Lessons 9–10 · 2 lessons · P. Habibullayev, A. Boydedayev, A. Bahromov, K. Suyarov, J. Usarov, M. Yuldasheva. Physics Grade 9, revised and expanded 3rd edition. G‘afur G‘ulom Publishing House, Tashkent, 2019
9
Equations of state of an ideal gas
Textbook: pp. 30–32
GoalKnow and use the ideal gas equation of state (Mendeleev–Clapeyron) and the Clapeyron equation.
New words
equation of state · holat tenglamasimacroscopic parameters · makroskopik parametrlarMendeleev–Clapeyron equation · Mendeleyev–Klapeyron tenglamasiClapeyron equation · Klapeyron tenglamasi
Explanation
The state of an ideal gas is described by three macroscopic parameters: pressure p, volume V and absolute temperature T. Substituting n = N/V = νN_A/V into p = nkT gives pV = νN_A kT = νRT, and since ν = m/M, pV = (m/M)·RT. This is the Mendeleev–Clapeyron equation, linking the gas’s mass, molar mass, pressure, volume and temperature. If two of the three parameters are known, the third can be found, e.g. V = mRT/(Mp), ρ = pM/(RT). For a constant mass (m = const) and two states pV/T = const, i.e. p₁V₁/T₁ = p₂V₂/T₂ (the Clapeyron equation). In calculations T is in kelvins, p in pascals, V in m³ and M in kg/mol.
Worked examples
A 10 l (0.01 m³) vessel holds 2 g of helium (M = 0.004 kg/mol) at 300 K. Pressure: p = mRT/(MV) = 0.002·8.31·300/(0.004·0.01) ≈ 1.25·10⁵ Pa ≈ 125 kPa.
A 5 l vessel holds gas at 400 K and 166.2 kPa. The amount of substance is ν = pV/(RT) = 166 200·0.005/(8.31·400) = 831/3324 = 0.25 mol.
Class activity
“Parameter cards”: from cards marked p, V, T, m, M a group draws two at random; the teacher gives the others; the group chooses the formula and expresses the unknown (without calculating). Paper and pencil only.
Practice
1
Write the Mendeleev–Clapeyron equation and say what each letter means.
pV = (m/M)RT: p – pressure, V – volume, m – mass, M – molar mass, R – universal gas constant, T – absolute temperature.
2
Estimate the density of CO₂ (M = 0.044 kg/mol) at 300 K and 100 kPa.
about 1.8 kg/m³
3
A gas goes from 300 K, 100 kPa, 6 l to 600 K and 150 kPa. What is the new volume in litres?
8
4
Why is the temperature taken in kelvins in the equation of state?
Because gas pressure and molecular energy are proportional to the absolute temperature. If the Celsius temperature were used, the formula would give zero pressure at 0 °C, which is false.