Download Lecture_3 - Department of Mathematics

USSC2001 Energy
Lecture 3 Thermodynamics of Heat
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
http://www.math.nus/~matwml
Tel (65) 6874-2749
1
TUTORIAL 3
1. In problem 2, tutorial 2, (i) show that angles a1, a2
opposite sides with lengths 1m,2m are not determined
but the ratio sin(a1)/sin(a2) is determined and compute
it, (ii) let M denote the mass of the object on the side
having length 2m and express the change of total
gravitational potential energy if the system has a
‘virtual displacement’ in which the object with mass M
moves by distance d downwards, (iii) explain the
“Principle of Virtual Work” and use it to compute the
value of M if the system is in equilibrium, (iv) discuss
Simon Stevinus and use his method to compute M.
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Stevin.html
http://3quarksdaily.blogs.com/3quarksdaily/2005/05/monday_musing_s.html
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WHAT IS THERMODYNAMICS?
Read chapter 19 (handouts) of Halliday, Resnick and
Walker, study Review&Summary and Problems.
Deals with thermal (internal) energy and involves the
concept of temperature, an elusive property of objects
that alters apparent properties, including lengths &
volumes and electrical resistance, any of which can be
used to make a thermoscope (not yet a thermometer).
Zeroth Law of Thermodynamics: If bodies A and B
are each in thermal equilibrium with a third body T,
then they are in thermal equilibrium with each other.
3
DEFINING TEMPERATURE
The triple point of water http://en.wikipedia.org/wiki/Triple_point
273.16 K degrees Kelvin
611.73 Pa
Pascals
We define the temperature of a gas by


p(T )

T  273.16 K 
lim
gas0,vol const p ( 273.16 K )


Here we use the fact that T is the same for ALL gases.
4
CONSTANT-VOLUME GAS THERMOMETER
The ingenius mercury thermometer shown below
(page 428 in HRW) can measure T at constant volume
p atm  1.01 10 Pa
3
3
mercury  13.6 10 kg / m
5
Questions How is constant volume maintained at
different temperatures? How is density measured?
p
T
Gasfilled
bulb
h
p  patm  gh
Reservoir
that can be
raised and
lowered
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TEMPERATURE SCALES
All scales are inter-related by affine functions
f(x)  ax  b, a, b { Real Numbers }
And therefore determined by their values at
absolute zero and the triple point of water


Absolute zero  0K  273.15 C  459.67 F


Triple point  273.16 K  0.01 C  32 .02 F
Question Compute the values of a and b for the six
affine functions that convert
K  C, K F, C K, C F, F K, F C
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IDEAL GAS LAW
Amadeo Avogado 1776-1856 suggested that all
gases contained the same number of molecules
for a fixed volume, pressure and temperature
pV
kT
molecules =
where N A  6.02  10
32
k  1.38  10
23
pV
RT
moles
= # molecules in a mole
J / K = the Boltzmann constant
R  8.314 joules / (mole  K) = the gas constant
Question How are k and R related?
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TUTORIAL 3
2. Newton’s 3rd Law states: When 2 bodies (particles)
interact, the forces on the bodies from each other are
always equal in magnitude and opposite in direction.
The (linear) momentum of a body is defined to be the
product of its mass times its velocity and the
momentum of a system is the ‘sum of its parts’ (i) use
Newton’s laws to show that when 2 bodies interact (eg
in a collision) the system momentum is conserved, (ii)
compute the average pressure that a molecule with
kinetic energy E_kin exerts on a cubic container with
volume V, (iii) combine this and the ideal gas law to
show average molecular kinetic energy = 3kT/2
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TEMPERATURE AND HEAT
Thermal or internal energy consists of kinetic and
energies associated with their random motions and,
especially for solids and liquids, the potential energy
due to their proximity.
Heat Q is thermal energy transferred to a system
from its environment, Q > 0, Q < 0 when the system
temperature is lower, higher than environment’s, it
can be associated with a change of temperature
Q  cm(Tf  Ti )
where m=mass and c = specific heat capacity of a
material (c=4190J/(kg K for water at 14.5C) or with
a change of phase (heats of fusion and vaporization).
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HEAT TRANSFER MECHANISMS
Heat can be transferred by conduction
dH
Thot  Tcold
kA
dt
L
convection, and radiation
dH
4
4
   A(Tenv  T )
dt
Questions What are the constants in these equations?
10
i
n
s
u
l
a
t
i
o
n
lead shot
pi
W
WORK AND HEAT
Vf
W   dW   pdV
Vi
pressure
pi
state
diagram
Q
thermal reservoir
pf
volume
Vi
Vf
W (and Q) depend on the thermodynamic process,
described by a path, not only on initial&final states 11
THERMODYNAMIC PROCESSES
As shown on p 438-439 in HRW, W = the work done
by a system is path dependent, this is also true for Q =
heat transferred to the system since, as stated in lines
6-8 from bottom page 435 that gases have different
values for their specific heats under constant-pressure
and under constant-volume conditions.
Question Compute W for constant p and constant T
Vf
W p const   pdV  p (Vf  Vi )
Vi
Vf nRT
Vf
WT const  
dV  nRT ln
Vi
V
Vi
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FIRST LAW OF THERMODYNAMICS
There exists an internal energy function
E int  E int (V , p)
such that during any thermodynamic process
E int  Q - W
The first law is illustrated for adiabatic (Q=0),
constant volume (W = 0), and closed cycle or cyclic
processes on page 441 of HRW. Free expansion on
page 442 differs from all other processes why?
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SECOND LAW OF THERMODYNAMICS
There exists an entropy function
S  S (V , p )
such that during any thermodynamic process
S  Sf - Si  
(Vf , p f )
(Vi , p i )
dQ
T
or, equivalently, such that
Sf
Q   T dS
Si
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TUTORIAL 3
3. Combine the formular dE int  dQ - dW with
the ideal gas law and the eqn. dE  n C dT
int
V
where n  moles of a quantity of gas
and CV  molar specific heat at constant volume
to show that for an ideal gas
Vf
Tf
S  Sf - Si  n R ln
 n CV ln
Vi
Ti
4. Find out what a Carnot Cycle is and how it differs
from a Stirling Cycle. What is more efficient?
5. What is free energy and how does it explain
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the thermodynamics of chemical reactions?