Unit 61: Engineering
Thermodynamics
Lesson 3: The Gas Laws
Objective
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• The purpose of this lesson is to continue to
explain the Gas laws
The Gas Laws
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• 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
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• 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
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• The value of the constant depends upon the
mass of gas and the temperature.
Pressure, P
Increasing
Temperatures
0
Volume, V
Boyle’s Law
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pV
Increasing
tempertures
0
Pressure, p
Boyle’s Law
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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
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• Measurements show show that the law is
obeyed only when the density of the gas is
low.
Charles’s Law
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• 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
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• 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
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• 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
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• 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
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• 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
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The Ideal Gas Equation
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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Important Notes
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• 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
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• 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
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Equations of State of a
Non-ideal Gas
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• 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
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Equations of State of a
Non-ideal Gas
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• 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
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• 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.