Tài liệu Chapter XV The First Law of Thermodynamics pdf

 Concerning to thermal phenomena, there exits a new form of energy called “heat”: Heat can be transferred from one to other systems For a system with volume held constant, the effect

GENERAL PHYSICS II

Electromagnetism

&

Thermal Physics

Chapter XV

The First Law of Thermodynamics

§1 Heat, work and paths of a thermodynamic process

§2 The first law of thermodynamics

§3 Kinds of thermodynamic processes

§4 Thermodynamic processes for an ideal gas

 We knew that the concepts of mechanical work and energy play an

important role in studying mechanical phenomena

 Concerning to thermal phenomena, there exits a new form of energy

called “heat”:

Heat can be transferred from one to other systems

For a system with volume held constant, the effect of heat is to

change the temparature of a system

In general cases, for a system there exist, at the same time, transfer

or exchange of heat and mechanical work

→ the GOAL of thermodynamics is the study of the relationships involving

heat, mechanical work, the laws that govern energy transfers

§1 Thermodynamic systems and processes:

1.1 Thermodynamic systems, heat and work:

 In any study of heat, work transfer we must define exactly what are

the objects under consideration:

A thermodynamic system is any collection of objects that is

regarded as a unit and that may have the potential to exchangeenergy with other bodies beside the system

All the other bodies which have energy exchanges with the

considered system are called surroundings or environment

 Then we must fix the convention on the symbol

for heat and work:

We will always denote

 by Q the quantity of heat added to the system

 by W the mechanical work done by the system

Therefore Q and W are understood as algebraic

values, they can be positive, negative or zero

systemsurroundings

systemsurroundings

system

surroundings

systemsurroundings

1.2 Calculation of work done during volume changes:

 A typical example of a thermodynamic system

is an amount of gas enclosed in a cylinder

with a movable piston (Such a system is the

central part of heat engines: locomotive,

engine of a car, refrigerator,…)

dx

 When a gas expands, it does work on its

environment For a small displacement dx,

the work done by the gas is:

dWby = F dx = p A dx = p (A dx)= p dV

A

 Consider the expansion of gas of from an initial state (with the volume

V1 ) to a final state (the volume V2) The system (gas) passes through

a series of intermediate states We assume the changes of states are

slow enough, then every intermediate state can establish equilibrium,

and has determined values of p, V, T

Note that when the gas expands, V2 > V1 → Wby > 0 , and when

the gas is compressed, V2 < V1 → Wby < 0 (it means that the

surroundings does work on the gas)

 In a p-V diagram, the equilibrium intermediate states are represented

by the points on a curve, and the work is represented as the area underthe curve

 The work done by the gas during the whole change V1 → V2 is

1.3 Paths between thermodynamic states:

 When a thermodynamic system changes from an initial state to a finalstate, it passes through a series of (equilibrium) intermediate states

However, with the same initial and final states, the system can pass invery different ways On a P-V diagram, every way corresponds to a

curve which is called the path between thermodynamic states

 Examples: Two different paths between the states 1 and 2 :

1 → 3 : keep the pressure constant

at p 1 while the gas expands

1 → 4: reduce the pressure

at the constant volume V 1

4 → 2: keep the pressure

constant at p 2 while the gas

expands to the volume V 2

V 1

 It is important to remark that with the same intial and final states:

The work done by the system depends on the intermediate states,that is, on the path,

Like work, the heat which the system exchanges with the

surroundings depends also on the path

In an isothermal expansion of the gas

we must supply an input heat to keep

constant temperature

Gas can expand in ancontainer which is isolatedfrom surroundings (no heatinput)

§2 The first law of thermodynamics:

2.1 Internal energy of a system:

 The internal energy of a system is the energy that the system owns

We can define:

(Note that the internal energy does not include potential energy arisingfrom the interaction between the system and its surroundings, for

example, system and gravitaitonal field)

Internal energy = ∑kinetic energies of constituent particles

+ ∑potential energies between them

 For an ideal gas we know how can calculate the internal energy Butfor any real system, the calculation of the internal energies by this waywould be very complicated

 We have another way Practically, in the study of thermodynamicalprocesses, we can determine not just the interal energy U , but the change in internal energy ΔU

We can choose by convention the internal energy of the system

at any reference state, and then knowing ΔU we can determine

U at all other states

(Recall that the potential energy of a particle in a gravitational field,

or the potential energy of a charge in the static electric field are

defined with the precision to an adding constant)

Having the concept of the internal energy, we can formulatethe first law of thermodynamics

2.2 Formulation of the first law of thermodynamics:

 Consider a change of state of the system from an initial value U1

to a final value U2 , then ΔU = U2 – U1

If the change is due to the addition of a quantity of heat Q with

no work done → the inernal energy increases, and ΔU = Q

If the system does work W by expanding and no heat is added,the internal energy decreases, we have ΔU = - W

 The first law of thermodynamics states that when both heat transferand work occur, the total change in internal energy is

ΔU = Q - W

Note: Always remember the convention on the signs of Q and W

given before !!!

§3 Kinds of thermodynamic processes:

We know that there are many different paths between thermodynamicstates We will study four specific kinds of thermodynamic processes

which are important in practical applications

3.1 Adiabatic process:

 Definition: Adiabatic process is defined as

one with no heat transfer

into or out of a system, Q = 0.

3.2 Isochoric process:

 Definition: This is a constant-volume process

 Example: A gas in a closed constant-volume

container

 When the volume of a thermodynamic system is

constant, it does no work on its surroundings

 Since the system does no work → all the energy (heat) added

remains in the system → the iternal energy increases

3.4 Isothermal process:

 Definition: This is a constant-temperature

process

 To keep temprature constant, the system must

exchange heat with the surroundings, and the

exchange must be slowly that thermal equilibrium

 In general, in a isothermal process, none of ΔU, Q, W is zero

 Only in the case of an ideal gas, the internal energy U ~ T

→ ΔU = 0 in a isothermal process

§4 Thermodynamic processes for an ideal gas:

 In this section, by applying the 1st law of thermodynamics we study

in more details thermodynamic processes for an ideal gas

 For an ideal gas, with the help of kinetic-molecular model, we know that

the internal energy of an ideal gas depends only on its temperature, not

on its pressure or volume.

 Owing to the explicit relation between the internal energy U and

temperature T we can find explicit equations which relate heat, workand internal energy

4.1 Constant-volume and constant-pressure heat capacities

of an ideal gas:

 We knew the concept of heat capacity of an ideal gas in a

constant-volume process Now consider more general cases of

thermodynamic process

 The general definition of heat capacity is the following equation:

where ΔQ is the quantity of heat added to the system for increase

ΔT in temperature

This definition can give rise different heat capacities which depend

on the paths of thermodynamic process

 The constant-volume heat capacity is defined by

Notes:

 Here we replace ΔQ by ΔU because no work done in the process

 If we understand CV as molar constant-volume heat capacity, then

ΔQ is the heat added per mole

 The constant-pressure heat capacity:

For a constant-pressure process the effect of the heat added to thesystem is twofold: to increase the internal energy and to do work

• Applying the 1st law we can write

• At the limit ΔT → 0 :

• In the case of an ideal gas, U depends only on T , we have

• Using the equation of state of an ideal gas we obtain the relation

for the molar heat capacities CP and CV :

CP = CV + R

(See experimantal values of CV and CP given in textbook, p 740,

tab 19.1)

4.2 Isothermal processes for an ideal gas:

 For a fix amount of ideal gas, from the

state equation thermal processes are

represented by P-V curves shown in

the picture

These curves are called isotherms.

 Calculate the work done by the gas

in the isothermal expansion A → B

with a fixed temperature T = T0

(see the picture):

where K = n R, n is the number of moles

 For the case of an ideal, the internal energy U depends only on

temperature → ΔU = 0 in isothermal processes By the 1st law,

the heat added to the gas for keeping constant temperature equals

to the work done by the gas:

Q = W

4.3 Adiabatic processes for an ideal gas:

 According to the definition, in an adiabatic process no heat transfer

takes place → Q = 0

 Remember for an ideal gas, U depends only on temperature T, and

→ for an adiabatic expansion (or compression) of gas, from 1st law

we have

Q = ΔU + P ΔV = CV ΔT + P ΔV = 0

For n moles of ideal gas n CV ΔT = - P ΔV = - n(RT/V) ΔV

apply the state equationfor n moles of ideal gas

R T

V P

C C

C C

1 (  



V

dV T

dT

constant ln

) 1 (

ln T    V  ln  ln   1  constant

V T

constant )

1 1 1



  



V T V

It means that for an adiabatic process:  

2 2 1

• Adiabatic curves: blue dashed

• Isoterms: solid black

) (

) ( T2 T1 nC T1 T2nC

( P1V1 P2V2 P1V1 P2V2R

 P-V curves for adiabatic processes

are shown in the picture, together with

isoterms for comparison

 An adiabatic expansion (for

example A → D) causes a drop

in temperature (T2 → T1)

 The first law of thermodynamics: ΔU = Q - W

The change ofinternal energy The quantityof heat added

 The relation between heat capacities: CP = CV + R

 For an isothermal process of an ideal gas: ΔU = 0

 For an adiabatic process of an ideal gas:

1 )

( )

( 1 2 P1V1 P2V2 P1V1 P2V2

R

C T

T nC

W Q