15.2 Equation of state
- Syllabus
- 9702–2028–2029
- Topic
- 15.2
- Level
- A2
Learning objectives
15.2.1A gas obeying pV ∝ T, where T is the thermodynamic temperature, is known• understand that a gas obeying pV ∝ T, where T is the thermodynamic temperature, is known as an ideal gas15.2.2The equation of state for an ideal gas expressed as pV = nRT• recall and use the equation of state for an ideal gas expressed as pV = nRT, where n = amount of substance (number of moles) and as pV = NkT, where N = number of molecules15.2.3That the Boltzmann constant k is given by k = R / NA• recall that the Boltzmann constant k is given by k = R / NA
An ideal gas obeys pV proportional to thermodynamic temperature
An ideal gas is a gas that obeys pV ∝ T, where p is pressure, V is volume and T is thermodynamic temperature, across the states considered.
Forafixedamountofidealgas:pV/T=constantandp1V1/T1=p2V2/T2
| Additional fixed quantity | Resulting proportionality |
|---|---|
| V and amount fixed | p ∝ T |
| p and amount fixed | V ∝ T |
| T and amount fixed | pV = constant |
A sealed rigid sample heated from 300 K to 450 K has p₂/p₁=T₂/T₁=1.50. This p∝T result follows only because both amount and volume are fixed.
Do not define every ideal gas as merely p∝T: the defining relation contains pV. Use kelvin, not Celsius, in ratios. Real gases approximate ideal behavior best under appropriate conditions; the ideal equation is the model definition.
Choose the molar or molecular ideal-gas equation
| Given gas quantity | Equation | Constant |
|---|---|---|
| amount n in mol | pV = nRT | R = 8.31 J mol⁻¹ K⁻¹ |
| number N of molecules | pV = NkT | k = 1.38 × 10⁻²³ J K⁻¹ |
| Symbol | SI meaning/unit |
|---|---|
| p | pressure in Pa |
| V | volume in m³ |
| T | thermodynamic temperature in K |
| n | amount in mol |
| N | number of molecules, a count |
For n=0.0160 mol, T=282 K and V=1.87×10⁻⁴ m³, p=nRT/V=(0.0160)(8.31)(282)/(1.87×10⁻⁴)=2.01×10⁵ Pa.
If pV=270 J and kT=8.0×10⁻²¹ J per molecule, N=pV/(kT)=270/(8.0×10⁻²¹)=3.4×10²² molecules.
N=nNAandk=R/NA,soNkT=(nNA)(R/NA)T=nRT
n and N are not interchangeable: n is measured in mol and N is a molecule count. Match n with R or N with k, convert cm³/dm³ to m³, and always use kelvin.
Boltzmann’s constant links particle-scale temperature to molar gas constant
Boltzmann constant k=R/N_A, so kT is the thermal energy scale per particle while RT is the corresponding molar scale.
Use k when counting individual particles and R when working with moles; the two forms describe the same thermal physics at different scales.
The average translational energy scale of one molecule uses kT, whereas one mole uses RT.
k and R are not interchangeable numbers; their units and particle-count basis differ.