1. LECTURE UNIT 004
General Gas Law Equation (Equation of State)
ILLUSTRATION:
A thermodynamic substance which is an ideal gas
TS
An equation of state gives a relation between properties of a pure substance. The volume occupied by a given mass of this
substance can be determined using the pressure and temperature to which it is subjected to. From experimental observations it
has been established that the P-V-T behavior of gases at low density is closely given by the following equation of the state.
PV = mRT (in terms of mass)
or
PV = nRT (in terms of moles)
The term “state” indicates an equilibrium state, that is temperature and pressure are the same at all points of the system.
Where:
P = absolute pressure
V = volume
m = mass of gas Btu , kJ
R = gas constant, each ideal gas has its own gas constant,
lbm - oR kg - K
T = absolute temperature
R = universal gas constant for any ideal gas
Note: n = number of moles (mols)
A gas can be considered ideal if its pressure is very low and the temperature is much higher than its
critical temperature.
Ideal gases behave according to various ideal gas laws:
1. Boyle’s Law
If the temperature of the given quantity of a gas held constant (T=C), the volume of a gas varies inversely with the
absolute pressure during a change of state.
Initial state Final state
ILLUSTRATION:
Change of
1 state (T = C) 2
P1V1T1 P2V2T2
V 1
8
P
V2 P1
=
V1 P2
2. Charles Law
a.) If the pressure of a particular quantity of gas is held constant (P=C), then, with any change of state, the volume will
vary directly as the absolute temperature.
Initial state Final state
ILLUSTRATION:
Change of
1 state (P = C) 2
P1V1T1 P2V2T2
V T
8
V1 T1
=
V2 T2
“Efficiency is doing things right. Effectiveness is doing the right thing.”

2. b.) If the volume of a particular quantity of gas is held constant (V=C), then, with any change of state, the pressure will
vary directly as the absolute temperature.
ILLUSTRATION:
Initial state Final state
Change of
1 state (V = C) 2
P1V1T1 P2V2T2
P T
8
P1 T1
=
P2 T2
Let an ideal gas to undergo a change of state at random,
P
1
P1 a
P2 2
v1 v2 v
Process 1 - a (P=C)
V1 T1
=
Va Ta
Va
T a = T1 (1)
V1
Process a - 2 (V=C)
P2 T1
=
Pa Ta
Pa
T a = T2 (2)
P2
Equating equation (1) and (2)
Va Pa
T1 = T2
V1 P2
Where:
Pa = P1
Va = V1
Hence; P1V1 P2V2
= = mR = constant
T1 T2
“You can give without loving, but you cannot love without giving.”

3. The expression of the General Gas Law Equation if the number of moles (mols) is given;
PV = nRT (1)
The expression of the General Gas Law Equation if the mass is given;
PV = mRT (2)
Equating equation (1) and (2);
mRT = nRT
R
R= m
n
And from definition of mols;
mass m
mols = =
molecular mass M
m Units:
M= M = (atomic weight)( number of atoms)
n
R m = mass of gas, lbm, kgm
R=
M n = number of mols, lbmol, kgmol
M = molecular mass, lbm/lbmol, kgm/kgmol
For air:
lbf-ft kJ
R = 53.34 = 0.287
o
lbm- R kg - K
Btu kJ
Cp = 0.24 = 1.0062 kg - K
lbm-oR
Btu kJ
Cv = 0.1714 = 0.7186 kg - K
lbm-oR
Cp
k= = 1.4 (ratio of specific heat)
Cv
For any ideal gas other than air, use TABLE A-1 Appendix A, page 785, Engineering Thermodynamics
Fourth Edition by Burghardt and Harbach.
An element is defined by:
nCx No. of atoms
No. of mols Element
An compound element is defined by:
nCxHy
Common Elements Used:
Elements Atomic Weight (AW) Number of atoms M
C (Carbon) 12 1 12
H2 (Hydrogen) 1 2 2
O2 (Oxygen) 16 2 32
N2 (Nitrogen) 14 2 28
S (Sulfur) 32 1 32
Common Compound Elements Used:
Elements Atomic Weight (AW) * (Number of atoms) M
CO2 (Carbon Dioxide) (12)(1) + (16)(2) 44
CO (Carbon Monoxide) (12)(1) + (16)(1) 28
H2O (Water Vapor) (1)(2) + (16)(1) 18
SO2 (Sulfur Dioxide) (32)(1) + (16)(2) 64
SO (Sulfur Monoxide) (32)(1) + (16)(1) 48
“Good leadership is motivating and mobilizing others to accomplish a task or to think in ways
that are for the benefit of all concerned.”

4. Universal Gas Constant, R
According to Avogadro’s Law
n V t P
3 o 14.7 psia
1 lbmol 359 ft 32 F
1 kgmol 22.4 m3 0 oC 101.325 kPa
Solving for the universal gas constant, using: PV = nRT
PV
R=
nT
kJ
R = 8.314 SI units
kg - K
ft - lbf
= 1545 Eng’g units
lbm - oR
Btu
= 1.9859
lbm - oR
PROBLEM SET:
1. An unknown gas has a mass of 1.5 kg and occupies 2.5 m3 while at a temperature of 300 K and a pressure of 200 kPa. Determine the
ideal-gas constant for the gas. [ ]
2. A 6-m3 tank contains helium at 400 K and is evacuated from atmospheric pressure to a pressure of 740 mmHg vacuum. Determine (a)
the mass of helium remaining in the tank (b) the mass of helium pumped out. (c) The temperature of the remaining helium
falls to 10oC. What is the pressure in kPa? [ ]
3. For a certain ideal gas, Cp = 0.255 Btu/lbm-R and k = 1.3. Determine (a) M, (b) R, (c) Cv. [
]
4. A transportation company specializes in the shipment of pressurized gaseous materials. An order is received for 100 liters of a particular
gas at STP (32oF and 1 atm). What minimum volume tank is necessary to transport the gas at 80oF and maximum pressure
of 8 atm? [ ]
5. A rigid vessel initially contains helium at 105 kPaa and 15oC. Two kilograms of helium are then added to the contents so that the final
pressure and temperature are 200 kPaa and 20oC. Determine (a) the volume of the vessel and (b) the final mass of helium.
[ ]
6. A 1500-L tank contains nitrogen at a pressure of 400 kPag and a temperature of 320 K. Later, because of leak, it was found that the
gage pressure has dropped to 320 kPag and the temperature has decreased to 300 K, determine (a) the initial mass of the
nitrogen, (b) the remaining nitrogen and (c) the amount of nitrogen that has leaked out. [ ]
7. A 0.5 m3 carbon dioxide tank at atmospheric pressure and 27oC is placed near a furnace with the exhaust valve open. The temperature
inside the tank reaches 87oC. The exhaust valve is then closed and the tank is cooled to its initial temperature. Determine (a)
the initial mass of carbon dioxide, (b) the remaining mass of carbon dioxide in the tank and (c) the final pressure of carbon dioxide in
the tank. [ ]
8. Complete the following ideal gas table:
MW P kPa ρ kg/m3 R kJ/kg.K T (K)
a 26 125 250
b 2.22 0.181 420
c 450 1.68 0.244
9. A closed vessel A contains 0.085 m3 of O2 at PA = 3450 kPaa and a temperature of 50oC. This vessel is connected by a pipeline to
another closed vessel B which contains an unknown volume of O2 at 110 kPaa and 10oC. After the valve in the pipe is opened
the resulting pressure and temperature of the mixture are Pm = 1380 kPaa and tm = 25oC, respectively. What is the
volume of vessel B? [ ]
10. A typical adult breathes 30 in3 of air with each breath and has 25 breathes per minute. At 14.7 psia and 70oF, determine the mass of air
per hour entering a person’s lungs. This person now is skiing on a mountain where the air is 10oF and the pressure
is 13.1 psia. How many breaths per minute are required if the mass of air per hour entering the lungs is to be constant? [
]
11. For a certain ideal gas, R = 0.270 kJ/kg-K and k = 1.25. Determine (a) Cp (b) Cv (c) M. [ ]
“Imagination is the beginning of creativity.”

5. 12. A sealed tank contains oxygen at 27oC and a pressure of 2 atm. If the temperature increases to 100oC, what will be the pressure inside
the tank? [ ]
13. A spherical balloon containing helium at at atmospheric pressure and temperature of 21oC is used to carry a load of 10 metric tons.
What minimum balloon diameter is required for this purpose? [ ]
14. Air is contained in a vertical cylinder fitted with a frictionless piston and a set of stops, as shown below. The piston cross-sectional area
is 0.2 m2, and the air inside is initially at 200 kPa, 500oC. The air is then cooled as a result of heat transfer to the
surroundings. (a) What is the temperature of the air inside the cylinder when the piston reaches the stops? (b) The cooling
is now continued until the temperature reaches 20oC. What is the pressure at this state? [ ]
1m
Air
1m
15. Helium is assumed to obey the Beattie-Bridgeman equation of state. Determine the pressure for a temperature of 500oC and a specific
volume of 5.2 m3/kg. Compare with the ideal-gas equation of state. Refer to page 133, Engineering Thermodynamics Fourth
edition by Burghardt and Harbach. [ ]
“We can do anything we want as we stick to it.”