Thermochemistry
Name: _
AP Chemistry Lecture Outline
thermodynamics: the study of energy and its transformations
- thermochemistry: the subdiscipline involving chemical reactions and energy changes
Energy
kinetic energy: energy of motion; KE = mv2
- all p
1. A bank is offering 2.5% simple interest on a savings account. If you deposit $5000,
how much interest will you earn in one year?
2. To buy a car, Jessica borrowed $15,000 for 3 years at an annual simple interest rate
of 9%. How much interest will she p
APES- Water Exam
Name: _
Multiple Choice- Circle the correct answer
1. The amount of the Earths surface that is covered by water is approximately:
a) 12%
b) 36%
c) 50%
d) 75%
e) 93%
2. An area where salt and freshwater mix that has a very high level of pr
5TQ: Thermochemistry
Name: _
Text Questions from Brown, et. al.
1. What invariably accompany chemical reactions?
2. Give three examples of energy derived from chemical reactions.
3. What is thermochemistry?
5.1
4. How is energy commonly defined?
5. What i
CASO OLYMPIA MACHINE COMPANY
1. Antecedentes
Olympia Machine, ubicada en Providence, Rhode Island, es una
empresa fabricante y distribuidora de equipo y refacciones para la
industria qumica especializada.
El mercado de Olympia consiste en 1000 plantas q
the Oracle
On this issue:
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Sunpharmas first retail pharmacy was opened in 1998 and with the selling of the company in
2009, a completely new management team developed. Sunpharma organization included
twenty seven corporate employees in departments of purchasing, finance, ma
1. How often does the appraisal take place?
Twice a year-calendar year
2. In what form is the rating?
a. Numerical, what scale?
b. Display like star?
c. Buckets?
d. Comments?
Technivally 7 likert scale
3. How often are the goals and objectives set/reset?
1. How often does the appraisal take place?
2. In what form is the rating?
a. Numerical, what scale?
b. Display like star?
c. Buckets?
d. Comments?
3. How often are the goals and objectives set/reset?
4. Is there self-appraisal?
a. If yes, then it is agai
1. How often does the appraisal take place?
Once a year
2. In what form is the rating?
a. Numerical, what scale?
b. Display like star?
c. Buckets?
d. Comments?
3. How often are the goals and objectives set/reset?
You and councilor decided on a quarterly b
Shreetam Subhrankar Microsoft 8 : 45 PM
1.
2. How often does the appraisal take place?
mandatiry twice a year
recommended 4
rewards happen once a year
3. In what form is the rating?
a. Numerical, what scale?
b. Display like star?
c. Buckets?
d. Comments?
1. How often does the appraisal take place? Once a year
2. In what form is the rating?
a. Numerical, what scale?
b. Display like star?
c. Buckets?
d. Comments?
Sort of comments in three parts: exceeds, achieves, expect more
3. How often are the goals and
1. How often does the appraisal take place?
Once a year
2. In what form is the rating?
a. Numerical, what scale? Scale of 5
b. Display like star?
c. Buckets?
d. Comments?
3. How often are the goals and objectives set/reset? Once in the beginning. Then cor
6.2
RECTANGLES AND CUBES
161
10. Prove the uniqueness of the Dirichlet problem !u = f in D, u = g
on bdy D by the energy method. That is, after subtracting two solutions
w = u v, multiply the Laplace equation for w by w itself and use the
divergence theor
126
CHAPTER 5
FOURIER SERIES
a < x < b we have
!
!
N
!
!
"
!
!
f n (x)! 0 as N .
! f (x)
!
!
n=1
(5)
Definition. We say that the series converges uniformly to f (x) in [a, b] if
!
!
N
!
!
"
!
!
(6)
max ! f (x)
f n (x)! 0 as N .
axb !
!
n=1
(Note that th
5.5
COMPLETENESS AND THE GIBBS PHENOMENON
the constant f (x) = f (x) 1 and use formula (5) to get
!
d
KN () [ f (x + ) f (x)]
S N (x) f (x) =
2
or
!
"#
$ % d
g() sin N + 12
S N (x) f (x) =
,
2
139
(6)
where
g() =
f (x + ) f (x)
sin 12
(7)
Remember tha
5.3 ORTHOGONALITY AND GENERAL FOURIER SERIES
121
COMPLEX EIGENVALUES
What about complex eigenvalues and complex-valued eigenfunctions X(x)?
If f (x) and g(x) are two complex-valued functions, we define the inner product
on (a, b) as
( f, g) =
!
b
f (x)g(x
130
CHAPTER 5
FOURIER SERIES
The Fourier series of a continuous but nondifferentiable function f (x) is
not guaranteed to converge pointwise. By Theorem 3 it must converge to f (x)
in the L2 sense. If we wanted to be sure of its pointwise convergence, we
6.3
POISSONS FORMULA
169
A careful mathematical statement of Poissons formula is as follows. Its
proof is given below, just prior to the exercises.
Theorem 1. Let h() = u(x ) be any continuous function on the circle
C = bdy D. Then the Poisson formula (13
6
HARMONIC
FUNCTIONS
This chapter is devoted to the Laplace equation. We introduce two of its
important properties, the maximum principle and the rotational invariance.
Then we solve the equation in series form in rectangles, circles, and related
shapes.
6.3
POISSONS FORMULA
165
(Hint: Note that the necessary condition of Exercise 6.1.11 is satisfied. A
shortcut is to guess that the solution might be a quadratic polynomial in
x and y.)
2. Prove that the eigenfunctions cfw_sin my sin nz are orthogonal on t
140
CHAPTER 5
FOURIER SERIES
Theorem 5.4.4). This means that we assume that f (x) and f (x) are continuous except at a finite number of points, and at those points they have jump
discontinuities.
The proof begins as before. However, we modify the third st
6.1
LAPLACES EQUATION
159
Figure 3
variables is (x, y, z) (s, , z) (r, , ). By the two-dimensional Laplace
calculation, we have both
1
1
u zz + u ss = u rr + u r + 2 u
r
r
and
1
1
u xx + u yy = u ss + u s + 2 u .
s
s
We add these two equations, and cance
5.3 ORTHOGONALITY AND GENERAL FOURIER SERIES
119
(Work out the right side using the product rule and two of the terms will
cancel.) We integrate to get
$
! b
"
#
"
#$b
$
X 1 X 2 + X 1 X 2 d x = X 1 X 2 + X 1 X 2 $ .
(3)
a
a
This is sometimes called Green
132
CHAPTER 5
FOURIER SERIES
Proof. For the sake of simplicity we assume in this proof that f (x) and all
the X n (x) are real valued. Denote the error (remainder) by
!
$2
!2 # $
!
!
$
b$
"
"
!
$
!
$
cn X n ! =
cn X n (x)$ d x.
(14)
EN = ! f
$ f (x)
!
$
168
CHAPTER 6
HARMONIC FUNCTIONS
Figure 2
Therefore,
2
2
u(r, ) = (a r )
!
2
0
a2
h()
d
.
2
2ar cos( ) + r 2
(13)
This single formula (13), known as Poissons formula, replaces the triple of
formulas (10)(12). It expresses any harmonic function inside a c
146
CHAPTER 5
FOURIER SERIES
(d) Find A and B. (Hint: A + Bx is the beginning of the series. Take
the inner product of the series for (x) with each of the functions 1
and x. Make use of the orthogonality.)
5. Prove the Schwarz inequality for infinite seri
166
CHAPTER 6
HARMONIC FUNCTIONS
Figure 1
Dividing by R! and multiplying by r2 , we find that
! + ! = 0
(3)
r 2R + rR R = 0.
(4)
These are ordinary differential equations, easily solved. What boundary conditions do we associate with them?
For !() we natur
6.1
LAPLACES EQUATION
153
1. Electrostatics. From Maxwells equations, one has curl E = 0 and div E =
4, where is the charge density. The first equation implies E = grad
for a scalar function (called the electric potential). Therefore,
$ = div(grad ) = di
124
CHAPTER 5
FOURIER SERIES
space with an inner product, it can be replaced by a sequence of linear
combinations that are mutually orthogonal. The idea is that at each step
one subtracts off the components parallel to the previous vectors. The
procedure