1

The First Law of Thermodynamics Definitions

• system • isolated system • open system • closed system • properties • state of a system • change in state • path • process • cycle – cyclic process • state variable – state function

Each time you have a problem to solve in Thermodynamics you should pose the following questions to yourself:

• what is the system? • where is the boundary? • what is the initial state? • what is the final state? • what is the path of the transformation?

Work and Heat work – any quantity that flows across the boundary of a system during a change in its state and is completely convertible into the lifting of a mass in the surroundings Note that in this thermodynamic definition of work:

a. work appears only at the boundary of the system; b. work appears only during a change in system; c. work is manifested by an effect in the surroundings; d. work is an algebraic quantity ( it can be calculated from the change in PE of

the mass, Epotential = mgh)

2

e. W < 0, the height of the mass in the surroundings is lifted: work has been done by the system on the surroundings.

W > 0, the height of the mass in the surroundings is lowered. Work has been done on the system by the surroundings

Heat – a quantity that flows across the boundary of a system during a change in state by virtue of a difference in temperature between the system and its surroundings and flows from a point of higher to a point of lower temperature Note that in this thermodynamic definition of heat:

a. heat appears only at the boundary of the system; b. heat appears only during a change in state; c. the quantity of heat is manifested by an effect in the surroundings; d. the quantity of heat is equal to the number of grams of water in the

surroundings which are increased by one degree in temperature starting at a specified temperature under a specified pressure;

e. heat is an algebraic quantity. Heat is positive if a mass of water in the surroundings is cooled, in which case heat has flowed from the surroundings to the system. Heat is negative if a mass of water in the surroundings is warmed, in which case heat has flowed from the system to the surroundings

The quantities of heat and work which flow depend upon the process and therefore on the path connecting the initial and final states. Heat and work are called path functions. The First Law of Thermodynamics is a statement of the following universal experience: If a system is subjected to any cyclic transformation, the work produced in the surroundings is equal to the heat withdrawn from the surroundings. The First Law is also a statement of the conservation of energy, U:

dU = Dq + Dw or ∆U = q + w (system + surroundings) includes everything. If the system is returned to its original state, then the work produced in the surroundings must equal to the heat withdrawn from the surroundings. The change in internal energy of the system must be accounted for by the net heat + work appearing at the surroundings.

3

Internal Energy, U – The total energy content of a system due to the potential energy between molecules, due to the kinetic energy of molecular motions (translation in space as well rotations and vibrations), and due to the chemical energy stored in chemical bonds. U is a state function. Too difficult to measure; we can determine ∆U.

∆U ≡ Ufinal - Uinitial

In other words, for any change of state of the system, ∆Usystem + ∆Usurroundings= 0 “The energy of the universe is constant” - Clausius Work – a closer look

• the infinitesimal amount of work, dw done on the body by the force F is dw = Fx dx.

dw = Fx dx + Fy dy + Fz dz, if the infinitesimal displacement has components in all three dimensions The work, w done by F during displacement from x1 to x2 is the sum of the infinitesimal amounts of work done during the displacement:

∑= dxxFw )( . And this sum is equal to

∫=2

1

)(x

x

dxxFw

)( 12 xxFw −= for F constant • Units

1 J = 1 N m = 1 kg m2/s2 1 erg = 1 dyn cm 1 J = 107 ergs

• Conservative Forces: A conservative force is a force with the property that the work done in moving a particle between two points is independent of the path taken: gravitational, electrical, Hooke’s law force of a spring

For a conservative force we define the Potential Energy, V(x,y,z) as a function of x, y, z whose partial derivatives satisfy

4

xVFx ∂∂

−≡ yVFy ∂∂

−≡ zVFz ∂∂

−≡

• Work – energy theorem: w = K2 – K1 (one-particle system)

)(21)()()( 22

1

1111

0

1

o

v

v

t

t

t

t

r

r

r

r

vvmvdvmvdtdtdvmvdtrFdt

dtdrrFdrrFw

oooo

−====== ∫∫∫∫∫where, K = ½ mv2

• Mechanical Principle of the Conservation of Energy

Since rdUrF∂

−≡)( then 211)()()( KKrUrUdrrFw o

r

r

i

o

−=−== ∫

So, Emehc = K + U If only conservative forces act Emech remains constant. The following figure shows compression work done on the system.

This system alters its volume against an opposing external pressure and if the displacement is dI, then

5

Power is defined the rate at which work is done. P = dw/dt and SI unit of power is the watt, W = J s-1

6

7

Heat Capacity The response of system to heat flow is described by heat capacity Material-dependent and extensive quantity

dTDq

TTqC

ifT

=−

=→∆ 0

lim

SI units: J K-1 The value depends on the experimental condition Cv or CP Why Dq and Dw and not dq and dw? – Exact and Inexact Differentials

An exact differential integrates to a finite difference, i

y

yf yydy

f

i

−=∫ , which is

independent of the path of integration. In contrast, PdV is the kind of expression that cannot be regarded as the differential of any function of the state of the system; it is an inexact differential. P is a function of V and T, so the integral PdV is meaningless unless there exists a functional relationship between T and V – path.

If the path is specified then the line integral ∫∫ −==f

i

f

i

V

V

V

Vext dVPDww can be

evaluated! The reason for Dq will be explained in short time! The test for exactness The total differential dz of a quantity z can be determined by the differentials dx and dy in two other quantities x and y. In general,

dz=M(x,y)dx+N(x,y)dy, M and N are functions of x and y

8

To show the test for exactness, consider a function z that has an exact differential. If z has an exact value in the x-y plane, then it is a function of x and y. If z=f(x,y), then

dyyzdx

xzdz

xy

∂∂

+

∂∂

=

On comparison of the last two equations we can conclude that

dxxzyxM

y

∂∂

=),( and dyyzyxN

x

∂∂

=),(

Since yxxy y

zxx

zy

∂∂

∂∂

=

∂∂

∂∂ then

yx xN

yM

∂∂

=

∂∂

Euler’s criterion for exactness

Consider, P, T and Vm We will find that Thermodynamics enables to relate many thermodynamic properties of substances to partial derivatives of the above physical quantities.

Pm

TV

)(∂∂

, Tm

PV )(∂∂

, TmV

P)(

∂

∂, mVT

P )(∂∂

, PmV

T)(

∂

∂, mVP

T )(∂∂

By utilizing the relation yy zxxz )/1)( ∂∂=∂∂ we see that three of the above six are the reciprocals of the other three.

And TmV

P)(

∂

∂P

m

TV

)(∂∂

mVPT )(∂∂

= -1

Hence, there are only two independent partial derivatives Pm

TV

)(∂∂

and Tm

PV )(∂∂

.

The other four can be calculated from these two and need not be measured.

9

thermal expansitivity, (α)

Pm

mnP T

VVT

VV

PT )(1)(1),( , ∂∂

≡∂∂

≡α

isothermal compressibility, (κ) Tm

mnT P

VVP

VV

PT )(1)(1),( , ∂∂

−≡∂∂

−≡κ

α and κ tell you how fast the volume of a substance increases by temperature and decreases with pressure

Problem: Show that =∂∂

mVTP )(

κα

Choosing a system of fixed mass, we can describe the state of the system by specifying T and V. Then we can write the energy U as a function of T and V.

U = U(T,V) and dVVUdT

TUdU

TV

∂∂

+

∂∂

= - total derivative

What the subscript “v” (at constant V) really says? If the temperature of the system increases by an amount dT and the volume increases by an amount dV, then the total increase in energy is the sum of two contributions.

dTTU

V

∂∂

is the increase in energy resulting from the temperature increase

alone. Changes in State at Constant Volume

dVVUdT

TUdU

TV

∂∂

+

∂∂

= If the volume is constant as the system undergoes

a change in state, then dV = 0. So, dTTUdU

V

∂∂

=

10

dTTUU

V

T

T

f

i

∂∂

=∆ ∫ when dV = 0.

Also, since dV = 0 the integral ∫ =−=f

i

V

Vexternal dVPw 0

The First Law says ∆U = q + w which leads to ∆U = qV This allows us to write dU=DqV

And VV

DqdTTUdU =

∂∂

= , which relates the heat withdrawn from the

surroundings to the increase in temperature dT at constant volume. Heat capacity of the system at constant volume (CV)

V

VV T

UdT

DqC

∂∂

=

≡ For an infinitesimal change dU = CV dT and this can be

integrated to dTCUf

i

T

TV∫=∆ - constant volume change of any system