Download dw = -p dV and more Study notes Physical Chemistry in PDF only on Docsity!
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
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
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
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
