# The First Law of Thermodynamics

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27 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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1

The First Law of Thermodynamics

Experimentally, if we can only measure the energy entering or leaving a system,
then we can only know about
energy changes
. Therefore we re
-
cast the First Law
as

The change in the energy of the system

the internal energy change,
D
E
, is equal
to the negative of the energy change of the surroundings.

¢

U

=
energy entering
-

energy leaving

The First Law of Thermodynamics is a piece of accounting. We need to recognise and
account for all forms of energy entering or leaving a system.

How do we recognise energy? as
heat

and
work
.

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.:
A
A
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2

The classical expression of the First Law

The First Law of thermodynamics is most often written in terms of the change in
the internal energy of a system as

Work is “usable energy”

the capacity to push a piston or a turbine, or to exert a
force,
F
, over a distance,
d
x
.

Work,
w

=
F

x
d
x
=
P

x
A

x
d
x
=
p
D
V

Which gives

Heat into the system
increases its internal
energy.

Work done by the
system decreases
its internal energy.

Where the force is
pressure x area.

Increasing volume lowers
E
int
.

3

Thermodynamics is generally concerned
with the transformation from one
Equilibrium State to another Equilibrium
State.

A transformation is called reversible if
every point along the path between the
two states is also at equilibrium.

Transformations can proceed along
different paths with the same endpoints.
The work done (by the system) and
thermal energy exchanged (between the
system and surroundings) along the
different paths may be different yet the
final states (A&B) and the final state
functions (U,P,T,v,n) are the same.

Transformations of an Ideal Gas

4

Rudolf Clausius
-

1850

The

first

law

of

thermodynamics
:

the

internal

energy

of

a

system

can

be

changed

by

doing

work

on

it

or

by

heating/cooling

it
.

¢

U =
q + w

The

conserved

quantity

is

not

Heat
:

The

conserved

quantity

is

Energy

which

can

be

changed

by

having

heat

flow

into/out

of

the

system

or

by

work

being

done

on/by

the

system
.

A

change

in

the

energy

of

a

system
:

Is

by

heat

leaving

or

entering

the

system

or

work

being

done

or

by

the

system

5

What is Energy

q

Internal energy

Tentative definition:

The internal energy U of a system is the sum of the kinetic

energies of all of its constituent particles, plus the sum of

all the potential energies of interactions among these

particles.

q

The first law of

thermodynamics

q

Infinitesimal changes in state

6

PV work on gasses?

Piston in a cylinder moves by
dl

due to expansion of gas at pressure
p

Force on piston:

Incremental work done:

p

7

What is Heat?

Up to mid
-
1800’s heat was considered a fluid
that flowed from hot objects to cold objects.

A better definition is that it is the energy that
flows from hot bodies to cold bodies in order
to establish thermal equilibrium. This is
called
thermal energy.

The term
Heat (Q)

is properly used to
describe
energy in transit
, that is thermal
energy transferred into or out of a system.
Heat only makes sense in the context of the
process of the transformation from one state
to another. It is not a
state

or equilibrium
property of the system.

8

The First Law of Thermodynamics

Thermodynamic transformations

q

is
+

if energy is
gained

by the system from the flow of heat.

q

is

if energy is
lost

by the system by heat flow

w

is
+

if
work

is done
on

the system.

w

is

if
work

is done
by

the system.

Signs for heat (
q
) and work (
w)

in thermodynamic transformations

9

Coupling between System
and Surroundings

Thermal reservoir

-

The orange walls insulate the
system completely (no heat can pass
through)

Isochoric
-

Literally means no work. For
our purposes it means the Volume is
constant.

Isobaric
-

The force on the piston from
above is constant (for example
atmospheric pressure)

Isothermal
-

The thermal reservoir keeps
the system at a constant temperature

Free Expansion
-

The piston offers no
resistance (there is vacuum above the
brown piston)

10

Internal energy of an ideal gas

q

Free expansion

Partition is broken and gas is

Allowed to expand in an adiabatic

(no heat transfer container)

Controlled expansion

(Isothermal Reversible)

the same final state

No heat exchanged

Heat exchanged

Different paths

the same initial state

No work done

Work done

¢
U is the same for both processes (Why? What is the internal energy of an ideal gas?

11

State Variables

The Physical State of a system is completely
described by a small number of Macroscopic Variables
(N,V,T,U, concentrations… we will add more later on)

These variables are called state variables and include
pressure, volume, temperature, amount of substance,
Internal Energy

In a chemical reaction it is the amounts
(concentrations) of the products and reactants, their
phases (gas, liquid, solid), Temperature and Pressure.

12

a

b

c

i

f

State Functions for a Mountain
Trekker

13

Is work a state function?

P
1

P
2

V
1

V
2

Work on red path is w
red

=
-
p
2
(V
2
-
V
1
)
.

i

f

Ideal Gas

Work is not a state function

Work on blue path is w
blue

=
-
p
1
(V
2
-
V
1
)

Thus w
red

< w
blue
.

Work done on system differs for different paths from
i
to
f
.

14

Thermodynamics is concerned with equilbrium states.

An equilibrium state is one in which macroscopic properties

(E,V,P,T,n
1
,n
2
,...) do not change with time. Properties are

Either
extensive

(E,V,n) or
intensive

(T,P)

Equilibrium States

i

f

State space

15

Assume the contrary and take
¢

U
a

>
¢

U
b

.

Go forward on
a

(gaining
¢ U
a
) and backward along
b

(losing
-
¢

U
b
)

During each cycle there is a net gain in energy (
¢

E
a

-

¢

E
b

) > 0,
available to do work.

How nice! This would solve energy crisis! Perpetual motion machine!

Not possible, therefore,
¢

U
a

=
¢

U
b
. Energy is conserved.

Energy is a state function

Energy is a State Function if and only if
¢

U
a

=
¢

U
b

i

f

a

b

16

Is Heat a state function?

¢

U

is a state function but
w

is not, thus q is not a state function
.

i

f

a

b

17

Consider a transformation of an ideal gas that increases
its temperature.

Δ
U (E)

depends only on

Δ
T

18

Constant Volume Heat Capacity for Diatomic Ideal Gas

19

The First Law of Thermodynamics

Experimentally, if we can only measure the energy entering or leaving a system,
then we can only know about
energy changes
. Therefore we re
-
cast the First Law
as

The change in the energy of the system

the internal energy change,
Δ
E
, is equal
to the negative of the energy change of the surroundings.

¢

U

=
energy entering
-

energy leaving

The First Law of Thermodynamics is a piece of accounting. We need to recognise and
account for all forms of energy entering or leaving a system.

How do we recognise energy? as
heat

and
work
.

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.:
A
A
A
A
A
A
A

20

The classical expression of the First Law

The First Law of thermodynamics is most often written in terms of the change in
the internal energy of a system as

Work is “usable energy”

the capacity to push a piston or a turbine, or to exert a
force,
F
, over a distance,
d
x
.

Work,
w

=
F

x
d
x
=
P

x
A

x
d
x
=
p
D
V

Which gives

Heat into the system
increases its internal
energy.

Work done by the
system decreases
its internal energy.

Where the force is
pressure x area.

Increasing volume lowers
E
int
.

21

Thermodynamics is generally concerned
with the transformation from one
Equilibrium State to another Equilibrium
State.

A transformation is called reversible if
every point along the path between the
two states is also at equilibrium.

Transformations can proceed along
different paths with the same endpoints.
The work done (by the system) and
thermal energy exchanged (between the
system and surroundings) along the
different paths may be different yet the
final states (A&B) and the final state
functions (U,P,T,v,n) are the same.

Transformations of an Ideal Gas

22

Different paths in the

Expansion of a gas

Consider the expansion of an ideal gas with various coupling
between it (the system) and the surroundings

Free expansion (No external Pressure)

Isothermal (Constant Temperature)

Isobaric (Constant pressure)

Isochoric (No work)

Each of these represents different coupling

Between the system and the surroundings.

Thermal reservoir

23

Free expansion

U = KE + PE

U = q + w

w= p
ext
dV , p
ext
=0, w=0

No temperature change in
surroundings so Joule measured
that q=0

Therefore

U =0, the internal
energy is independent of pressure
and volume and is only dependent
on Temperature

Joule originally did a free expansion experiment and concluded
that no heat was transferred between the system and its
surroundings.

24

Free expansion

Joule concluded that there was no
change in energy. Is this true for
an ideal gas?

He was clearly working on a real
gas.

What could have gone wrong?

Joule originally did a free expansion experiment and found
that no heat was transferred between the system and its
surroundings.

r

U
(
r
)

van der Waals

attraction

U
(
r
)

U
(
r
)

25

Joule did his experiments with relatively
dilute gases so that their behavior was
similar to an ideal gas. How would you
expect a denser gas to behave?

Vanderwaals attractions between the
particles.

Would they become more or less
favorable?

If the expansion was done in an adiabatic
container what would happen? Would
the final temperature be higher, lower or
remain the same?

If the Temperature was controlled with a
bath would heat be transferred into or out
of the system?

26

Internal energy of Real Gas

Joule originally did a free expansion experiment and concluded
that virtually no heat was transferred.

However, for non
-
ideal gas some
temperature change occur.

The internal energy U is the sum of the
kinetic and potential

energies for all the particles that make
up the system. Non
-
ideal

gas have attractive intermolecular
forces. So if the internal energy

is constant, the kinetic energies
generally decrease (think of the
vanderwaals potential). Therefore as the
temperature is directly related to
molecular kinetic energy, for a non
-
ideal
dilute gas a free expansion results in a
drop in temperature.

r

van der Waals

interaction

U
(
r
)

What about for a dense gas?

27

Reversible Process

Pressure decreases by very small amount

Volume increases by very small amount.

Allow system to equilibrate

Remove one grain of sand at a time.

In limit of infinitesmal changes, system moves through

equilibrium states during transition
--
>reversible process

Grains of sand

A reversible process is one where every step of the path is at
equilibrium with its surroundings

28

Examples of Reversible and
Irreversible

Jumping into a pool? (from the perspective of
the water)

Getting into a pool slowly?

Free expansion of a gas

Isothermal expansion (piston must move very
slowly for bath)

Moving a glass slowly

29

Isothermal Reversible Process: T constant
: Ideal gas

For an ideal gas
Δ
U= q + w =0

-
w = q

30

Isothermal Reversible Process: T constant
: Ideal gas

For an ideal gas
E=q+w=0

w=
-
q

In rev process P
ext
=P
gas
=P=nRT/V

w=
-
P
1
V
-
P
2
V
-

31

32

33

34

Does z exist? Apply Euler’s rule of mixed differentials

35

36

37

i

f

a

b

Heat is not a State Function

Path Dependent

Inexact differentials

Can’t use
¢

Lot

s of paths lot

Exact Differential

Path Independent

Can use
¢

38

Joule’s Experiment

Work can be directly transformed into an equal
amount of heat through friction

Same goes for electrical work
.

How can we heat up
the water in a glass?