Lecture_1_Final_MSE - Arizona State University

flinkexistenceMechanics

Oct 27, 2013 (3 years and 1 month ago)

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1

Introduction
First Law of Thermodynamics

Thermodynamic Property

Measurable quantity characterizing the state


of a system



Temperature



Volume



Pressure



Voltage



Magnetic field strength



Energy, etc.

Notable exceptions to this are Work and Heat



It doesn’t make any

sense to say that some object has a certain amount of work or heat. These quantities


can’t always easily be distinguished!

h

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Introduction
First Law of Thermodynamics

Thermodynamic equilibrium



mechanical equilibrium



thermal equilibrium



chemical equilibrium

The
state

of a system refers to the condition of the system as

described by the thermodynamic parameters characterizing

the system.


Equation of state


A functional relationship among the thermodynamic parameters characterizing

the system.


This reduces the number of independent

parameters characterizing the system from

three to two.

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Mechanical Work Terms


W = pdV; hydrostatic work


=
-

dZ; electrical work


=
-
HdM; magnetic work


=
-
g
dA; surface work


=
-
Fds



General form



W = Intensive

x
d

(
Extensive
)


Intensive variable
-

mass independent

Extensive variable
-

mass dependent



W

=
S
i

X
i
dx
i
=
pdV
-


dZ


HdM

+…

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If F and s are in the same direction, W > 0.

If F and s are in the opposite direction, W < 0.

Work

Massless, frictionless piston cylinder arrangements

vacuum

Free expansion

F = 0

W = 0

d
W =
-
Fd
z

F = p
ex
A

W =
-

p
ex
(
V
f
-
V
i

)

expansion against a constant

atmospheric pressure, P
ext

P
ext

vacuum

expansion against mg

F = mg

W =
-
Fd
z

Hydrostatic Work

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Heat

Historically, the concept of heat was difficult to quantify because

it is an energy flow based on microscopic actions rather than

Correlated macroscopic action.

If energy “enters a system” by some correlated motion, i.e., the

movement of a piston, we consider that as work.

If energy enters a system owing to some microscopic transport,

i.e., hot piston uncorrelated motion of molecules, we consider that

as heat.

If,

E+
*

≠ 0, the remainder of the energy change



is due to heat.
Sign convention


E+⁗‽†

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Heat Transport

Quite generally all transport processes can be described by an

equation of the form,
v =
m

x

F,
where
v

is a generalized velocity

or flux of transport and
F

is a generalized force.

This works for thermal transport, mass transport, electrical

transport, etc. All transport equations take the following form,

j =
-
k

grad

.

In the case of heat transport the heat flux is given by,

j
heat

=
-
k

grad
T

or

The minus sign is there because the flux is always in a direction

of decreasing temperature.

7

First Law of Thermodynamics

First Law
-

Conservation of Energy

dE

d
(
KE
) +
d
(PE) +
dU

dU

change in internal energy


Q

heat added to the system


W

work done by the system

dU
is an
exact differential

whereas


Q

and

W

are not.
*What’s and exact

differential? Is it possible to somehow turn a differential

quantity that isn’t exact in to an exact differential?

For example
-

In a conservative

field the quantity



is path independent, i.e.,







Note that


leads

directly to Maxwell the
reciprocal relations.




* Q is positive for heat flow in to the system


W is positive for work done by the system.

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Adiabatic Work

P
1
, V
1

P
2
, V
2

Quasistatic compression

of an ideal gas.

Adiabatic walls

What’s the physical manifestation

of the increase in internal energy?

p

V

V
f

V
i

p = constant/V
g

Sign convention:

The work is taken negative if it increases the energy in the system. If the volume
of the system is decreased work is done on the system, increasing its energy; hence the positive sign
in the equation


. The unusual convention was established to fit the behavior of heat
engines whose normal operation involves the input of heat and the output of work. So according to
this unfortunate convention work is positive if the engine is doing its job. Note that some textbooks
e.g.,
Equilibrium Thermodynamics

by Adkins will write the first law as which is
more in line with results in the
opposite sign convention

for work. In this convention positive work is
work going in to a system raising its energy. So, the manner in which you write the first law
establishes the sign convention that you use.

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Adiabatic Work

P
2
, V
2

P
1
, V
1

Composite system

p

Different adiabatic paths between two states of a fluid.




1A2. An adiabatic compression followed by electrical work at

constant volume performed via a “heater” of negligible thermal

capacity immersed in the fluid.



1B2. The same process but in reverse order



1C2. A complex route requiring simultaneous electrical and

mechanical work.

V

2

1

A

B

C

If a system is caused to change to an initial
state to a final state by adiabatic processes, the
work done is the same for all adiabatic paths
connecting the two states.

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Non adiabatic Work

P
1
, V
1

P
2
, V
2

Quasistatic isothermal compression

of an ideal gas.

Diathermal walls

Heat reservoir at fixed T

Heat reservoir at fixed T

Q

p

V

V
f

V
i

p = nRT/V

T



Heat


The heat

flux to a system in any process is

simply the difference in internal energy

between the final and initial states plus

the work done in that process.




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Simple thermodynamic system


A system that is macroscopically homo
-

geneous, isotropic, and uncharged, that is

large enough so that surface effects can

be neglected, and that is not acted on by

external electric, magnetic or gravitational

fields.


All thermodynamic systems have an enor
-

mous number of atomic coordinates. Only

a few of of these survive the statistical

averaging associated with a transition to a

macroscopic description. Certain of these

coordinates are “mechanical” in nature.

Thermodynamics is concerned with

consequences

of atomic coordinates that,

by virtue of the
coarseness

of observation,

do not appear in a macroscopic description

of the system.

Thermodynamic Equilibrium


In all systems there is a tendency to

evolve toward states in which the

properties are determined by intrinsic

factors and not by previously applied

external influences. Such simple

terminal states are time independent

and are called equilibrium states.


Equilibrium states of simple systems

are characterized completely by the

internal energy
U
, the volume
V
, and

the mole numbers
N
1
, N
2
, …

of the

chemical components.

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Thermodynamic Parameters


Measurable macroscopic quantities


associated with the system, e.g.,
p, V,

T, H, etc.

Ideal Gas Temperature Scale

pV/nk

T

100

divisions

Fp H
2
0

Bp H
2
0

0 K

pV

=
nkT

Equation of State


Functional relationship among the

thermodynamic parameters for a system

in equilibrium. If
p,V

and
T

are the

parameters,



f
(
p, V, T
) = 0,


This reduces the number of independent

parameters. Any point lying on this

surface represents a state on equilibrium.

p

V

T

Surface representing the

equation of state.

Equilibrium point

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Mathematical digression

Suppose that three variables are related as in an equation of state:

This expression can be rearranged to yield any one variable in

Terms of the other two independent variables,

Differentiating this expression,

(A)

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The terms in brackets are partial derivatives which are defined

In a manor analogous to normal derivatives

Also since,
z =z

(
x, y
)

Substituting in equation
(A)

for
dz
,

(B)

Mathematical digression

15

This result is true whatever pair of variables we choose as independent.

In expression
(B)

suppose,
dy = 0
, then

This is the
reciprocal theorem

which allows us to replace any partial

derivative by the reciprocal of the inverted derivative
with the same

variable(s) held constant
.

Mathematical digression

16

In expression
(B)

suppose,
dx = 0
, and
dy


0
, then,

by repeated application of the reciprocity theorem.

Taylor series expansion in 1 variable.

x

f
(
x
o
)

x
o


x = x
A

-

x
o

Mathematical digression

17

Taylor series expansion in 2 variables,
x = x(y,z
).


z

y


y


z

1

2

A

B

Given
x

at
y
1
, z
1
, we want to evaluate the value

of the function
x

at point 2. Proceeding from 1

to A,

And then from point A to 2,

Substitution for
x
A

from the 1
st

relation,

Mathematical digression

18

If we had proceed from 1 to B to 2,

Since the result must be the same, the last 2 terms in each of the

Expressions must be equal,

Mathematical digression

19

Exact differentials

If
x
is a function of
y

and
z
, we can write an expression for an

infinitesimal change in
x

owing to infinitesimal changes in
y

and
z
,

Here
Y

and
Z

correspond to the partial derivatives. In principle,
dx


can always be integrated since all we need are the initial and final

states.

If a quantity is not an exact differential,

W, then in order to integrate

To get the work, we need to know the path.

(C)

Mathematical digression

20

Since
dx

is an exact differential,

Which just expresses the condition
(C)

on the previous page.

Mathematical digression

21






Internal Energy


Macroscopic systems have a definite and precise energy energy subject to
a conservation principle. The energy of the universe is the same

today as it was eons ago.


Thermodynamics is concerned with measuring energy differences of a

system owing to external influences. We can easily determine the energy

difference between two equilibrium states of a system by enclosing the

system with
adiabatic

walls so that energy change can only occur by

doing some form of mechanical work on the system.

For a simple hydrostatic system (
pdV

work term) the change in internal energy

is describable in terms of any two of the variables p, V and T since the third is

by an equation of state (for an ideal gas,
pV = nRT
):

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Heat


The heat

flux to a system in any process is simply the difference in
internal energy between the final and initial states plus the work done in
that process,


.

Heat is what is adsorbed by a system if its temperature changes while no

work is done. If

Q

is the small amount of heat adsorbed in a system causing

a temperature change of
dT

the ratio

Q

/
dT

is called the
heat capacity
C
, of

that system. The
specific heat
c
, is the heat capacity per unit mass. For

example the heat capacity per mole of substance or the molar heat capacity

is defined as;

Specific heat

In the case of a hydrostatic system the heat capacity is unique when constraints

of either constant pressure or volume are imposed so that


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Here the symbol
H

is a thermodynamic potential or “energy” called the
enthalpy

which I will define shortly.

The second of the set of equalities is easily demonstrated just by writing out the

first law,

and substituting for
dU


obtaining


At constant volume,


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If we try to do the same thing for
C
p

we run in to a problem because of
the form of the internal energy function,

At constant pressure

You can see that this is messy compared to the simple equation for
C
V
. It is

convenient to define a new function
H
, such that
dH

=
d(U+pV).
Then

Introduction of the enthalpy function

allows for a simple relation for
C
p
.