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Ideal Gas
S.GunabalanAssociate ProfessorMechanical Engineering DepartmentBharathiyar College of Engineering & TechnologyKaraikal - 609 609.e-Mail : [email protected]
Part - 2
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Ideal Gas Equation of stateFor perfect gas, the ideal gas equation is
( Molar Volume )
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Ideal Gas Equation of state
mas of gas in Kg
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Ideal Gas Equation of state
( Molar Volume )
= = m3 = Volume (V)
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Ideal Gas Equation of stateFor perfect gas, the ideal gas equation is ( Molar Volume )
= = m3 = Volume (V) Specific Volume –(m3 /Kg)Density - Kg/ m3 Molar Volume
specific Volume V – Volume m3
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Ideal Gas Equation of state ( Molar Volume )
= = m3 = Volume (V)
Molar Volume specific Volume V – Volume m3
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Units of Pressure
• atmospheric pressure (1 Atm.)= 1.013 bar, = 101325 Pascal (Pa = N/m²); = 760 millimeters of mercury absolute
(mmHgA) = 760 Torr (1 Torr = 1 mm HgA)
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Gas Constant RGas Molar Weight ( M)Kg/Kmol Gas Constant (R )KJ/KgK
Air 28.97 0.287
Nitrogen 28.01 0.297
Oxygen 32 0.260
Hydrogen 2.016 4.124
Helium 4.004 2.077
Carbon dioxide 44.01 0.189
Steam 18.02 0.461
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Note:
The specific volume of a substance is the ratio of the substance's volume to its mass. It is the reciprocal of density and is an intrinsic property of matter.
Substance Name Density Specific Volume
Kg/m3 m3/Kg
Air 1.2 0.83
Ice 916.7 0.00109
Water (liquid) 1000 0.00100
Salt Water 1030 0.00097
Mercury 13546 0.00007
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Note: Molar volume
• The molar volume is the volume occupied by one mole of a substance at a given temperature and pressure.
• It is equal to the molar mass (M) divided by the mass density (ρ).
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Specific heat
• The specific heat - the amount of heat required to raise a unit mass of the substance through a unit rise in temperature.
• For gases– At constant pressure
• Cp
– At constant Volume• Cv
This classic relationship between the specific heats of an ideal gas is called Mayer’s equation
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Specific heat
• The specific heat - the amount of heat required to raise a unit mass of the substance through a unit rise in temperature.
• The product of mass and specific heat (mCv) is called the heat capacity(Cv) at constant volume (J/K)
• The latent heat is the amount of heat transfer required to cause a phase change in unit mass of a substance at a constant pressure and temperature
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Questions -1
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P = ?
V = Givenm = > M (Molar Weight) x one K Mole of N2
= 28.01 = m
R
• specific volume ( of the gas
Gas Molar Weight ( M)Kg/Kmol
Air 28.97
Nitrogen 28.01
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= 1.4 given – (1)
Mayer’s equation
• – Two Equation two Variables
– R – is known from part a).
R
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c)
Vessel capacity is same so V1 = V2
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specific internal energy (kJ/kg) =specific enthalpy (kJ/kg) =Increase in specific entropy (kJ/kgK) =Magnitude and sign Heat transfer (kJ) = + w
= m + p(V2-V1)
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• Given data• Vessel capacity (Volume)
– V = 3 m3
• Contain 1 K Mole of N2 At T = 90 0C
Answera) Calculate the pressure () and specific volume
( of the gas
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From above table one K Mole of N2 = 28.01 = m
R 0.2968 KJ/KgKSubstituting in the ideal gas equation
Kelvin
945.58959 KN/m2
Gas Molar Weight ( M)Kg/Kmol
Air 28.97
Nitrogen 28.01
J = Nm
and
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• specific volume ( of the gas
b) If the ratio of specific heat is 1.4, evaluate value of Cp and Cv
• ratio of specific heat is 1.4
Mayer’s equation
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• = 1.4
• 0.297 KJ/KgK • 0.297 • = 0.7425 KJ/KgK• = 1.4 • = 1.4 • = 1.0395 KJ/KgK
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C) Subsequently, the gas cools to atmospheric temperature of 20 0C; evaluate the final pressure of the gas.T2 = 20 0C
P2 = ?
Vessel capacity is same so V1 = V2
Substituting the values
= 763.2445
From Given dataVessel capacity (Volume) V1 = 3 m3
T1 = 90 0CCorresponding calculated 945.58959 KN/m2
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d) Evaluate the increase in specific internal energy, Enthalpy , the increase in specific entropy and magnitude and sign of heat transfer.specific internal energy (kJ/kg) = = 0.7425 KJ/KgK
= -51.975 KJ/Kgspecific enthalpy (kJ/kg) = = 1.0395 KJ/KgK
= -72.765 KJ/Kg
T1 = 90 0CT2 = 20 0C = 0.7425 KJ/KgK = 1.0395 KJ/KgK
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Increase in specific entropy (kJ/kgK) =Constant volume V1 = V2
= = = - 0.06908kJ/kgK
T1 = 90 0CT2 = 20 0C = 0.7425 KJ/KgK = 1.0395 KJ/KgK
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Magnitude and sign Heat transfer (kJ) = + w
= m + p(V2-V1)
Constant volume V1 = V2
p(V2-V1) = 0
= 28.01 Kg x 0.7425 KJ/KgK
= 28.01 Kg x -51.975 KJ/Kg = - 1455.82 KJWhen work is done by a system, it is arbitrarily taken to be positive, andwhen work is done on a system, taken to be negative
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Resulta)• 945.58959 KN/m2
b) = 0.7425 KJ/KgK = 1.0395 KJ/KgKc) = 763.2445 d)
= -51.975 KJ/Kg = 72.765 KJ/Kg = - 0.06908kJ/kgK = -1455.82 KJ
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Exercise-1
• 1 Kg mole of N2 is contained in a vessel of volume 2.5 m3 at 100 . – i) find the mas, the pressure and specific volume of the
gas.– ii) if the ratio of the specific heat is 1.4, evaluate the
value of Cp and Cv– iii) Subsequently, the gas cools to atmospheric
temperature of 30 , evaluate the final pressure of the gas– iv) Evaluate the specific internal energy, the increase in
specific enthalpy, increase in specific entropy and sign of heat transfer
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Question-2
A cylinder of 60 liter capacity contains CO2 at 10 bar and 20 . Determine mas of the gas, molar volume, density and specific volume of the gas.(Nov 2011)
V = 60 liters of CO2
P = 10 barT = 20 °C
DetermineMass of the gas (m)Molar Volume ()Density ()Specific volume ()
Molar Volume specific Volume V – Volume m3
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Procedure
DetermineMass of the gas (m)
Molar Volume ()
Molar Volume
Molar Volume specific Volume V – Volume m3
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Procedure
DetermineDensity ()
Specific volume ()Molar Volume specific Volume V – Volume m3
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Ans:Mass of the gas (m)
P V⤫ m⤫ R T
Given dataV = 60 liters of CO2
P = 10 barT = 20 °C
atmospheric pressure (1 Atm.)1 Atm = 1.013 bar, 1 Atm = 101325 Pascal (Pa = N/m²);
Unit conversion1 bar = 1x 105 N/m2
1 m3 = 1000 litersJ = Nm
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DetermineMass of the gas (m)
P = 10 bar V = 60 liters ⤫ m⤫ R T = 20 °C
Given dataV = 60 liters of CO2
P = 10 barT = 20 °C
Gas Molar Weight ( M)Kg/Kmol
Gas Constant (R) KJ/KgK
Air 28.97 0.287
Nitrogen 28.01 0.297
Oxygen 32 0.260
Hydrogen 2.016 4.124
Helium 4.004 2.077
Carbon dioxide 44.01 0.189
Steam 18.02 0.461
atmospheric pressure (1 Atm.)1 Atm = 1.013 bar, 1 Atm = 101325 Pascal (Pa = N/m²);
Unit conversion1 bar = 1x 105 N/m2
1 m3 = 1000 litersJ = Nm
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DetermineMass of the gas (m)
P = 10 bar = 10 x 105 N/m2
V = 60 liters = 60/1000 m3
⤫ M = find⤫ R = cal for the substitution T = 20 °C = 20 +273 K
Gas Molar Weight ( M)Kg/Kmol
Gas Constant (R) KJ/KgK
Air 28.97 0.287
Nitrogen 28.01 0.297
Oxygen 32 0.260
Hydrogen 2.016 4.124
Helium 4.004 2.077
Carbon dioxide 44.01 0.189
Steam 18.02 0.461
atmospheric pressure (1 Atm.)1 Atm = 1.013 bar, 1 Atm = 101325 Pascal (Pa = N/m²);
Unit conversion1 bar = 1x 105 N/m2
1 m3 = 1000 litersJ = Nm
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⤫ R = cal for the substitution for Carbon dioxideR KJ/KgKMass of the gas (m)
10 10 5 N/m2 m3 = 10 10 5 N/m2 m3 = ---------- kg
Gas Molar Weight ( M)Kg/Kmol
Gas Constant (R) KJ/KgK
Air 28.97 0.287
Helium 4.004 2.077
Carbon dioxide 44.01 0.189
Steam 18.02 0.461
P = 10 bar = 10 x 105 N/m2
V = 60 liters = 60/1000 m3
⤫M = find⤫R = cal for the substitutionT = 20 °C = 20 +273 K
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DetermineSpecific volume ()
Density ()
Given dataV = 60 liters of CO2
P = 10 barT = 20 °C
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DetermineMolar Volume
10 10 5 N/m2 (20+273) K = 0.410488
Given dataV = 60 liters of CO2
P = 10 barT = 20 °C
Result
= 0.410488
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Exercise -2
Find the molecular weight and gas constant for the gas, whose specific heats are as followsCp = 1.967 KJ/Kg KCv = 1.507 KJ/Kg K
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Exercise -2
– Mayer’s equation
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Reference• Rajput, R. K. 2010. Engineering thermodynamics. Jones and Bartlett
Publishers, Sudbury, Mass.• Singh, O. 2003. Applied thermodynamics. New Age International (P) Ltd.,
Publishers, New Delhi.• Nag, P. K. 2002. Basic and applied thermodynamics. Tata McGraw-Hill,
New Delhi.