molecular speed; m/s
m: Mass of one molecule, not molar mass; kg
T: Absolute gas temperature; K
k_{\mathrm B}: Boltzmann constant; J/K
Signs and operations: The negative exponential penalizes high kinetic energy. The integral from zero to infinity is one; the curve height is not a probability.
Conditions: Dilute, nonrelativistic classical gas in equilibrium; velocities are measured relative to the gas bulk motion.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

WORKED EXAMPLE

Turn a graph height into a molecule count

A normalized speed density is approximately constant at 0.0015 s/m over a 40 m/s-wide bin. In a sample containing 50,000 molecules, estimate the number in that bin.

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One gas, two temperatures

Inspect a plotted point: u = v/a = 0

T₀: 0
2T₀: 0

Original normalized Maxwell curves. u = v/a, where a is the most probable speed at T₀. The area under each complete curve is one; only u from 0 to 4 is shown. At 2T₀ the peak shifts by √2 and drops by 1/√2. Lines connect the plotted samples.

STOP & THINK

At the same temperature, a gas sample doubles its molecule count without changing its normalized speed distribution. What happens to the area under F(v)?

It doubles
It stays one
It becomes the mean speed

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: A distribution predicts a fraction of molecules, not one speed
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
F(v)=4\pi\left(\frac{m}{2\pi k_{\mathrm B}T}\right)^{3/2}v^2e^{-mv^2/(2k_{\mathrm B}T)}
Meaning: Integrate F over a speed interval to obtain a probability.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

08 / Gases: state variables, molecular speeds and collisions

Count energy terms before choosing a heat capacity

A molecule can store energy while its centre moves, while it rotates, and while its atoms vibrate relative to one another. Temperature measures an equilibrium state; it does not promise that all this energy is translational. Pressure in the dilute ideal-gas model still comes from translational momentum striking the walls.

The classical equipartition rule assigns an average kBT/2 to each independent quadratic energy term that is thermally active. Three translation terms give 3kBT/2 per molecule. For a linear diatomic molecule, two ordinary rotational terms can add kBT. A fully active vibrational mode adds two quadratic terms: one kinetic and one potential.

The word active matters. Quantum energy spacing can suppress rotational or vibrational contributions at a given temperature. Use the model stated in the problem; do not decide that every diatomic gas has one universal heat capacity at every temperature. Chemical dissociation and changes of composition require a different energy accounting.

When f active quadratic terms remain constant over the temperature range, U changes as f nRT/2. The heat capacity at fixed volume follows by differentiating with respect to T. Adding the ideal-gas expansion work at constant pressure gives Cp = Cv + R. More active modes mean more energy is needed for the same temperature rise, and a smaller gamma.

C_V=\frac f2R,\quad C_p=\left(\frac f2+1\right)R,\quad\gamma=1+\frac2f

These are molar capacities for a fixed-composition ideal gas with a constant active-mode count.

Explain this equation

Read it in words

Count f active quadratic energy terms. Half f times R is the molar constant-volume capacity; add R for constant pressure.

Symbols and units

$f$: Number of active quadratic energy terms per moleculedimensionless
$R$: Molar gas constantJ/(mol K)
$C_V$: Molar heat capacity at fixed volumeJ/(mol K)
$C_p$: Molar heat capacity at fixed pressureJ/(mol K)
$\gamma$: Ratio Cp/Cvdimensionless

Signs and operations

The additional R accounts for boundary work at constant pressure. The factor one half belongs to each quadratic term, not each atom.

When it applies

Classical equipartition for the active modes, ideal gas, fixed composition and approximately constant capacities. Integrate Cv(T) when modes activate gradually.

Make it concrete

With f = 5, Cv = 2.5R, Cp = 3.5R and gamma = 1.4. A fully active vibrational mode changes f to 7.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Equation (LaTeX): C_V=\frac f2R,\quad C_p=\left(\frac f2+1\right)R,\quad\gamma=1+\frac2f
Read it as: Count f active quadratic energy terms. Half f times R is the molar constant-volume capacity; add R for constant pressure.
Symbols:
f: Number of active quadratic energy terms per molecule; dimensionless
R: Molar gas constant; J/(mol K)
C_V: Molar heat capacity at fixed volume; J/(mol K)
C_p: Molar heat capacity at fixed pressure; J/(mol K)
\gamma: Ratio Cp/Cv; dimensionless
Signs and operations: The additional R accounts for boundary work at constant pressure. The factor one half belongs to each quadratic term, not each atom.
Conditions: Classical equipartition for the active modes, ideal gas, fixed composition and approximately constant capacities. Integrate Cv(T) when modes activate gradually.

Help me explain the equation in words, predict how changing a quantity affects the result, and apply it to a small numerical example.

WORKED EXAMPLE

Add capacities before taking their ratio

An ideal, nonreacting mixture contains 1 mol of a monatomic gas and 2 mol of a diatomic gas with rotation active and vibration frozen. Find the total constant-volume capacity in units of R and the mixture gamma.

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STOP & THINK

One harmonic vibrational mode becomes fully active in a diatomic ideal-gas model. How much does the molar Cv increase?

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: Count energy terms before choosing a heat capacity
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
C_V=\frac f2R,\quad C_p=\left(\frac f2+1\right)R,\quad\gamma=1+\frac2f
Meaning: These are molar capacities for a fixed-composition ideal gas with a constant active-mode count.

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09 / Gases: state variables, molecular speeds and collisions

Mean free path counts distance between collisions

A molecule in a dilute gas travels in short segments between collisions. Mean free path is the average length of those segments. It is not the straight-line displacement after many collisions, because directions repeatedly change.

In a hard-sphere model with molecular diameter d, two molecular centres collide when they approach within d of each other. A moving centre therefore sweeps an effective collision cross-section πd², not π(d/2)². Treating all other molecules as stationary would predict a path scale 1/(n_Nπd²).

Other molecules are moving too. Averaging their relative motion in the equilibrium hard-sphere model adds a factor √2 to the collision rate, giving ℓ = 1/(√2 n_Nπd²). This formula assumes a dilute, identical-particle gas with an effective collision diameter; mixtures and real interaction potentials need more detailed collision models.

Using a finite collision diameter alongside an ideal equation of state is an approximation, not a contradiction to be ignored: the gas is dilute enough that the excluded volume and interaction energy have negligible effects on bulk state properties, while occasional collisions still establish equilibrium and limit free flights.

\ell=\frac{1}{\sqrt2\,\pi d^2n_N}=\frac{k_{\mathrm B}T}{\sqrt2\,\pi d^2p}

The second form uses the ideal-gas number density p/(k_BT). Here ℓ denotes mean free path, avoiding confusion with a wave’s wavelength.

WORKED EXAMPLE

Use collision diameter and number density consistently

A dilute hard-sphere gas has number density 1.0 × 10²⁵ m⁻³ and effective molecular diameter 3.0 × 10⁻¹⁰ m. Estimate the mean free path.

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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: Mean free path counts distance between collisions
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
\ell=\frac{1}{\sqrt2\,\pi d^2n_N}=\frac{k_{\mathrm B}T}{\sqrt2\,\pi d^2p}
Meaning: The second form uses the ideal-gas number density p/(k_BT). Here ℓ denotes mean free path, avoiding confusion with a wave’s wavelength.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

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10 / Gases: state variables, molecular speeds and collisions

State what is held fixed before predicting collision changes

At fixed temperature and effective diameter, increasing pressure increases number density and shortens mean free path in inverse proportion. At fixed pressure, increasing temperature lowers number density and lengthens mean free path in proportion to T.

For a sealed rigid container, number density is fixed even when temperature changes. In the constant-diameter model, mean free path then stays the same while molecular speeds increase. Collisions happen more frequently in time, even though the average distance between them is unchanged.

This distinction separates collision distance from collision time. A characteristic mean time between collisions is ℓ divided by the mean molecular speed under the same equilibrium model. Use the ordinary mean speed for this rate relation, not automatically the rms speed.

Effective collision diameters can depend on temperature for real interaction potentials. The simple proportionalities therefore belong to the stated hard-sphere approximation. A question supplying a constant d authorizes that approximation; an experiment may require checking it.

WORKED EXAMPLE

Explain how collisions change without changing mean free path

A sealed rigid vessel contains a dilute hard-sphere gas. Its absolute temperature is raised by a factor of four, and the effective molecular diameter is taken as constant. Compare pressure, rms speed, mean free path and characteristic collision time.

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: State what is held fixed before predicting collision changes
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

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11 / Gases: state variables, molecular speeds and collisions

Compare a molecular length with the apparatus length

A gas can often be treated as a continuous fluid when the apparatus length is much larger than mean free path. If the channel width or object size approaches the molecular path length, molecules may interact with boundaries before enough intermolecular collisions occur to support the usual local fluid description.

The ratio ℓ/L is called the Knudsen number. It is a different dimensionless comparison from the Reynolds number: one compares a molecular length with a system length, while the other compares inertial and viscous flow effects. A low Reynolds number does not by itself guarantee the continuum approximation.

This provides a final model audit. Use the gas law for an appropriate state, kinetic theory for equilibrium velocity statistics and collision models for paths or rates. Check the required density, temperature and length scales before combining them.

STOP & THINK

A channel width becomes comparable to the gas mean free path. What needs reconsideration?

Only the choice of pressure units
The continuum-fluid approximation near boundaries
The arithmetic definition of a mole

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Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Gases: state variables, molecular speeds and collisions
Focus: Compare a molecular length with the apparatus length
Learning goals:
- Use absolute pressure, Kelvin temperature and consistent mole/mass units.
- Distinguish mass density, number density and partial pressure.
- Derive the pressure relation from molecular momentum transfer.
- Compare molecular speeds and translational energies at a common temperature.
- Distinguish vector, mean-speed and rms averages.
- Explain the collision cross-section and mean-free-path model.
- Predict collision changes under fixed-pressure and fixed-volume constraints.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

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BRING THE IDEAS TOGETHER

What should stay with you

The ideal-gas law requires absolute pressure and temperature.
A fixed-volume relation also needs fixed gas amount unless amount changes are included.
Pressure depends on mean squared thermal speed, not mean vector velocity.
Equal-temperature species share mean translational energy but can have different rms speeds.
Mean free path is a travelled distance between collisions, not net displacement.
At fixed number density and collision diameter, heating changes collision time but not mean free path.
Continuum validity and flow regime require different dimensionless checks.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

Two ideal gases share pressure and temperature. Gas B has three times the molar mass of gas A. What is ρ_B/ρ_A?

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SELF CHECK 2

At fixed temperature and collision diameter, pressure doubles. What is the new-to-old mean-free-path ratio?

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SELF CHECK 3

A gas is heated from 20°C to 40°C. Does its rms speed increase by √2?

Yes, because the stated temperature doubles
No; the relevant ratio is 313.15/293.15
No; rms speed never depends on temperature

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SELF CHECK 4

Two ideal-gas species at the same temperature have different molecular masses. Which is equal per particle?

Rms speed
Mean translational kinetic energy
Rms momentum

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SELF CHECK 5

In a sealed rigid hard-sphere gas, temperature rises while diameter is treated as constant. What happens to the mean free path?

It decreases solely because speeds increase
It remains unchanged because number density is unchanged
It must increase in proportion to temperature

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APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

A gas at rest has zero average molecular velocity. Does that mean its molecules have zero average kinetic energy?

Compare your reasoning

No. Opposite velocity directions cancel in the vector average. Squared speeds are nonnegative, so their average can remain large and determine a nonzero translational kinetic energy.

Revisit the explanation →

A few days later: change the situation

A gas mixture contains different species. How should you find its heat-capacity ratio?

Compare your reasoning

For a shared temperature rise, add each amount times its molar Cv to find total storage capacity. Add ntotalR to get total Cp under the ideal fixed-composition model. Divide the totals; do not average the species ratios.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

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