Applications of First Law of Thermodynamics to Ideal Gases: Isothermal & Adiabatic, Slides of Law

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Applications of the First Law to ideal gases

RT

pv

What

we

know Equation of state

Internal energy is functiononly of temperature

dT c

du

v

dT c

dh

p

R

c

c^

v

p^

=

−

Specific heats are related

pdv

dw

−

"expansion work" The fist law

dw

dq

du

=

vdp

dq

dh

Isothermal vs. Adiabatic

An

isothermal process

in one in which

the initial and final temperatures arethe same.

Isothermal processes are not

necessarily adiabatic.

An

adiabatic process

in one in which no

heat is exchanged between the system andits surroundings.

dT

0

dq

Reversible Changes

• A

reversible change

is one that can be

reversed by an infinitesimal modificationof a variable.

gas

env

p

p

• In a reversible expansion or compression

If we want to integrate

pdv

dq

=

pdv

dw

−

only expansionwork"

∫^

≠

0

dw

What do we need????

Path!!!

v

v

A

B

Lets consider the path of anisothermal reversible expansionEquation of state is satisfiedduring all the stages of theexpansion

RT

pv

isothermal expansion

pdv

dq

RT

pv

RT v

p

dv^ v

RT

dq

∫

=

Δ

v^21 v^

dv v

RT

q

⎞ ⎟ ⎟ ⎠

⎛ ⎜⎜ ⎝

=

Δ

2 1

ln

v v

RT

q

p RT

v

⎞ ⎟⎟ ⎠

⎛ ⎜⎜ ⎝

=

Δ

1 2

ln

p p

RT

q

⎞ ⎟ ⎟ ⎠

⎛ ⎜⎜ ⎝

= ⎞ ⎟⎟ ⎠

⎛ ⎜⎜ ⎝

=

Δ

1 2

2 1

ln

ln

p p

RT

v v

RT

q

If we want to integrate

isothermal expansion

vdp

dq

dh

Constant pressure process

0

dp

dq

dh

dT c

dh

p

dT c

dq

p

=

The amount of heat required toraise the temperature of thegas from T

1

to T

2

at constant

pressure

(^

) 1

2

T

T

c

q

p^

dT c

du

v

dT c

dh

p

=

R

c

c^

v

p^

=

−

pdv

dw

Constant pressure

Adiabatic

process

Process in which NO HEAT isexchange between the system andits environment

0

=

dq

dw

dq

du

=

dw

du

=

Fist law for areversible adiabaticprocess

An adiabatic compressionincreases the internalenergy of the system

w q q

pdv

du

−

Adiabatic process

Poisson’s

equations pdv

dT c

v^

RT

pv

RT v

p

dv

RT v

dT c

v^

dv v R

dT T c

v^

Lets assumeconstant

v c

There is a final and initial state

Lets consider a reversible adiabatic

EXPANSION

for an ideal gas

0

=

dq

∫

∫

(^21)

(^21)

v v

T T v^

dv v

R

dT T

c

(^21)

(^2 )

ln

ln

v v

R

T T

c

v

cv R

v v

T T

1 2

2 1

During an adiabaticexpansion of a gas, thetemperature decreases

∫

∫

(^21)

(^21)

v v

T T v^

dv v

R

dT T

c

Poisson’s

equations Reverse process

adiabatic

Compression

Work is done on the gas and the temperature increases

A given pressuredecrease produces asmaller volume increasein the adiabatic caserelative to theisothermal case

RT

pv

isothermal expansion Vs reversible adiabatic expansion

P, v diagram

Temperature also decreasesduring the adiabatic expansion

Dry

Adiabatic

Processes

in

the

Atmosphere

For reversible adiabaticprocesses for an ideal gas

c^ p R

p p

T T

⎞ ⎟ ⎟ ⎠

⎛ ⎜⎜ ⎝

0

0

Orographic

Lifting

Frontal Lifting

Low-level

convergence

Vertical

Mixing

Lifting Processes

p T

0

dq

cp R

p p T

⎞ ⎟⎟⎠

⎛ ⎜⎜⎝

0

θ

If for

mb

1000 0

= p^

the temperature is

, θ

c^ p R

p p

T T

⎞ ⎟ ⎟ ⎠

⎛ ⎜⎜ ⎝

0

0

p R c

Where

for dry air is

(^286). 0 2 7

5 2

=

=

=

=^

R R R

R

c

R

R c

v

p

Potential Temperature

Temperature a parcel of gas (e.g. dry air)would have if compressed (or expanded) inan reversible adiabatic process from astate,

and

, to a a pressure of

mb

1000 0

= p

p

T

Also a state variable,

invariant during a

reversible adiabatic process: Conservative quantity!!!!

c^ p R

p p

T

⎞ ⎟⎟ ⎠

⎛ ⎜⎜ ⎝

0

θ

• km C Ο 6 = Γ

Consider aTemperature profilewith For

mb

1000 < p^

T > θ

⇒

Adiabatic compressionto lower the parcel

mb

1000 > p^

T < θ

⇒

mb

1000

p^

T

θ

⇒

T^2

T^1 Adiabatic expansion torise the parcel

dT^ dz − = Γ

Lapse rate