Chapter 2: Internal Energy (U), Work (w), Heat (q ...scienide2.uwaterloo.ca/~nooijen/website_new_20_10_2011/Chem254...Winter 2013 Chem 254: Introductory Thermodynamics Chapter 2: Internal

Winter 2013 Chem 254: Introductory Thermodynamics

Chapter 2: Internal Energy, Work, Heat and Enthalpy 13

Chapter 2: Internal Energy (U), Work (w), Heat (q), Enthalpy (H) ................................................ 13

Heat Capacities ......................................................................................................................... 16

Calculating ΔU, ΔH, w, q in Ideal Gas ........................................................................................ 18

Isothermal Compression ........................................................................................................... 21

Reversible Process (limiting process) ....................................................................................... 22

Isothermal Expansion ............................................................................................................... 22

Chapter 2: Internal Energy (U), Work (w), Heat (q), Enthalpy (H)

Internal Energy (excludes motion and rotation of vessel)

Look at isolated part of universe

system EnvironmentU U U

Total = isolated

First law of thermodynamics:

- Total U for isolated system is constant

- Energy can be exchanged between various components

- Energy forms can be interconverted

Eg. Chemical En Heat Work

0total system environementU U U



Work

In classical mechanics, move object a distance d with force F in direction of

displacement is work N m = J

F mg d h

w mgh (kg m s-2 m = N m = J)

cosw mgd cosh

d

h

w mgd mghd

General formula

w F dL Line integral

PV work (constant external pressure)

m applies constant force F

PA

1 2( ) ( )ext

Fw mgh Fh Ah P V V

A

( )ext final initialw P V V Joules, or L Bar (1 L Bar = 100 J)



More general formula for PV work, P does not need to be constant

f

i

V

extV

w P dV

Sign Convention : Work done on the system raises internal energy of system ( 0w )

Work done by the system lowers the internal energy ( 0w )

Other forms of work:

- electrical work

w Q Q is charge in coulombs

difference in potential (in Volts or J/C)

Run a current over

Q I t I is current (in Amps or C/s)

w I t

Important: Work is associated with a process, with change. Work is transitory. You

cannot say that a system contains that amount of energy or heat

Heat: associated with a process going from State 1 State 2

systemU q w q is heat; w is the work

Heat is exchanged between system and environment

0q : system loses energy

0q : system gains energy

system environmentq q

note: system environmentT T for heat to flow

Isolated system



outer innerT T (regulate)

So there is no flow of heat

0system environmentU U

0innerU

Beaker + Lab +… = environment (isolated)

0IU ; 0IIU

0I IIU U

Chemical Energy 2 2 2Butane + O CO H

Note : 0IIU even if temperature increases!

Why? Chemical energy of butane is converted to heat.

Heat Capacities

The amount of energy (heat) required to raise the temperature of 1 gram of substance

by 1 oC. Heat capacity of water is 4.18 J/g K = 1 calorie

1) Heat capacity is dependent on heat

Eg. 10 oC 11 oC and 80 oC 81 oC, require slightly different energies

2) At least 2 types of heat capacity

a) Keep volume constant VC

b) Keep pressure constant PC

3) Heat capacity is proportional to amount of substance

Molar heat capacities : ,P mC , ,V mC

n moles : ,V V mC nC , ,P P mC nC

4) General formula



f

i

T

V VT

q C dT

If VC is constant over temperature range:

f f

ii

T T

V V V V f iTTq C dT C T C T T

( )V Vq C T

And ( )P Pq C T

Which is larger PC or

VC ?

Relation for PC and

VC for ideal gas?

2 1V V ; 2 1T T

2 1( )P P extU q w q P V V PV nRT

2 1( )PU q nR T T

P Pq C T ; V VU q C T

P VC T C T nR T

P VC C nR or , ,P m V mC C R

Therefore PC is larger than

VC . At constant P , the system also does PV work when

raising T . (analysis for ideal gas)

No work because V is constant

VU q w Vq

VU C T

Bomb calorimetry

system surrounding Calorimeter

V V V measureq q C T

reactionU



system surroundings

P Vq q

system Calorimeter

P V measureq C T

system

P reactionq H

True definition of Enthalpy

( )H U PV 2 2 1 1PV PV PV ; for 1 1 2 2PV PV ;

H U PV

At constant Pressure

2 2 1 1H U PV PV

H U P V

( ) ( )P PH q w P V q P V P V

P PH q C T

Completely general : ,U H are function of state

specify , ,T V P

2 2 2 1 1 1( , , ) ( , , )U U T P V U T P V

2 2 2 1 1 1( , , ) ( , , )H H T P V H T P V

Change in ,U H are the same for both paths

Change in ,q w are different for different

paths

Calculating ΔU, ΔH, w, q in Ideal Gas

1) Calculating ,U H is easy if T is known

( ) ]f

f

ii

T T

V V T V f iT

U U T U C dT C T C T T

VU C T for any process

( ) .....H H T



PH C T for any process (if PC is constant)

We know P VC C nR

Special cases:

Isothermal Process T is constant 0T ; 0U H

2) Work: extw P dV PV work only

- Constant V 0i fV V w ; Vq q U

- Constant Pext ( )f

i

V

ext ext f iV

w P dV P V V ; Pq q H

Isothermal reversible process: (Reversible process: delicate, see later)

1

extP nRTV

nRT is constant

lnf f

ii

V V

VV

dVw nRT nRT V

V

ln ln lnf

f i

i

VnRT V V nRT

V

lnf

i

Vw nRT

V

3) Heat

Adiabatic process : 0q by definition

U q w ; U w

Adiabatic Reversible Process

0q , U w , ext

nRTP

V

f

i

V

V

nRTU w dV

V

,V m

nRTnC dT dV

V

,V m

dT nRnC dV

T V



,

f f

V mi i

dT dVnC nR

T V

, ln lnf f

v m

i i

T VnC nR

T V

,

ln lnf fv m

i i

T VC

R T V

For adiabatic reversible process:

,

ln lnf fv m

i i

T VC

R T V

OR

,ln ln

fP m i

i f

TC P

R T P

OR ,

,

ln lnf P m i

i V m f

V C P

V C P

1) ,

ln lnf fv m

i i

T VC

R T V

,

,

ln ln lnV m

R

Cf f f

i V m i i

T V VR

T C V V

,V m

R

Cf f

i i

T V

T V

2) ,

ln lnf fv m

i i

T VC

R T V

,

ln lnf fV m i

i f i

T nRTC P

R T P nRT

lnf i

i f

T P

T P

ln lnf i

i f

T P

T P

,

1 ln lnfV m i

i f

TC P

R T P

,

ln lnfV m i

i f

TC R P

R T P

,ln ln

fP m i

i f

TC P

R T P



Adiabatic Isobaric Process

Constant external pressure AND 0q

Isothermal Compression

Constant external pressure 0f f iw P V V

0q w (because 0U because isothermal)

What is work in 2-step process?

2 int int inti f fw P V V P V V

2 1w w ; 2 1q q



Conclusion: w and q depend on details of process, not only on initial and final state.

Repeat for 3 step, 4….

5 4 3 2 1w w w w w ; 5 4 3 2 1q q q q q

The more steps, the less w and less heat

Reversible Process (limiting process)

ext gasP P at each step

ext

nRTP

V

Isothermal Reversible Process

f f

i i

V V

extV V

dVw P dV nRT

V

ln | lnf

i

V f

V

i

VnRT nRT

V

work, q is minimal

Isothermal Expansion



1 0f f iw P V V ;

1 1 0q w

2 1w w ; 2 1q q

3 2 1w w w ; 3 2 1q q q

More processes more work ( w ), more heat ( q )

5 4 3 2 1w w w w w ; 5 4 3 2 1q q q q q

Limiting Expansion Work exp

l imit l imit

compression ansionw w

limit lnf f

i i

V V f

extV V

i

VdVw P dV nRT nRT

V V



Grains of sand : I can run process either way

The thermodynamic work is the same both ways for reversible process

Irreversible Process (Big chunks of mass)

Follows arrows in reverse: add mass, piston rises? ; removes mass, piston lowers?

This is absurd, hence:

Why do irreversible processes run in one way and not another?

What is special about irreversible?

Documents

Chapter 2: Internal Energy (U), Work (w), Heat (q ...scienide2.uwaterloo.ca/~nooijen/website_new_20_10_2011/Chem254...Winter 2013 Chem 254: Introductory Thermodynamics Chapter 2: Internal