General Physics IGeneral Physics I

Lecture 25: Heat and The 1st Lecture 25: Heat and The 1st Law of ThermodynamicsLaw of Thermodynamics

Prof. WAN, Xin

xinwan@zju.edu.cnhttp://zimp.zju.edu.cn/~xinwan/

Latent Heat in Phase ChangesLatent Heat in Phase Changes

Latent HeatLatent Heat

The latent heat of vaporization for a given substance is usually somewhat higher than the latent heat of fusion. Why?

Mechanical Equivalence of HeatMechanical Equivalence of Heat

The amount of energy transfer necessary to raise the temperature of 1 g of water from 14.5oC to 15.5oC.

HeatHeat

Heat is defined as the transfer of energy across the boundary of a system due to a temperature difference between the system and its surroundings.

Heat transfer– Conduction, convection, radiation

Heat Conduction: MicroscopicHeat Conduction: Microscopic

Q

l l

UL, NL UR, NR

T

Heat Conduction: MacroscopicHeat Conduction: Macroscopic

dx

dTA

t

Qt

Fourier heat conduction law

Remind you of Ohm’s law?

Energy Transfer Through Two SlabsEnergy Transfer Through Two Slabs

Mean Free PathMean Free Path

dd

v

d

v

d

Average distance between two collisions

During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.

Mean Free PathMean Free Path

pd

Tk

dnvtdn

vtl B

VV222

1

Tknp BV

Tkpn BV /

vtdnz V2

Volume of the cylinder

vtdV 2Average number of collisions

Mean free path

During time interval t, a molecule sweeps a cylinder of diameter 2d and length vt.

Mean Free PathMean Free Path

pd

Tk

dntvdn

vtl B

VV222 22

1

)2(

vtdnz V2

Average number of collisions

Mean free path Relative motion vv 2

Q&A: Collision FrequencyQ&A: Collision Frequency

Consider air at room temperature. – How far does a typical molecule (with a

diameter of 2 10-10 m) move before it collides with another molecule?

Q&A: Collision FrequencyQ&A: Collision Frequency

Consider air at room temperature. – How far does a typical molecule (with a

diameter of 2 10-10 m) move before it collides with another molecule?

Q&A: Collision FrequencyQ&A: Collision Frequency

Consider air at room temperature. – Average molecular separation:

Q&A: Collision FrequencyQ&A: Collision Frequency

Consider air at room temperature. – On average, how frequently does one

molecule collide with another?

m

kT

m

kTv ~

8

l

vf

Expect ~ 500 m/s

Expect ~ 2109 /s

Try yourself!

Kinetic TheoryKinetic Theory

Q

l l

UL, NL UR, NR

T

md

f

m

1~

2

TlvT

Pf Tltht

22

1

TCUUQ VLR 2

1

2

1

dx

dTA

t

Qt

ldx

dT BkAlV

Nf

2

Energy exchange across plane A

tt for Air at Room Temperature for Air at Room Temperature

m1025.2 7l m/s500v

500m/sm1025.2300K

N/m10

2

5

2

1

22

1 75

tht lv

T

Pf

K)W/(m047.0

From earlier lecture

A factor less than 2 larger than the measured value of 0.026. Not bad after so many crude approximations.

Transport in ComparisonTransport in Comparison

Phenomena Imbalance Things being transported

Experimental observation

Unit of Coefficient

Thermal conduction

temperature energy W/m·K

Viscosity velocity momentum N·s/m2

Diffusion density particle m2/s

Charge conduction

voltage charge -1m-1

dy

dvAF

dx

dTA

t

Qt

dx

dnDAI n

x )(

dx

dVAI e

x )(

Internal Energy, Heat & WorkInternal Energy, Heat & Work

Heat is defined as the transfer of energy across the boundary of a system due to a temperature difference between the system and its surroundings.

Energy can also be transferred to or from the system by work.

Internal energy is all the energy of a system that is associated with its microscopic components —atoms and molecules —when viewed from a reference frame at rest with respect to the object.

Work in Thermodynamic ProcessesWork in Thermodynamic Processes

PdVPAdyFdydW

Quasi-static assumption: the gas expands slowly enough to allow the system to remain essentially in thermal equilibrium at all times.

Work done by the gas

f

i

V

VPdVW

Work in Thermodynamic ProcessesWork in Thermodynamic Processes

PdVPAdyFdydW

Work done by the gas

f

i

V

VPdVW

The work done by a gas in the expansion from an initial state to a final state is the area under the curve connecting the states in a PV diagram.

Warning:Warning: Sign Convention Sign Convention

Historically, people are interested in the amount of work done by the expansion of gas, say, to drive a steam engine. The common treatment is

– Positive work: gas expands– Negative work: gas compressed

In mechanics we use the opposite sign, unfortunately.

But some books follow the same convention in thermal physics as in mechanics.

Trust your common sense!

Work Depends on the PathWork Depends on the Path

)()(iff

a VVPW

)()(ifi

b VVPW

)()()( bca WWW

The work done by a system depends on the initial, final, and intermediate states of the system.

Ideal GasesIdeal Gases

Tkmv B2

3

2

1 2

Tkf

NU B2 BV Nk

f

T

UC

2Vfixed

TNkpV B

Experiments found Kinetic theory found

2

2

1

3

2mv

NVp

Generalized equipartition theorem (can be proved based on statistical principles)

Isothermal vs Free ExpansionIsothermal vs Free Expansion

An energy reservoir is a source of energy that is considered to be so great that a finite transfer of energy from the reservoir does not change its temperature.

An adiabatic process is one during which no energy enters or leaves the system by heat.

Isothermal ExpansionIsothermal Expansion

i

fB V

VTNkWQ ln

i

fB

V

V

BV

V V

VTNkdV

V

TNkPdVW

f

i

f

i

ln

0U at fixed T

(Adiabatic) Free Expansion(Adiabatic) Free Expansion

0WQ0U

Energy transfer by heat, like work done, depends on the initial, final, and intermediate states of the system.

- Is it possible to show the process on the PV diagram?

- Is the process reversible?

The 1st Law of ThermodynamicsThe 1st Law of Thermodynamics

Although Q and W both depend on the path, the quantity Q-W is independent of the path change.

The change in the internal energy U of the system can be expressed as:

The infinitesimal change:

WQU

PdVdQdU

reminding you that it is path dependent

Discussion on the 1st LawDiscussion on the 1st Law

The 1st law is a statement of energy conservation (now with the internal energy included).

The internal energy of an isolated system remains constant. In a cyclic process,

– The net work done by the systemper cycle equals the area enclosed by the path representing the process on a PV diagram.

WQU ,0

Discussion on the 1st LawDiscussion on the 1st Law

On a microscopic scale, no distinction exists between the result of heat and that of work.

The internal energy function is therefore called a state function, whose value is determined by the state of the system.

– In general,

),( VTUU

Digression on Multivariate CalculusDigression on Multivariate Calculus

If we take energy and volume as parameters, how comes heat is path dependent?

In mathematical language, dU + pdV is an inexact differential.

– In multivariate calculus, a differential is said to be exact (or perfect), as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q.

PdVdUdQ

Inexact DifferentialInexact Differential

12ln)2,2(

)2,1(

)2,1(

)1,1(

dy

y

xdx

0lnln),( fyxyxf

2ln21)2,2(

)1,2(

)1,2(

)1,1(

dy

y

xdx

dyy

xdxdg Assume

Note: is an exact differential.

Integrating factor

y

dy

x

dx

x

dgdf

Isobaric ProcessesIsobaric Processes

TNkTC BVTNkPV B

BVP NkCT

QC

Pfixed

TCU V)( if VVPW

VPTCWUQ V

isobaric

f

fC

CNkfCV

PBV

22/

Isobaric vs Isovolumetric ProcessesIsobaric vs Isovolumetric Processes

0W

isovolumetric

TCU V

TCUQ V

)( if VVPW

TCU V

TCWUQ P

isobaric

Molar specific heat: RCC VP

Degrees of Freedom, AgainDegrees of Freedom, Again

BV Nkf

C2

BP Nkf

C2

2

f

fC

CV

P2

f 3 5 7

1.67 1.4 1.28