# The First Law

Mechanics

Oct 27, 2013 (4 years and 8 months ago)

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Thermodynamics:

The First Law

자연과학대학

화학과

박영동

교수

Classical thermodynamics

Thermodynamics: the first law

2.1 The conservation of energy

2.1.1 Systems and surroundings

2.1.2 Work and heat

2.1.3 The measurement of work

2.1.4 The measurement of heat

2.1.5 Heat influx during expansion

2.2 Internal energy and enthalpy

2.2.6 The internal energy

2.2.7 Internal energy as a state function

2.2.8 The enthalpy

2.2.9 The temperature variation of the enthalpy

Thermodynamic Systems, States and
Processes

Objectives are to:

define thermodynamics systems and states of systems

explain how processes affect such systems

apply the above thermodynamic terms and ideas to the laws of
thermodynamics

Thermodynamic universe

the system
is

the region of
interest; its region is defined by
the boundary
.

the rest of the world is its
surroundings.

The surroundings are where
system.

The universe
consists of the
system and surroundings.

system

surroundings

The Universe

Various Systems

system

matter

energy

open(
열린

)

allowed

allowed

closed(
닫힌

)

forbidden

allowed

isolated(
고립

)

forbidden

forbidden

Some other thermodynamics terms

state,

state functions,

path, process,

extensive properties,

intensive properties.

closed

open

isolated

A State Function and Paths

The altitude is a state property, because
it depends only on the current state of
the system. The change in the value of a
state property is independent of the path
between the two states.

The distance between the initial and final
states depends on which path (as
depicted by the blue and red lines) is
used to travel between them. So it is not
a state function.

pressure, temperature, volume, ...

heat, work

--

Heat and work are forms of energy
transfer and
energy is conserved
.

The First Law of Thermodynamics

U

=
Q

+
W

work done

on

the system

change in

total internal energy

to system

State Function

Process Functions

Calculating the change in internal energy

We see that the person

s internal energy falls by 704 kJ. Later, that energy will
be restored by eating.

Suppose someone does 622 kJ of work on an exercise
bicycle and loses 82 kJ of energy as heat. What is the
change in internal energy of the person? Disregard any
matter loss by perspiration.

Solution

w

=

622 kJ (622 kJ is lost by doing work

on the bicycle
)
,

q

=

82 kJ (82 kJ is lost by heating the surroundings).

Then

the first law of thermodynamics
gives

us

U

=
q

+
w =
(
-
82 kJ )+ (
-
622 kJ) =
-
704 kJ

Work, and the expansion(
p
-
V
) work

F

F

dW

p
ex
=F/A

Increase in volume,
dV

(
p
-
V
work)

Work =
force

distance

force acting on the piston =
p
ex

×

Area

distance when expand =
dy

+y

Total Work Done

To evaluate the integral, we
must
know how the pressure
depends (functionally) on
the
volume.

We will consider the following cases

0. Constant volume work

1.
Free expansion

2.
Expansion against constant pressure

3.
Reversible isothermal expansion

0. Work for the constant volume process

For the constant volume process,
there is no
p
-
V

work

Heat and Internal Energy

U

=
q

+
w =
q
V

For the constant volume process,
there is no p
-
V work, so

q

to the system, the temperature
of the system increases by

T

, and the internal
energy increases by

U.

Constant volume heat capacity

Internal Energy of a Gas

A

pressurized

gas

bottle

(
V

=

0
.
05

m
3
),

contains

helium

gas

(an

ideal

monatomic

gas)

at

a

pressure

p

=

1
×
10
7

Pa

and

temperature

T

=

300

K
.

What

is

the

internal

thermal

energy

of

this

gas?

molar constant volume heat
capacity of monatomic gases

(at 1
atm
, 25
°
C)

Monatomic gas

C
V, m

(J/(
mol∙K
))

C
V, m
/
R

He

12.5

1.50

Ne

12.5

1.50

Ar

12.5

1.50

Kr

12.5

1.50

Xe

12.5

1.50

Temperature and Energy
distribution

The temperature is a parameter that
indicates the extent to which the
exponentially decaying Boltzmann
distribution reaches up into the higher
energy levels of a system. (a) When the
temperature is low, only the lower energy
states are occupied (as indicated by the
green rectangles). (b) At higher
temperatures, more higher states are
occupied. In each case, the populations
decay exponentially with increasing
temperature, with the total population of
all levels a constant.

A constant
-
volume bomb
calorimeter.

The constant
-
volume heat capacity is the
slope of a curve showing how the
internal energy varies with temperature.
The slope, and therefore the heat
capacity, may be different at different
temperatures.

For the free expansion process,
there is no
p
-
V

work

U

=
q

+
w = q

2. Work for expansion against constant
pressure

U

=
q

+
w = q
-

p

V

q
p

=

U

+
p

V=

(
U

+
p V
)

If we define Enthalpy as,
H

U +
pV

q
p

=

H

Enthalpy change is the heat given to the
system at constant pressure.

C
p
,
C
v

for an ideal gas

For an ideal gas,
U

and
H

do not
depend on volume or pressure.

For an example, for an ideal
monatomic gases,
U

= (3/2)
nRT

Heat capacity of CO
2

and N
2

The heat capacity with
temperature as expressed
by the empirical formula
Cp,m
/(J K
-
1

mol
-
1
) =
a
+
bT
+c
/
T
2
. The circles
show the measured
values at 298 K.

molar heat capacities,

Cp,m
/(J K
-
1

mol
-
1
) =
a
+
bT
+c
/
T
2

a

b
/(l0
-
3
K
-
1
)

c
/(10
5
K
2
)

@273K*

@298K*

@350K*

Monatomic

gases

20.78

0

0

2.50

2.50

2.50

Other

gases

Br
2

37.32

0.5

-
1.26

4.30

4.34

4.39

Cl
2

37.03

0.67

-
2.85

4.02

4.09

4.20

CO
2

44.22

8.79

-
8.62

4.22

4.47

4.84

F
2

34.56

2.51

-
3.51

3.67

3.77

3.92

H
2

27.28

3.26

0.5

3.47

3.47

3.47

I
2

37.4

0.59

-
0.71

4.40

4.42

4.45

N
2

28.58

3.77

-
0.5

3.48

3.50

3.55

NH
3

29.75

25.1

-
1.55

4.15

4.27

4.48

O
2

29.96

4.18

-
1.67

3.47

3.53

3.62

H
2
O(
l
)

75.29

0

0

9.06

9.06

9.06

C(
s
, graphite
)

16.86

4.77

-
8.54

0.81

1.04

1.39

*calculated results are given in unit of R(8.314 J K
-
1

mol
-
1
)

An exothermic process

When hydrochloric acid reacts with
zinc, the hydrogen gas produced
must push back the surrounding
atmosphere (represented by the
weight resting on the piston), and
hence must do work on its
surroundings. This is an example of
energy leaving a system as work.

Zn(s) + 2 HCl(aq) → ZnCl
2
(aq) + H
2
(g)

exothermic and endothermic processes

Work and Heat

Work is transfer of energy that causes or
utilizes uniform motion of atoms in the
surroundings. For example, when a
weight is raised, all the atoms of the
weight (shown magnified) move in unison
in the same direction.

Heat is the transfer of energy that causes
or utilizes chaotic motion in the
surroundings. When energy leaves the
system (the green region), it generates
chaotic motion in the surroundings
(shown magnified).

The expansion(
p
-
V

) Work

When a piston of area
A

moves out
through a distance
h
, it sweeps out a
volume Δ
V

=
Ah
. The external pressure
p
ex

opposes the expansion with a force
p
ex
A
.

3. Work for reversible isothermal expansion
for an ideal gas

Reversible Isothermal Expansion
Work for a perfect gas

The work of reversible, isothermal
expansion of a perfect gas. Note that for
a given change of volume and fixed
amount of gas, the work is greater the
higher the temperature.

Molecular Basis for Heat Capacity

The heat capacity depends on the
availability of levels. (a) When the levels
are close together, a given amount of
energy arriving as heat can be
the populations and hence the
temperature that occurs in the Boltzmann
distribution. This system has a high heat
capacity. (b) When the levels are widely
separated, the same incoming energy has
to be accommodated by making use of
higher energy levels, with a consequent
greater change in the ‘reach’ of the
Boltzmann distribution, and therefore a
greater change in temperature. This
system therefore has a low heat capacity.
In each case the green line is the
distribution at low temperature and the
red line that at higher temperature.

Temperature dependence of
enthalpy(H) and internal energy(U)

The enthalpy of a system increases as its temperature is raised. Note
that the enthalpy is always greater than the internal energy of the
system, and that the difference increases with temperature.

Temperature dependence of
enthalpy(H) and internal energy(U)

Note that the heat
capacityies

depend on
temperature, and that
C
p

is greater than
C
V
.

a DSC

(differential scanning calorimeter)

A
thermogram

for the protein
ubiquitin
. The protein retains its
native structure up to about 45
°
C
and then undergoes an endothermic
from B.
Chowdhry

and S.
LeHarne
,
J.
Chem. Educ.

74, 236 (1997).)

sample

reference

C
p

and

C
V

The Maxwell relations

U

=
q

+
w =
w

U

= C
V

T=

q

=
0

If
the heat capacity is
independent of temperature,

C
V

dT

=
-
pdV

For an adiabatic process for an ideal gas,

For an ideal monatomic gas

of

n

moles,

calculate
q, w,
Δ
U
for each process.

w
12
=
-
p
b
(
V
b
-
V
a
)

Δ
U
12

= C
V
(
T
2
-
T
1
)

= C
V
(
p
b
V
b
/
nR

-

p
b
V
a
/
nR
)

=
C
V
p
b
(
V
b

-

V
a
)

/
nR

q
12
=
Δ
U
12

-

w
12

T
2

=
p
b
V
b
/
nR

T
1

= T
3

=
p
b
V
a
/
nR

=
p
a
V
b
/
nR

p
b
V
a

=
p
a
V
b

w
2 3
= 0

Δ
U
23

= C
V
(
T
3
-
T
2
)

= C
V
(
p
b
V
a
/
nR

-

p
b
V
b
/
nR
)

=
C
V
p
b
(
V
a

-

V
b
)

/
nR

=
-

Δ
U
12

q
12
=
Δ
U
23

w
31
=
-
nRT
3

ln
(
V
a
/
V
b
) =
-
p
b
V
a

ln
(
V
a
/
V
b
)

Δ
U
31

=
0

q
12
=
-

w
31