The Thermodynamic Behavior of Gases
Variables and Constants
Ideal GasAn ideal gas is a gas in which the volume occupied by the gas particles is negligible compared to the volume occupied by the gas itself. There is little or no interaction between individual gas particles. Most gases behave as ideal gases as long as their temperature is not near the liquefication point and the pressure is not significantly higher than standard atmospheric pressure.
1. Temperature, T...Kelvin is required in all gas law equations.
Variables
2. Volume, V...space occupied by a gas. A gas always fills any container into which it is placed.
Units : m3 , liter −LK 1m3 =1000L
K=C+273
3. Pressure, P...due to collisions between gas particles and the walls of the container. Defined as the total force of all of the collisions divided by the surface area of the container. P = F
AUnits :
Nm2 =Pascal(Pa),
torr=mmHg,lbin2
,
atmosphere(atm)
Standard Atmospheric Pressure
1.013 ×105 Pa=760 torr=1atm=14.7 lbin2
4. Amount of gas
a) mass -m, kg
b) moles, n
c) molecules, N
1mole =1gram−atomic(molecular)−mass
=6.023 ×1023particles
Example :
1mole H2O
=18gH2O
=.018kgH2O
=6.023 ×1023moleculesof H2O
Constants
Universal Gas Constant, R =8.31 Jmole• K
Boltzman Constant, k =1.38 ×10−23 JK
Avagadro' s Number, N0 =6.023×1023 particlesmole
Definitions
Monatomic gas... gas which is composed of single atomsHe, Ne, Ar, Kr, Xe, and Ra
Isobaric process…thermodynamic process in which the pressure is held constant
Isochoric process…thermodynamic process in which the volume is held constant
Isothermal process…thermodynamic process in which the temperature is held constant
Adibatic process…thermodynamic process in which there is no heat flow.
Jkg⋅K
specific heatK
cp = specific heat at constant pressure
cv = specific heat at constant volume
⎧ ⎨ ⎩ ⎪
molar specific heatK
Cp
Cv
⎧ ⎨ ⎩ ⎪
C =c× ,molecular mass kg Jmole⋅K
Special Case - Monatomic Gases
Cp = 52 R
Cv = 32 R
⎧ ⎨ ⎪
⎩ ⎪
Why is the constant pressure molar specific heat greater than the constant volume molar specific heat?
adiabatic gas constant, γ=CpCv
Special Case - Monatomic Gases
γ= 53
The state of a gas is determined by the values of the four variables: T, V, P and n (or m, or N). Once values of the four are given a unique state of the gas has been defined. Any three of the four variables are independent...can be arbitrarily set, the fourth variable is dependent and uniquely determined through an equation of state.
An equation of state determines the relationship between the four variables of a gas. It causes one of the four variables to be dependent.
The thermodynamics of gases is concerned with determining the states of a gas , the heat flow-Q, the work done-W, and the change in thermal energy-U during thermodynamic processes.
Sign conventions for Work , Heat Flow, and Change in Thermal Energy
Work, WK
+ done on the gas
− done by the gas
⎧ ⎨ ⎩
Heat Flow, QK
+ into the gas
− out of the gas
⎧ ⎨ ⎩
Change in Thermal Energy, UK
+ increase
− decrease
⎧ ⎨ ⎩
Gas Laws and the
First Law of Thermodynamics
Ideal Gas Law
a) PV =nRTb) PV =NkT
a) n constantK closed system
PiViTi
=PfVfTf
Special Cases of the Ideal Gas Law
b) n,T constantK isothermal process
PiVi =PfVf → Boyle' s Law
c) n, P constantK isobaric process
ViTi
=VfTf
→ Charles'Law
d) n,V constantK isochoric process
PiTi
=PfTf
→ Gay−Lussac' s Law
Dalton’s Law of Partial Pressures
If an ideal gas is a homogeneous mixture of non-reacting gases (e.g. air), the total pressure is equal to the sum of the partial pressures of the component gases. The partial pressure of a component gas is the pressure it would exert if it were in the container alone.
Pi =niRT
V K ( )partial pressure of gas i
Ptotal = Pii∑
Ptotal =niRT
Vi∑ =RT
V nii∑
Thermal Energy of an Ideal Gas
The thermal energy, U of an ideal gas is a function of the Kelvin temperature alone. Therefore, in an isothermal process (T constant) the thermal energy must remain constant.
If T =0, thenU=0
The First Law of Thermodynamics
E + ΔU = Q + WIf the total mechanical energy, E remains constant E = 0:
U = Q + W
Mathematical Description of the Thermodynamic Processes in Ideal
Gases
Isobaric Process, P constantViTi
=VfTf
Q =mcPT
Q =nCPT
⎫ ⎬ ⎭L T =Tf −Ti
W =−PV =−P Vf −Vi( )U = Q + W
Monatomic Gases Only
a) Q = 52 nRT
b) U =32 nRT
Isochoric Process, V constantPiTi
=PfTf
Q =mcVT
Q =nCVT
⎫ ⎬ ⎭K T =Tf −Ti
W =0U = Q
Monatomic Gases Only
a) Q =32 nRT
b) U =32 nRT
Isothermal Process, T constant
PiVi =PfVf
Q =−W
W =−nRT lnVfVi
⎛ ⎝ ⎜ ⎞
⎠ ⎟
U = 0
Adiabatic Process, Q = 0
PiViTi
=PfVfTf
PiViγ =PfVf
γK γ=CPCV
⎧
⎨ ⎪ ⎪
⎩ ⎪ ⎪
Q =0
W =PiVi
γ
γ−1 ⎛
⎝ ⎜
⎞
⎠ ⎟ 1Vf
γ−1 − 1Vi
γ−1
⎛
⎝ ⎜
⎞
⎠ ⎟ U = W
Monatomic Gases Only
a) γ=53
b)W =32 nRT
Graphical Representation of the
Thermodynamic Processes in Ideal
Gases
P-V Diagram of Isobaric and Isochoric Processes
Volume
Pi
Pf
ViVf
(Pi, Vi , Ti )
Isobaric expansion to Vf , and Tf .
Isobaric Process(Pi, Vf , Tf )
Isochoric process to Pf , and Tf .
Isochoric Process
(Pf , Vi , Tf )
P-V Diagram of Isothermal and Adiabatic Processes
Volume
Pi
P1,f
ViVf
P2,f
(Pi, Vi , Ti )
Isothermal Expansion to P2,f and Vf .
(P2,f ,Vf , Ti )
Adiabatic Expansion to P1,f , Vf and Tf.
(P1,f ,Vf , Tf )