Unit 61: Engineering Thermodynamics Lesson 3: The Gas Laws Objective NDGTA • The purpose of this lesson is to continue to explain the Gas laws The Gas Laws NDGTA • Experiments have established three gas laws that describe the thermal behaviour of gases: Boyle’s Law, Charles’s Law and Charles’s Law of Pressures (the Pressure Law) Boyle’s Law NDGTA • This relates the pressure and volume of a gas and emanated from the work undertaken by Boyle around 1600. • “The pressure of a fixed mass of gas is inversely proportional to its volume it the temperature is constant.” p ∞ 1/V Thus p = constant / V i.e. pV = constant Boyle’s Law NDGTA • The value of the constant depends upon the mass of gas and the temperature. Pressure, P Increasing Temperatures 0 Volume, V Boyle’s Law NDGTA pV Increasing tempertures 0 Pressure, p Boyle’s Law NDGTA Note: Each point on these graphs represent a constant temperature for a fixed mass of gas i.e. these are termed isotherms. p Increasing tempertures 0 Volume, V Boyle’s Law NDGTA • Measurements show show that the law is obeyed only when the density of the gas is low. Charles’s Law NDGTA • The connection between the volume change with temperature of a fixed mass of gas kept at constant pressure was published by Charles in 1787 (and independently by Gay-Lussac in 1802). • “The volume V of a gas of fixed mass and pressure, measured at temperature θ, is related to its volume V0 at the ice point by… V = V0(1 + αθ) • …where α is the cubic expansivity of the gas at constant pressure and has roughly the same value (1/273) for all gases at low pressure.” Pressure Law NDGTA • A similar relationship is found to exist between the pressure p of a fixed mass of a gas kept at constant volume and the temperature θ… p = p0(1 + βθ) • …where p0 is the pressure at the ice point and β is a constant know as the pressure coefficient of gas – it has practicably the same value as α. The Ideal Gas Scale NDGTA • Lord Kelvin (G.P. Thompson) suggested that the standard scale of temperature should be based on an imaginary ideal gas with properties that were those of a real gas at low pressures i.e. it obeyed Boyle’s Law. • He proposed that the product of the pressure and the volume of this ideal gas be used as the thermometric property. The Ideal Gas Scale NDGTA • The simplest procedure is then to say that if p1 and p2 are measures and V1 and V2 are the corresponding volumes of a fixed mass of gas at temperatures T1 and T2 on this scale then these temperatures are defined as… T1 = p1V1 T2 p2V2 Thus if Volume is kept constant, then… T1 = p1 T2 p2 The Ideal Gas Scale NDGTA • Thus if an unknown temperature T can be determined if the volume remains constant by knowing the temperature and pressure at the triple point… T = 273.16(p1 / p2) The Ideal Gas Equation • From pervious… T1 = p1V1 T2 p2V2 • Rearranging give… p1V1 = p2V2 T1 T 2 • Thus for an ideal gas… pV = constant T NDGTA The Ideal Gas Equation NDGTA • Note:… pV = constant T • …is not based upon experiment but incorporates two definitions: Boyle’s Law and and the temperature of an ideal gas as a quantity proportional to pV. The Ideal Gas Equation NDGTA • The value of the constant depends on the mass of gas and experiments with real gases at low enough pressures show that it has the same value R for all gases if one mole is considered… pV = RT (this is called an equation of state) • R is called the universal molar gas constant and has a value of 8.31 J.mol-1.K-1 The Ideal Gas Equation NDGTA • For air M = 28.97 kg/kmol, so that R is 0.287 kJ/kg.K (i.e. 287 N.m/kg.K) • Other forms of the ideal gas equation are… PV = mRT; P = ρRT; PV = NRuT Important Notes NDGTA • Note 1: The mole is the amount of substance which contains the same number of elementary entities as there atoms in 12grams of carbon-12. Experiment show this to be 6.02 x 1023 – a value denoted by L and called Avogadro’s constant. • Note 2: the relative molar mass Mr… Mr = mass of a molecule of substance x 12 mass of carbon 12-atom Important Notes NDGTA • Note 3: the molar mass Mm may thus be the mass of a substance (in kg) containing L molecules i.e. for oxygen for example… Mm = 32 x 10-3 kg.mol-1. • Note 4: for n moles of gas… pV = nRT where V is the molar volume… V = molar mass (in kg.mol-1) = Mm (m3.mol-1) density (in kg. m-3) ρ Important Notes NDGTA • Note 5: one mole at s.t.p. has a volume of of 22.4 x 10-3 m3 (22.4 x 103 m3) • The compressibility factor Z helps in determining whether or not the ideal-gas equation should be used. Z is defined as… Z = Pv/RT • For the ideal gas equation to be use Z = 1 (or very nearly 1). If Z does not equal 1 then the above equation must be used. Important Notes NDGTA Important Notes NDGTA • The compressibility factor can be determined for any gas by using a generalised compressibility chart – Handout Appendix G-1 in Schaum: Thermodynamics for Engineers. • In the chart the reduced pressure PR and reduced Temperature TR must be used… PR = P/Pcr; TR = T/Tcr; vR = v/(RTcr/Pcr) Where Pcr and Tcr are the critical point pressure and temperature respectively Important Notes NDGTA • A car’s tyre has a volume of 0.6 m3 and is inflated to a gauge pressure of 200kPa. Calculate the mass of air in the tyre if the temperature is 20oC. • Assume air to be at ideal gas conditions. • Thus PV = mRT, where P and T are absolute pressure and absolute temperature respectively. Important Notes Noting that Patm = 100kPa Then P = 200 + 100 = 300 kPa And T = 20 + 273 = 293 K Then m = PV / RT i.e. m = (3 x 105)(0.6)/(287)(293) = 2.14 kg NDGTA Equations of State of a Non-ideal Gas NDGTA • There are many equations of state that can be used to account for a non-ideal gas i.e. when the pressure may be relatively high (say >4mPa) or when the temperature is near the saturation temperature. • One such equation of state is van der Waals equation which tries to take account of the volume of the gas molecules themselves and the attractive forces between molecules Equations of State of a Non-ideal Gas • Van der Waals equation of state… P = RT - a (v – b) v2 Where a and b are constants NDGTA Equations of State of a Non-ideal Gas NDGTA • An improved equation of state is the RedlichKwong equation… P = RT a (v – b) v(v + b)√T Where a and b are constants but have different values Example NDGTA • Given the table (Table B8 in Appendices in Schaum) calculate the pressure of steam at a temperature of 500oC and density of 24kg/m3 using (a) the ideal-gas equation (b) van der Walls equation (c) Redlich-Kwong equation (d) the compressibility factor. – Note the gas constant for steam is 0.462 kJ/kg.K. – Note use the compressibility chart to determine TR and PR.