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 )