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dw = -p dV, Study notes of Physical Chemistry

Change in energy for a chemical reaction carried out at constant volume is directly equal to the heat evolved or absorbed. We know DE = q + w a) What is w?

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bg1
work = dw = F
F ¥ dL = (Ap
Apext
ext) ¥ dL = p
pext
ext ¥ (A
AdL)
But, AdL = V2 - V1 = dV (infinitesimal volume change)
dw = pext(V2 - V1) = pextdV
dL
A
T1T2
Gas pext=F / A
Æ
Definition of work using calculus:
Infinitesimal work done dw by infinitesimal change in
volume of gas dV:
But sign is arbitrary, so choose
dw = -pextdV (w < 0 is work done by gas, dV > 0)
dw
dw = -
= -p
pext
extdV
dV
(Note! p is the external
external pressure on the gas!)
Movable Piston
Movable Piston
Bonus * Bonus * Bonus
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe

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work = dw = F

F ¥ dL = ( Ap

Ap

ext

ext

) ¥ dL = p

p

ext

ext

A

AdL)

But, AdL = V

2

- V

1

= dV (infinitesimal volume change)

dw = p

ext

(V

2

- V

1

) = p

ext

dV

dL A

T

1

T

2

Gas

p

ext

=F / A

Æ

Definition of work using calculus:

Infinitesimal work done dw by infinitesimal change in

volume of gas dV:

But sign is arbitrary, so choose

dw = -p

ext

dV (w < 0 is work done by gas, dV > 0)

dw

dw

p

p

ext

ext

dV

dV

(Note! p is the

external

external pressure on the gas!)

Movable Piston

Movable Piston

Bonus * Bonus * Bonus

dw

dw = -

p

p

extext

dV

dV

Total work done in any change is the sum of little

Total work done in any change is the sum of little

infinitesimal increments for an infinitesimal change

infinitesimal increments for an infinitesimal change dV

dV .

Ú
Ú

dw

dw

Ú
Ú

p

p

extext

dV

dV = w (work done by the system )

= w (work done by the system )

Two Examples :

Two Examples :

( 1 ) pressure = constant =

( 1 ) pressure = constant = p

p

externalexternal

V changes v

V changes v

ii

Æ
Æ

v

v

ff

w

w = -

p

p

ext

ext

dV

dV = -

p

p

ext

ext

dV

dV

p

p

ext

ext

v

v

f

f

  • v
  • v

i

i

p

p

ext

ext

D
D
V
V

fi

fi

v

i

v

f

Ú

v

i

v

f

Ú

Irreversible

Irreversible expansion if

expansion if p

p

ext

ext

p

p

gas

gas

That is if,

That is if, p

p

gasgas

nRT

nRT /V

/V

p

p

externalexternal

Expansion At

Expansion At

constantPres

constantPres

sure

sure p

p

extext

=P
=P

22

Graphical representation of p

ext

dV

V V

V V

ii

= V= V

11

VV

ff

= V= V

22

P

P

11

P

P

22

no work

no work

P

P

P=

P= nRT

nRT /V

/V

Isothermal reversible

expansion

V

V

V

V

i

i

= V

= V

1

1

V

V

f

f

= V

= V

2

2

P P

P = P = nRTnRT/V/V

p

p

extext

nRT

nRT /V

/V

constant

constant

PV= PV=

const const

PV=const is PV=const is

a hyperbola

a hyperbola

Compare the shaded

area in the plot above

to the shaded area in the

plot for a reversible

isothermal expansion with

p

p

extext

p

p

gasgas

nRT

nRT /V

/V

shaded area = -w

shaded area = -w

shaded area = -w

shaded area = -w

P

P

22

P

P

1

1

Work done is NOT independent of path : Change the State of

a gas two different ways:

Consider n moles of an ideal gas

w = 0 for 2nd step since V = const

Final condition: T

f

= 300 K, V

f

= 1 liter, p

f

= 4 atm.

Initial condition: T

i

= 300 K, V

i

= 2 liter, p

i

= 2 atm.

Step 2: Warm at constant V: 2 atm, 1 liter, 150 K Æ

4 atm, 1 liter, 300 K.

Step 1 : 2 atm, 2 l , 300K

cool at

2 atm, 1 l , 150K

const - p

compress

ææ æææÆ

w = - p

ext

( V

f

- V

i

) for the first step, p

ext

= const = 2 atm

w = - 2 atm ( 1 - 2 ) l = 2 l -atm

w

tot

= 2 l -atm

Path 1 consists of two steps:

DV=0 for

this step

DV≠0 for

this step

Heat : Just as work is a form of energy, heat is also a form of

Heat : Just as work is a form of energy, heat is also a form of

energy.

energy.

Heat is energy which can flow between bodies that are in

Heat is energy which can flow between bodies that are in

thermal contact. thermal contact.

In general heat can be converted to work and work to heat -- can

In general heat can be converted to work and work to heat -- can

exchange the various energy forms.

exchange the various energy forms.

Heat is also NOT a state function. The heat change occurring

Heat is also NOT a state function. The heat change occurring

when a system changes state very definitely depends on the path.

when a system changes state very definitely depends on the path.

Can prove by doing experiments, or (for ideal gases) can use heat

Can prove by doing experiments, or (for ideal gases) can use heat

capacities to determine heat changes by different paths.

capacities to determine heat changes by different paths.

The First Law of Thermodynamics

The First Law of Thermodynamics

I) Energy is a state function for any system :

I) Energy is a state function for any system :

State 1

State 1

State 2

State 2

Path a

Path a

Path b

Path b

E
E

aa

E
E

bb

E
E

aa

and

and

E
E

bb

are

are

both for going

both for going

from 1

from 1 Æ

Æ

If

If

E not a state function then:

E not a state function then:

Suppose

Suppose D

D
E
E

a

a

D
D
E
E

b

b

  • now go from state 1 to state 2 along path a,
  • now go from state 1 to state 2 along path a,

then return to 1 along path b.

then return to 1 along path b.

D
D
E
E

a

a

D
D
E
E

b

b

Totally empirical law. The result of observations in many,

Totally empirical law. The result of observations in many,

many experiments.

many experiments.

q and w are NOT state functions and do depend on the path

q and w are NOT state functions and do depend on the path

used to effect the change between the two states of the system.

used to effect the change between the two states of the system.

dE

dE

dq

dq

dw

dw

q > 0 for heat added to the system

q > 0 for heat added to the system

w > 0 for work done on the system (

w > 0 for work done on the system ( dV

dV < 0)

dw = -p

ext

dV (w < 0 is work done by system, dV > 0)

D
D

E = q+w (Here is where choice of sign for w is made)

E = q+w (Here is where choice of sign for w is made)

D
D

E is a state function independent of the path.

E is a state function independent of the path.

Taking a system over different paths results in

Taking a system over different paths results in

same

same

D

D

E but different q, w:

E but different q, w:

q

q

a

a

w

w

a

a

q

q

a

a

q

q

b

b

q

q

c

c

all different,

all different,

D
D
E = E
E = E

2

2

- E
- E

1

1

Same for

Same for

Paths a, b, c

Paths a, b, c

11

22

q

q

a

a

w

w

a

a

q

q

b

b

w

w

b

b

q

q

c

c

w

w

c

c

w

w

a

a

w

w

b

b

w

w

c

c

all different,

all different, but

but

E
E

2

2

E
E

1

1

D =
D
E
E

q

q

c

c

w

w

c

c

q

q

b

b

w

w

b

b

Change in energy for a chemical reaction carried out at constant

Change in energy for a chemical reaction carried out at constant

volume is

volume is directly

directly equal to the heat evolved or absorbed.

equal to the heat evolved or absorbed.

We know

We know D

D

E = q + w

E = q + w

a) What is w? 1st let us carry the change above

a) What is w? 1st let us carry the change above out at

out at

constant volume

constant volume :

N
N

2

2

+ O
+ O

2

2

Æ
Æ
2NO
2NO

Then no mechanical work is done by the gases as they react to form

Then no mechanical work is done by the gases as they react to form

NO because they are not coupled to the world ---

NO because they are not coupled to the world ---

no force moving through a distance --- nothing moves

no force moving through a distance --- nothing moves Æ

Æ

w = 0.

w = 0.

D
D
E =
E =

q

q

vv

If

If q

q

vv

then

then D

D
E > 0
E > 0

and energy or heat is absorbed by the system.

and energy or heat is absorbed by the system.

This is called an

This is called an endoergic reaction

endoergic reaction .

If

If q

q

vv

then

then D

D
E < 0
E < 0

and energy or heat is evolved by the system.

and energy or heat is evolved by the system.

This is called an

This is called an exoergic

exoergic reaction

reaction .

Can we find or define a new state function which is equal to the heat

Can we find or define a new state function which is equal to the heat

evolved by a system undergoing a change at constant pressure rather

evolved by a system undergoing a change at constant pressure rather

than constant volume?

than constant volume?

i.e. is there a state function =

i.e. is there a state function = q

q

p

p

Yes!

Yes!

H

H

E +

E +

pV

pV

will have this property

will have this property

Note E, p, V are state

Note E, p, V are state fcts

fcts .

\
\

H must also be a state

H must also be a state fct

fct .

Let us prove

Let us prove D

D
H =
H =

q

q

p

p

: (for changes carried out at constant p)

: (for changes carried out at constant p)

D
D

E = q + w

E = q + w

D
D
H =
H =

q

q

p

p

  • w + p

  • w + p D

D

V, since p=const

V, since p=const

dH

dH

p

p

dq

dq

p

p

dw

dw

pdV

pdV ;

dw

dw = -

pdV

pdV

w = - p

w = - p D

D

V for changes at const p

V for changes at const p

dH

dH

dq

dq

pp

pdV

pdV

pdV

pdV

dq

dq

pp

Æ

Æ

\
\
D
D
H =
H =

q

q

pp

  • p
  • p D
D

V + p

V + p D

D
V
V

Æ

Æ

D
D
H =
H =
D
D
E +
E +
D
D

pV

pV )

D

D

H =

H =

q

q

p

p

dH

dH

dq

dq

p

p