Relationships between WORK, HEAT, and ENERGY

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Introduction to Thermodynamics, Lecture 3-5 Prof. G. Ciccarelli (2012)

Relationships between WORK, HEAT, and ENERGY

Consider a force, F, acting on a block sliding on a

frictionless surface

Mdv

dtF

Mdv

dtF F v

F Mdv

dtM

dv

dx

dx

dtM

dv

dxv

Fdx Mvdv

Fdx M vdv

FdxMv Mv

x

= ; mass velocity in x direction

Integrating both sides from block position 1 to 2

W KE1 2

1

2

1

2

1

22

2

1

2

2 2

M

x FRICTIONLESS

SURFACE F

x1 x2



∫Fdx is the energy transferred to the block in the process

work done W1-2 by force F

Consider an object falling in a gravitational field

W1 2 F dy mg dy mg h h

1

2

1

2

1 2( ) ( )

Gravity has the potential to do work and the quantity

mgh is therefore called the potential energy

Work done by gravity results in a drop in potential energy

of the object

since W KE1 2 (see previous example)

m

h1 h2

y

F=mg



mg h hmv mv

( )1 2

2

2

1

2

2 2 PE KE

Mass PE is converted to KE via the work done by gravity

Energy Transfer by Work

In general, work done is evaluated by

W1 2 F ds

1

2

Work is a means of transferring energy, it does not refer

to what is being transferred or stored within the system.

The value of W1 2 depends on the details of the

interaction taking place between the system and the

surroundings during a process, e.g., F(s), and not just the

initial and final state

By definition a state property is evaluated at a specific

time and is independent of the process

energy is a property of the system

work is not a property of the system



The differential of a property is “exact” since it is

independent of details of the process, e.g.,

dE E E1

2

2 1

Differential of work is “inexact,” the following integral

can’t be evaluated without knowing details of the process

W W1

2

not 2

112 WWW

The work done over a period of time is:

2

1

2

1

2

1

dtvFdtdt

sdFsdFW

where v

is velocity

The rate of energy transfer by work is called power and

is denoted by W . In general,

vFW



Expansion and Compression Work

Consider the expansion of the gas in a piston-cylinder

assembly ( Pp is average pressure on piston face)

Pp

W1 2

F ds P A dx P dV

p p1

2

1

2

1

2

( )

For a slow or quasi-equilibrium process all the states

through which the system passes are considered

equilibrium states and thus the intensive properties, i.e.,

pressure, are uniform throughout the system gasP PP , so

W1 2

P dVgas

V

V

1

2

x x1 x2

Piston Area A

Pp



Graphical Interpretation:

State 1

State 2

Processpath Pressure

Volume

P1

P2

V1 V2 dV

δW=PdV

x1 x2



W PdV

W PdVV

V

shaded area

W total area under curve1 21

2

1

2

Consider two processes with the same start and end state

Since the area under each curve is different the amount of

work done for each path is different.

2 1 )()(2

1

2

1

path

V

Vpath

V

V

PdVPdV

State 1

State 2

Path 2

P1

P2

V1 V2

Path 1



Work done depends on the path taken and not just the

value of the end states.

Work is not a property!

Polytropic Compression and Expansion

The pressure-volume relationship can be described by

PVn= constant c n= constant

The work done is:



W

but

W

W

1 2

1 2

1 2

PdVc

VdV cV dV

cV

nc

V V

n

c=PV PV

PVV V

n

PV PV V

n

PV

V

V

nV

V

n

V

V

n

V

V n n

n n

n

n n n n

1

2

1

2

1

2

1

1

22

1

1

1

1 1 2 2

2 2

2

1

1

1

2 2 2 2 1

1

2 2

1 1

1 1

( ) ( )

PV

nn1 1

11



For n=1 P = c/V

W

W P

1 2

1 2 1

PdVc

VdV c V

c V V cV

V

VV

Vn=

V

V

V

V

V

V

1

2

1

2

1

2

2 1

2

1

1

2

1

1

ln

ln ln ln

ln

Special case:

For n = 0 P = c constant pressure process

W 1 2

PdV P V V nV

V

( )2 1

1

2

0



Spring Potential Energy

F – spring force = kx

k – spring constant (N/m)

x – displacement from

relaxed position

W

Spring PE

1 2

F ds kx dx

kx

k x x

kx

x

x

1

2

1

2 2

2

2

1

2

2

2

1

2

1

2

1

2

( )

( )

The spring potential energy can be grouped in with

gravitational potential energy.

F=

x



Other forms of Energy

In engineering, the change in total energy of a system is

considered to be made up of macroscopic contributions

such as changes in KE and gravitational PE of the system

as a whole relative to an external coordinate frame and

Internal Energy, U.

E2- E1= (KE2- KE1) + (PE2- PE1) + (U2- U1)

Consider the vigorous stirring of a fluid in a well

insulated tank

Energy is transferred into the system via work by the

paddle wheel, results in an increase in the system energy.

E2- E1= (KE2- KE1) + (PE2- PE1) + (U2- U1)= W

Electric motor

Fluid

WELL INSULATED

W system



This transferred energy does not increase the KE or PE of

the system.

The change in system energy can be accounted for in

terms of internal energy of the fluid.

Changes in internal energy for solids, liquids, and gases

are evaluated using empirical data, e.g. U = f(T)

Microscopic Interpretation of Internal Energy

Energy is attributed to the motions and configuration of

the individual molecules, atoms and subatomic particles

making up the matter in the system.

Energy on molecular level associated with:

- Translation

- Rotation

- Vibration

- Molecular bonds

Energy on atomic level:

- Electron orbital states

- Nuclear spin

- Nuclear binding



Conservation of Energy for Closed System

A closed system can interact with its surroundings via

work as well as thermally

Energy can be transferred between the system and the

surroundings by thermal (heat) interactions

A process that involves work interactions but does not

involve thermal interactions is called an adiabatic

process

A process that involves thermal interactions is called a

nonadiabatic process

It has been shown experimentally that the net work done

by, or on, a closed system undergoing an adiabatic

process depends solely on the end states and not on the

details of the process.

E2 – E1 = -Wad

Sign convention for energy transfer by work:

Work done by the system is positive

Work done on the system is negative



For a quasi-equilibrium adiabatic gas compression or

expansion process the value of the polytropic exponent n

is fixed (n =1.4 for air) and thus the area under the curve

(work done) depends only on the end states

Consider an adiabatic process and nonadiabatic process

between the same two end states 1 and 2

1

2

Adiabatic path PV1.4 = const (air)

P1

P2

V1 V2

1

2

Adiabatic (only work) PV1.4 = const.

P1

P2

Nonadiabatic (work and heat) PVn = const.



Since the area under the two curves is different the work

done for each path is different, so Wad Wnonad

Since the end states for both processes are the same the

system would experience exactly the same energy change

in each of the processes, so

(E2 – E1)ad = (E2 – E1)nonad = E2 – E1

We know the energy change for the adiabatic process is

E2 – E1 = -Wad

But since Wad Wnonad we can infer that

E2 – E1 -Wnonad

Since energy must be conserved the net energy transferred

to the system in both processes must be the same. It

follows that the heat interaction in the nonadiabatic

process must involve energy transfer. The amount of

energy transferred to the closed system by heat is Q

E2 – E1 = -Wnonad + Q

The First Law of Thermodynamics states:

V1 V2



E2 – E1 = Q - W

Energy Transfer by Heat

The quantity Q in the First Law accounts for any energy

transferred to a closed system during a process by means

other than by work.

Such energy transfer Q is induced only as a result of a

temperature difference between the system and the

surroundings and occurring in the direction of decreasing

temperature, e.g. heat transfer: conduction, convection,

radiation

Sign convention for energy transfer by heat:

Heat transfer to the system is positive

Heat transfer from the system is negative

Consider the immersion of a lump of hot metal initially at

Tm into a colder fluid at Tf

Tm Tm > Tf



Because the metal is at a higher temperature than the fluid

energy is transferred from the metal to the fluid, Q is

negative.

Since there is no work done and the change in KE and PE

is negligible, the amount of heat transferred from the

metal to the fluid is equal to the decrease in the metal

internal energy,

WQPEKEU

U2 - U1 = (-Q) or Q = U1 – U2

Just like work, heat is not a property and the amount of

energy transfer depends on the process details, therefore

Q1 2 Q1

2

Tf

TmTf Q

Tf



The rate of heat transfer is denoted by Q and the total

energy transferred via heat over a period of time is

Q1 2 Qdt1

2

Relationships between WORK, HEAT, and ENERGY

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