Lesson summary
15.2.2 Ideal Gas Equation — AS PHYSICS 9702 notes: Use pV = nRT and pV = NkT and apply the proportionalities of an ideal gas.Ideal gas — a gas obeying pV ∝ T, where p is in Pa, V in m³ and T in kelvinConstant volume — p ∝ T, because faster molecules with more kinetic energy hit t…
- Constant pressure
- Constant temperature
- What the symbols mean
- Definition of an ideal gas
- Unit conversions
- Avogadro's hypothesis
Section 1 of 10Ideal gas behaviour
Ideal gas
- Ideal gas a gas obeying pV ∝ T, where p is in Pa, V in m³ and T in kelvin
Constant volume
- Constant volume p ∝ T, because faster molecules with more kinetic energy hit the walls more often and harder
Constant pressure
- V ∝ T, because faster molecules move further apart and the gas expands
Constant temperature
- Constant temperature p ∝ 1/V, because a smaller container gives more collisions per unit area
Temperature
- Temperature always use kelvin in these equations
Section 2 of 10The equation of state
1Ideal gas equation
- Ideal gas equation pV = nRTWhat the symbols mean: p = pressure (Pa) · V = volume (m3) · n = amount of gas (mol) · R = 8.31 J mol−1 K−1 · T = temperature (K = °C + 273)
- p is the pressure in Pa
- V is the volume in m³
- n is the amount in mol
- R is the molar gas constant, 8.31 J mol⁻¹ K⁻¹
- T is the thermodynamic temperature in K
2Definition of an ideal gas
- Definition of an ideal gas a gas which obeys pV = nRT at all pressures, volumes and temperatures
3Unit conversions
- 1 litre = 1.0 × 10⁻³ m³, 1 cm³ = 1.0 × 10⁻⁶ m³ and 1 atm = 1.01 × 10⁵ Pa
4Avogadro's hypothesis
- Avogadro's hypothesis at the same p and T, equal volumes of any ideal gas contain equal numbers of molecules
- At 273 K and 1.01 × 10⁵ Pa one mole occupies 0.0224 m³
Section 3 of 10In terms of the number of molecules
pV = NkT
- N = nN_A,
- and k = R/NA = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constantWhat the symbols mean: p = pressure (Pa) · V = volume (m3) · N = number of molecules · k = Boltzmann constant (1.38 × 10−23 J K−1) · T = temperature (K)
- and k = R/NA = 1.38 × 10⁻²³ J K⁻¹ is the Boltzmann constant
Choosing the form
- Choosing the form use pV = nRT for moles and pV = NkT when counting molecules
Number density
- N/V = p/(kT), about 2.5 × 10²⁵ molecules per m³ at room conditions
Meaning of k
- Meaning of k it links microscopic energy to macroscopic temperature and is small because the energy rise per molecule per kelvin is tiny
Section 4 of 10Fixed amount of gas
Combined relation
- Combined relation p₁V₁/T₁ = p₂V₂/T₂ when n is constant
Pressure law
- Pressure law at constant volume, p₁/T₁ = p₂/T₂
Changing amount
- Changing amount if gas leaks or is pumped, apply pV = nRT to each state separately
Section 5 of 10Density and graphs
Density and graphs
Gas density
- ρ = m/V = pM/(RT), with M in kg mol⁻¹What the symbols mean: ρ = density (kg m−3 or g cm−3) · m = mass (kg) · V = volume (m3)
- At fixed temperature, doubling pressure doubles density
p against V at constant T
- p against V at constant T a hyperbola
pV against p at constant T
- pV against p at constant T a horizontal line at nRT, higher for larger T
V against T at constant p
- V against T at constant p a straight line through the origin with gradient nR/p
Real gases
- Real gases they deviate from these shapes at high pressure
Section 6 of 10Problem-solving
Identify what is constant
Convert to SI
- Convert to SI kelvin, pascals, cubic metres and kilograms
Decide on moles (R) or molecules (k)
Rearrange symbolically before substituting
Check the answer
- Check the answer doubling the pressure at fixed volume doubles T, and leaking gas in a rigid container lowers p in proportion to n
- Gauge pressure
- add atmospheric pressure to obtain the absolute pressure used in the equation
Section 7 of 10Link with kinetic theory
Comparison
- Comparison pV = NkT with pV = ⅓Nm⟨c²⟩ shows mean translational kinetic energy is proportional to T
Constants
- R = N_Ak, so either form gives identical results with consistent units
Section 8 of 10More from the specification
More from the specification
Ideal Gas Equation of State
- The macroscopic behaviour of an ideal gas confined in a cylinder of fixed volume is governed by pV = nRT, relating pressure p, volume V, amount of substance n in moles, gas constant R,
- and thermodynamic temperature T.
Pressure-Temperature Relationship
- At constant volume, gas pressure is directly proportional to thermodynamic temperature;
- increasing temperature increases the mean kinetic energy of gas molecules,
- resulting in more frequent and energetic collisions with cylinder walls.
Properties of Ideal Gases
- An ideal gas is assumed to consist of point particles with negligible molecular volume and zero intermolecular attractive forces except during instantaneous elastic collisions.
Ideal Gas Equation of State
- The macroscopic behaviour of an ideal gas confined in a cylinder of fixed volume is governed by pV = nRT, relating pressure p, volume V, amount of substance n in moles, gas constant R,
- and thermodynamic temperature T.
Pressure-Temperature Relationship
- At constant volume, gas pressure is directly proportional to thermodynamic temperature;
- increasing temperature increases the mean kinetic energy of gas molecules,
- resulting in more frequent and energetic collisions with cylinder walls.
Properties of Ideal Gases
- An ideal gas is assumed to consist of point particles with negligible molecular volume and zero intermolecular attractive forces except during instantaneous elastic collisions.
Directly proportional
- Two quantities increase together in a constant ratio, giving a straight line through the origin.
Section 9 of 10Key terms
| Term | Meaning |
|---|---|
| Ideal gas | a gas which obeys pV = nRT at all pressures, volumes and temperatures |
| Molar gas constant | R = 8.31 J mol⁻¹ K⁻¹ |
| Boltzmann constant | k = R/NA = 1.38 × 10⁻²³ J K⁻¹ |
| Equation of state | the equation relating p, V, n and T for a gas |
| Number density | the number of molecules per unit volume, N/V |
| Absolute pressure | pressure measured from zero, equal to gauge pressure plus atmospheric pressure |
Section 10 of 10Exam technique
1
Using Celsius temperatures — convert to kelvin first.
2
Leaving volume in litres or cm³ — convert to m³.
3
Using gauge pressure — add atmospheric pressure to get the absolute pressure.
4
Using R with N or k with n — pair R with n and k with N.