Lecture 4 Applications of the First Law (new version)

Applications of the First Law

(Lecture 4)

1

^st

semester, 2021

Advanced Thermodynamics (M2794.007900) Song, Han Ho

(*) Some materials in this lecture note are borrowed from the textbook of Ashley H. Carter.

Heat Capacity

è

Definition: the limiting ratio of the heat absorbed divided by the temperature increase

è

Specific heat capacity (or specific heat)

è

Specific heat at constant volume and at constant pressure derivative

not truly

lim

0

÷ = ®

ø ç ö

è æ

= D

®

D

dT

Q T

C Q

T

d Applications of the First Law

K) (J/kmol 1

K) (J/kg 1

×

÷ = ø ç ö

è

= æ

×

÷ = ø ç ö

è

= æ

dT q dT

Q c n

dT q dT

Q c m

d d

d d

K) (J/kg

and

= ç æ ÷ ö ×

÷ ö ç æ

= q

q c

c d d

3

Heat Capacity

Applications of the First Law

Solid copper at 1 atm

Various ideal gases at 1 atm

1. Various ideal gases show different limit behaviors.

2. Argon has constant specific heats.

3. There is almost constant difference between c_P and c_v.

(à Mayer’s equation)

è

Some examples

Mayer’s Equation

è

Let’s find the relationship between c

_v

and c

_P

for an ideal gas.

For a simple compressible substance, the first law is

In general,

Then,

But for an ideal gas, (Gay-Lussac-Joule exp.)

Then,

Applications of the First Law

Pdv q

w q

du = d - d = d -

T dT dv u

v du u

T v u u

v T

÷ ø ç ö

è æ

¶ + ¶

÷ ø ç ö

è æ

¶

= ¶

®

= ( , )

ò

= -

=

÷ = ø ç ö

è æ

¶

= ¶

^T

T v

v v

v

du c dT u u c dT

dT du T

c u

0 0

or

or

0

only )

( ÷ =

ø ç ö

è æ

¶

® ¶

=

v

T

T u

u

u

5

Mayer’s Equation

è

Continue on.

Using ideal gas EOS,

Substituting this equation,

Then,

Mayer’s equation

“Over the range of variables for which the ideal gas law holds, the two specific heat capacities differ by the constant R.”

è

The ratio of specific heat capacities (or specific heat ratio) Applications of the First Law

RdT vdP

Pdv RT

Pv = ® + =

R T c

c q

_v

P

P

÷ = +

ø ç ö

è æ º ¶ d

( c R ) dT vdP

q Pdv

dT c

q =

_v

+ ® d =

_v

+ -

d

v P

c

º c

g

Enthalpy and Heats of Transformation

è

The heat of transformation

l Heat transfer accompanying a phase change

l Phase change is an isothermal and isobaric process with changing volume.

l Applying the first law for a phase change, from phase 1 to phase 2,

Here, we define enthalpy(h), a state variable.

Then,

Applications of the First Law

( ) ( )

n) sublimatio fusion,

ion, vaporizat (i.e.

change phase

of heat latent

: where

)

( _{2 1 2 2 1 1}

1 2

l

Pv u

Pv u

l v

v P l u u

Pdv q

du

+ -

+

=

® -

-

= -

-

=

d

Pv u

h º +

P-v-T Surface for Water

7

Relationships involving Enthalpy

è

Let’s find some useful relationship using enthalpy for an ideal gas.

For S.C.S., the first law is

From the definition of enthalpy,

Then,

In general,

But for an ideal gas, (Joule-Thomson exp.)

Then,

Applications of the First Law

q Pdv

du Pdv

q

du = d - ® + = d

T dT dP h

P dh h

T P h h

P T

÷ ø ç ö

è æ

¶ + ¶

÷ ø ç ö

è æ

¶

= ¶

®

= ( , )

0

only )

( ÷ =

ø ç ö

è æ

¶

® ¶

=

P

T

T h h h

vdP dh

Pdv du

vdP Pdv

du Pv

u d

dh = ( + ) = + + ® + = - vdP

dh q = - d

ò

= -

=

÷ = ø ç ö

è æ

¶

= ¶

^T

T P P

P

P

dh c dT h h c dT

dT dh T

c h

0 0

or or

Work done in an Adiabatic Process

è

Let’s calculate work done in an adiabatic process for an ideal gas.

For a simple compressible substance of an ideal gas, the first law is

Then,

Work is given by the following expression for a constant specific heats,

Applications of the First Law

dT c

vdP vdP

dT c

q

dT c Pdv

Pdv dT

c q

P P

v v

=

® -

=

-

=

® +

=

d

d ® = - = - g

v P

c c Pdv

vdP

K const

v Pv dv P

dP = - g or

^g

= =

(

¹

) [

¹_{2 1}¹

] [

_{2 2 1 1}

]

1 1 1

1 1

1

²

1 2

1

v P v

P Kv

Kv Kv

dv v

K Pdv

w

^v

v v

v

- -

= - -

- =

=

=

= ò ò

^-g

g

^-g

g

^{-g -g}

g