Short Version : 18. Heat, Work, & First Law of Thermodynamics

draweryaleMechanics

Oct 27, 2013 (3 years and 8 months ago)

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Short Version : 18. Heat, Work, & First Law of Thermodynamics

18.1. The 1
st

Law of Thermodynamics

Either heating or stirring can raise
T

of the water.

Joule’s apparatus

1
st

Law of Thermodynamics
:

Increase in internal energy = Heat

added


Work done


Thermodynamic state variable

= variable independent of history.

e.g.,
U
,
T
,
P
,
V
, …

Not
Q
,
W
, …

PE

of falling weight




KE

of paddle




Heat

in water

18.2. Thermodynamic Processes

Quasi
-
static process
:

Arbitrarily slow process such that system always
stays arbitrarily close to thermodynamic equilibrium.

Reversible process
:

Any changes induced by the process in the universe (system
+ environment) can be removed by retracing its path.

Reversible processes must be quasi
-
static.

Irreversible process
:

Part or whole of process is not reversible.

e.g., any processes involving
friction, free expansion of gas ….

T
water

= T
gas

& rises slowly

system always in
thermodynamic equilibrium

Work & Volume Changes

面積

Work done by gas on piston

Isothermal Processes

Isothermal process
:
T

= constant.



Isothermal processes
on ideal gas

Constant
-
Volume Processes & Specific Heat

Constant
-
volume process
( isometric, isochoric, isovolumic ) :
V

= constant



C
V

= molar specific heat at constant volume


Ideal gas:
U = U
(
T
)



for all processes

isometric processes

only for isometric processes




Non
-
ideal gas:

Isobaric Processes & Specific Heat

Isobaric Process
: constant
P

isobaric processes

C
P

= molar specific heat at constant pressure

Ideal gas, isobaric :



Ideal gas

Isotherms

Adiabatic Processes

Adiabatic process
:
Q

= constant

e.g., insulated system, quick changes like combustion, …

Tactics 18.1.


adiabat
,
ideal gas

Prob. 66

Prob. 62

Adiabatic: larger

p


TACTIC 18.1
. Adiabatic Equation

Ideal gas, any process:



Adiabatic process:





Example 18.3
. Diesel Power

Fuel ignites in a diesel engine from the heat of compression (no spark plug needed).

Compression is fast enough to be adiabatic.

If the ignit temperature is 500

C, what compression ratio V
max

/ V
min

is needed?

Air’s specific heat ratio is


= 1.4, & before the compression the air is at 20

C.

Ideal Gas Processes

Cyclic Processes

Cyclic Process
: system returns to same thermodynamic state periodically.

Example 18.4
. Finding the Work

An ideal gas with


= 1.4 occupies 4.0 L at 300 K & 100 kPa pressure.

It’s compressed adiabatically to ¼ of original volume,

then cooled at constant
V

back to 300 K,

& finally allowed to expand isothermally to its original
V
.

How much work is done on the gas?

AB (adiabatic):

BC (isometric):

CA (
isothermal
):

work done by gas:

18.3. Specific Heats of an Ideal Gas




Ideal gas:

Experimental values

( room
T

):

For monatomic gases,




5/3, e.g.,
He, Ne, Ar,
….

For diatomic gases,




7/5 = 1.4, C
V

= 5R/2, e.g., H
2

, O
2

, N
2

, ….

For tri
-
atomic gases,




1.3, C
V

= 3.4R, e.g., SO
2

, NO
2

, ….



Degrees of freedom
(DoF) = number of independent
coordinates required to describe the system

Single atom: DoF = 3 (transl)

For low
T

( vib modes not active )
:

Rigid diatomic molecule : DoF = 5 (3 transl + 2 rot)

Rigid triatomic molecule : DoF = 6 (3 transl + 3 rot)

The Equipartition Theorem

Equipartition theorem

( kinetic energy version):

For a system in thermodynamic equilibrium, each degree of freedom of a
rigid molecule contributes ½
kT

to its average energy.

Equipartition theorem

( general version):

For a system in thermodynamic equilibrium, each degree of freedom described
by a quadratic term in the energy contributes ½
kT

to its average energy.

DoF (
f

)

C
V

C
P



Monatomic

3

3/2

5/2

5/3

Diatomic

5

5/2

7/2

7/5

Triatomic

6

3

4

4/3

Example 18.5
. Gas Mixture

A gas mixture consists of 2.0 mol of oxygen (O
2
) & 1.0 mol of Argon (Ar).

Find the volume specific heat of the mixture.

Quantum Effects

C
V

of H
2

gas as function of
T
.

Below 20 K hydrogen is liquid,

above 3200 K it dissociates into individual atoms.

Quantum effect
:

Each mechanism has a threshold energy.



E
transl

< E
rot

< E
vib

Translation

rotation+Translation

rotation+Translation+vibration

Reprise

Quasi
-
static process
:

Arbitrarily slow process such that system always stays arbitrarily close to thermodynamic equilibrium.

Reversible process
:

Any changes induced by the process in the universe (system + environment) can be removed
by retracing its path.

a


c : Free expansion with no dissipative work.

c


b

: Adiabatic.


a


d : Adiabatic.

d


b

: Free expansion with no dissipative work.


a


e : Adiabatic.

e


b

: Adiabatic dissipative work.

Insulated gas

1
st

law:

The net adiabatic work done in all 3 processes are
equal (shaded areas are equal).

Dissipative work
: Work done on system without changing its configuration, irreversible.