Chapter 19 – The First Law of Thermodynamics

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Chapter 19

The First Law of Thermodynamics

I.

Introduction

We have been looking at how various systems behave as a result of heat being
exchanged. Now we will first look at how
work

affects a thermodynamic system, then examine the
energy relationships
in a thermodynamic system.

II.

Work

A.

Mechanical work

look at the work done
by

a system (gas) as its volume changes.

Consider a cylindrical container with a moveable, frictionless piston with a cross
-
sectional area
A
.
Within the container is a gas at a pre
ssure (absolute)
p
. Suppose that the piston is pushed upward
a distance
dx
.

The work done
by

the gas is:
.

is the volume
dV
, so

When a graph of pressure
p

versus volume
V

is d
rawn, the graph is referred to as a
pV

diagram.

If the volume increases, then work is done
by

the system and the work is positive.

If the volume decreases, then work is done
on

the system and the work is negative.

p

dx

Pressure,
p

Volume,
V

V
i

V
f

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III.

Types of therm
odynamic processes and work done in the processes

Again consider an ideal gas inside a
cylinder that is fitted with a piston.

A.

Isobaric process

B.

Isochoric or isovolumetric process

C.

Isothermal Process

piston

gas

p

V

p

V

p

V

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IV.

Combine the Effects of Heat Transfer and Work

Consider a container that is fitted with a piston filled with a gas at some initial pressure, volume and
temperature. An amount of heat
Q

will be added to the system (gas) and the energy changes
will be
examined.

Before the heat is added, how much energy and what type of energy is associated with the system?

A.

Add heat to the system while keeping the volume constant. Where does the energy go? Note
that the temperature of the gas must

change.

B.

Add heat to the system and keep the temperature constant. Where does the energy go?
(Remember that the temperature is directly proportional to the translational kinetic energy,
TKE
.)

C.

Add heat to the system, in general. Fi
rst Law of Thermodynamics.

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D.

Interesting Note: If a system changes from an initial state
i

to a final state
f

along different paths
(
e
.
g
.,
Path

A

and
Path B
), the change in internal energy will be the same along those paths. And, in
fact, all
paths that go from
i

to
f
. That is,

U

=
U
f

-

U
i

.

From the first law, that means that
Q

W

is also constant going from
i

to
f
.

V.

The First Law of Thermodynamics as applied to various thermodynamic processes:

U

=
Q

-

W

and
dU

=

dQ

-

dW

A.

Isochoric process,
V

= constant and
dV

= 0

dU

=
dQ

-

pdV

since

since
dQ

=
nC
V
dT

dU

=
nC
V
dT

This result represents the i
ncrease in the internal energy of molecules as a result of the
temperature change

T

or
dT
.
This result is valid for all processes involving an ideal gas.

p

i

Path A

0

T
f

T
i

f

V

0

Path B

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B.

Isobaric process,
p

= constant and
dp

= 0

C.

Isothermal process,
T

= constant and
dT

= 0

D.

Q

= 0 and
dQ

= 0

E.

Cyclic process (start and end in the same state)

F.

Some examples:

1.

iaf
:

Q

= + 50 J

W

= + 20 J

ibf
:

Q

= + 36 J

p

V

i

f

a

b

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a.

Find the work done going from
i

to
b

to
f
,
W
ibf

.

b.

If the work done
going from
f

to
i
,
W
fi

=
-

13 J , then find the heat exchanged going from
f

to
i
,
Q
fi

.

c.

If the internal energy at point
i

is
U
i

= 10 J , then find the internal energy at
f
,
U
f

.

d.

If the internal energy at point
b

is
U
b

= 22 J ,

then find the heat exchanged going from
i

to
b
,
Q
ib

, and from
b

to
f
,
Q
bf

.

2.

For the following data, find the heat exchanged
going from state C to A,
Q
CA

:
Q
AB

=
20 J,
Q
BC

= 0,
and
W
BCA

= + 15 J.

p

V

A

B

C

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3.

Find the
net heat transferred in the cyclic

process.

VI.

Heat Capacities for Ideal Gases

A.

Monatomic Gas

1.

Remember that in an isochoric process, the molar heat capacity for a monatomic gas is

2.

What is the

molar heat capacity of the gas when heat is added isobarically? Add heat while
keeping the pressure constant. In this case the added heat energy goes into increasing the
kinetic energy (internal energy) of the molecules
and

in doing work on the environm
ent
(volume increases).

Heat Added = Heat (KE) + Heat (Work)

dQ

=
dU

+
dW

=
nC
V
dT

+
pdV

p

(atm)

V

(L)

0

1 2 3 4

30

20

10

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3.

Ratio of heat capacities,

B.

Remember the value of the molar heat capacity at a constant volume,
C
V
, for molecules with
f

degrees of freedom.

C.

Experimental values of

at 300 K

Monatomic

Diatomic

Polyatomic

Helium

1.67

Hydrogen

1.41

CO
2

1.30

Argon

1.67

Ox
ygen

1.40

SO
2

1.29

Nitrogen

1.39

C
2
H
6

1.20

Chlorine

1.35

NH
3

1.31

(C
2
H
5
)
2
O

1.07

VII.

Examine the adiabatic process again, and find relationships between the state variables.

A.

Start with the first law and the equation of state of an ideal
gas

dU

=
dQ

-

dW

and
pV

=
nRT

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Example: One mole of a monatomic gas expands adiabatically from 10 atm at 0
o
C to a pressure of 2
atm. Find the initial volume of the gas, the final volume, and the final temperature.

B.

Work done in an adiabatic process

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VIII. Examples of cyclic processes. Complete the following tables:

Example 1
. Shown in the diagram is a cyclic process for
n

moles of a diatomic gas with heat capacities
C
V

=

and
C
p

=
.

Process

Name

Q

(J)

W

(J)

U

(J)

S

(J/K)

1 to 2

2 to 3

3 to 1

1 to 2 to 3 to 1

State

p
(Pa)

V
(m
3
)

T

(K)

1

100

2

4

4

3

1

1

2

3

p

V

dT

= 0

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Example 2
. A cyclic process is shown for a diatomic gas with a
ratio of heat capacities

= 1.40.

State

p

(atm)

V

(L)

T

(K)

A

2

2

273

B

1

C

Proc
ess

Name

Q

(J)

W

(J)

U

(J)

S

(J/K)

A to B

B to C

C to A

A to B to C to A

p

V

A

B

C