Revisica
Sign in
ENGR5002 / Week 1
Week 1 Exercise 6Exercise sheetCurrent spec2:120 min

Week 1 Exercises, Exercise 6 (Pressure Variation in a Gas), all parts

At sea level, the pressure of air is p0=101325p_0 = 101325 Pa. Assume the atmosphere is at a uniform temperature of T=300T = 300 K. The universal gas constant is R=8.314 J mol−1 K−1R = 8.314\ \mathrm{J\,mol^{-1}\,K^{-1}}, and gravitational acceleration is g=9.81 m s−2g = 9.81\ \mathrm{m\,s^{-2}}.

Assume that air has the following molar composition:

  • 78% N2\mathrm{N_2} (MN2=28.02 g mol−1M_{N_2} = 28.02\ \mathrm{g\,mol^{-1}})
  • 21% O2\mathrm{O_2} (MO2=32.00 g mol−1M_{O_2} = 32.00\ \mathrm{g\,mol^{-1}})
  • 1% Ar (MAr=39.95 g mol−1M_{Ar} = 39.95\ \mathrm{g\,mol^{-1}})
Formulas you may need
  • Differential hydrostatic equation: dpdz=−ρg\dfrac{dp}{dz} = -\rho g (zz upwards) (on the formula sheet)
  • Ideal-gas density: ρ=pMRT\rho = \dfrac{p M}{R T} (given in the question; learn this)
  • Mixture molar mass: Mave=∑xiMiM_{ave} = \sum x_i M_i (on the ENGR271 2026 exam formula sheet, mass transfer section)
  • Isothermal atmosphere: p=p0exp⁡ ⁣(−MgzRT)p = p_0 \exp\!\left(-\dfrac{M g z}{R T}\right) (learn the derivation; not on the sheet)
  1. (a)
    Calculate the average molar mass of the mixture using the given composition.
  2. (b)
    Derive an expression for p(z)p(z) assuming density ρ\rho is constant and equal to the one at sea level.
  3. (c)
    Derive an expression for p(z)p(z) assuming ρ=pMwRT\rho = \dfrac{p M_w}{R T} (ideal gas law, with TT = const).
  4. (d)
    Calculate the pressure in both cases at altitudes of 100 m, 500 m, 1 km, and 5 km.