Chemistry 4595--Physical Chemistry of Macromolecules in Solution
Homework #1
Due: Before Midterm #1
This is physical chemistry, so we begin by brushing up our math skills. This document is
part guide, part problem set. Please consult your favorite math texts as you work it. YOU
HAVE UNTIL MIDTERM #1 TO WORK ON THIS! If you never, ever saw some of the
math shown here, please tell me. You may wish to consider our Chem 4581 course
(Introduction to Mathematical Methods). The problems given below are selected to help
you brush up on math skills you will need. For the moment, it all probably looks like
boring old math, but you will get to see some applications as the course progresses.
Though mathematicians often forget it, many of our best math methods were discovered
by great scientists probing their physical world. Most of us have to learn math backwards
(without much concern for its basis in physical reality). This set lets you work on your
math skills simultaneously with real problems.
Caveat. The interpretations here are rather informal and, possibly, even misleading.
Also, this document is preliminary (i.e., it's new). I will appreciate your comments and
criticisms, and I advise you to check the info below against a real math textbook.
YOU CAN GET EXTRA CREDIT FOR IDENTIFYING ERRORS.
References. I relied heavily on "Calculus with Analytical Geometry" by Protter and
Morrey (Addison-Wesley, Reading, Massachusetts, 1971). Also, on "Elements of
Calculus" by Thomas (Addison-Wesley, Reading, Massachusetts, 1967).
Homework #1 / Math Guide -- Page 1
Vector math
1. Draw your own three-dimensional cartesian coordinate system and sketch the vector
u = 3iˆ  4 ˆj  6kˆ where iˆ, ˆj, kˆ are the unit vectors in an orthogonal system,as usual.
2. On the same system, sketch the vector v =  3iˆ  4 ˆj  6kˆ . Write an expression for the
vector sum v + u.
4. Sketch the sum v + u on your cartesian coordinate system.
(In this problem, I would prefer that you really draw the results; it is possible to make
these plots in MathCad or a similar program, but that won't help you take notes in
class when we start adding vectors to define polymer size).
5. Find the angle between the vectors v and u in the previous problem. (Hint: look up
"dot product")
6. Find the angle between the vector v and the x-axis. (i.e., -- project the vector onto x)
7. Write an expression for the inner (dot) product of the vectors m = aiˆ  bˆj  ckˆ and n
= diˆ  eˆj  fkˆ . (Please consider forming this scalar answer by computing mTn -i.e., use row-column form.
Summation notation and using sums:
3
Recall the summation notation. Example:
i = 1 + 2 + 3 = 6
i 1
10
1. Evaluate the sum
 ix
i
for x = 0.1
i 1

2. Approximate the sum
x
i
for any x < 1.
i 0
Partial derivatives
Recall that partial derivatives are evaluated while holding one variable as a constant.
 f 
Example: if f(x,y) = xy then    y is the partial derivative of f with respect to
 x  y
x holding y constant.
Partial derivatives are essential to physical chemistry, for we are often interested in
varying one quantity (e.g., a function of state like the Gibbs free energy) while holding
some one parameter (like number of molecules) constant. Try these problems:
 f 
1. If f(x,y) = ln(x2 + y2) then   
 x  y
Homework #1 / Math Guide -- Page 2
 f 
2. If f(x,y) = ln(x2 + y2) then   
 y  x
2
2
 f 
3. If f(x,y) = xye x  y then   
 x  y
 f 
4. Without doing any math, write   for the previous function.
 y  x
For functions that have several independent variables, the partial derivative concept may
be extended by holding several variables constant.
 f 
5. If f(x,y,z) = exyxsin(xy)cos(2xz) then   
 z  x , y
Partial derivatives with direction: gradient and divergence
Recall that the "del" operator
 has the characteristics of a vector and a derivative, and is




defined by   iˆ  ˆj  kˆ
where iˆ, ˆj , kˆ are orthogonal unit vectors. This operator
x
y
z

participates in three basic vector functions--gradient, divergence and curl. For this course,
we only need gradient and divergence, to be defined shortly. They describe how functions
of several variables--like the free energy or concentration of a chemical component-change with regard to those variables. For example, one can consider concentration in a
solution as a function of position: c = c(x,y,z). At a given point in space (a,b,c) direction
of maximum change is given by the gradient of c, evaluated at that point. If we wanted to
know how the concentration was changing in some other direction (let us imagine it to be
lie in-between the x- and y- axes) we would first compute the gradient of c, and then dot
product the resulting vector into a unit vector in the appropriate direction ( iˆ / 2  ˆj / 2 ).
For now, please accept that these functions are worth pursuing. Here are the definitions:
Gradient of a scalar function, f(x,y,z). You compute the gradient if you need to know how
a function is changing. Consider the result a "change vector".
f
f ˆ f
Definition: grad f =  f =   iˆ  ˆj
k
x
y
z


1. Given the function f(x,y,z) = x3 + 2y3 + z3 - 4xyz what is the gradient vector
 f at the

point (-1,1,2)?
2. Now compute the rate of change of the function f(x,y,z) in the x direction at the point (1,1,2). Do this by dotting  f into vector i.

Homework #1 / Math Guide -- Page 3
Divergence. Divergence is a dot product between the "del" operator and some vector.
Dot products between vectors result in a scalar function. You can consider the divergence
to be a measure of the total change at some point, arising from changes of some vector
occurring in all directions. For example, consider a velocity vector that represents the
flow of molecules. Suppose further that the velocity is steady (not turbulent). Then the
delivery of molecules at some point, coming from any direction, would be given by the
divergence of the vector, evaluated at that point. If the particular point in question is
neither a source nor a sink for molecules, then its divergence would be zero. Divergence
figures prominently in our understanding of what quantities are conserved. For now, let's
just define it and practice on one problem.
Divu =
  u =  iˆ x  ˆj y  kˆ z   iˆu
 



x

 ˆju y  kˆu z where a component like uy may,
in fact, be a function of x,y,z.
Compute
  u if u = (x2 + 1) iˆ + (y2 -1) ĵ
+ z2 k̂

Total differentials
 f 
 f 
Informal Definition: if f = f(x,y) then df    dx    dy is the total differential.
 x  y
 y  x
 G 
 G 
 dp
Example: if G = G(T,p) then dG  
 dT  
 T  p
 p  T
1. If G(T,p) in the above example is the Gibbs free energy, a function of temperature and
 G 
 G 
 each have a special
pressure, then the partial derivatives 
 and 
 T  p
 p  T
significance. What are they?
Second-order partial derivatives
Consider some function u(x,y). We can see how it varies with x, holding y constant….and
we can see how that quantity itself varies with x.
Example: u(x,y) = x3 + 3y2x
 u 
Then    2 x 2  3 y 2
 x  y
   u  
 2u
and      2  4 x
 x  x  
x
y y

   u  
 2u
You can also take cross-derivatives:     
 6y
 y  x  
yx
y x

Homework #1 / Math Guide -- Page 4
1. For the example above, show that the order of taking a double derivative doesn't
   u  
 2u
 2 u    u  


 
matter: i.e., show that



 y  x  
yx xy  x  y  x  y
y x

2. Show that the order of double partial differentiation doesn't matter for the function
f(x,y) = e xy sin( x)  e xy cos( y)
Exact differentials
Recall that exact differentials describe functions with path-independent properties, or
"state functions." For example, the state functions of thermodynamics form exact
differentials: E can be calculated, with the same result, by integrating dE over any path
that starts and ends at the same points. The value of E depends only on these starting
and ending points. The individual components of dE ( q and w --i.e., dE = q + w)
are not necessarily independent of path, and so they are not necessarily exact differentials.
Identifying exact differentials:
 f 
 f 
df = P(x,y)dx + Q(x,y)dy is exact if P=   and Q=   .
 x  y
 y  x
Following are two differentials. Which one is exact, and which is inexact?
1. df = (3x2 + 2y)dx+ (2x - 3y2)dy
2. df = (2x2 - 3y)dx+(2x - y3)dy
3. For the differential above that is exact, see if you can figure out the state function
f(x,y). (You may actually be able to guess it.)
Maxwell relations
The Maxwell relations result from the definition of exactness and the fact that order of
taking partial derivatives does not matter. If a differential df = P(x,y)dx + Q(x,y)dy is
 f 
 P 
2 f
2 f
 Q 
 f 
exact, then P=   and Q=   . Since
it follows that 

    .
xy yx
 x  y  y  x
 x  y
 y  x
Example: Consider the Helmholtz free energy, A(T,V) = E - TS where E is the total energy
and TS the product of entropy and absolute temperature. The total differential of A is:
 A 
 A 
dA  
 dV    dT .
 V T
 T V
But one can also write, from the definition, dA  dE  d (TS )  dE  TdS  SdT . Now, we
can write dE as q + w. This gives us: dA  w  q  TdS  SdT . Since A is a function
of state (it's made up of two others, E and S) we can consider any path. For convenience,
we choose a reversible path. Thus, q = TdS by the 2nd law of thermodynamics, and this
Homework #1 / Math Guide -- Page 5
cancels with the -TdS term. Let us further assume the only PV work w=-pdV, and we
get:
dA   pdV  SdT .
 A 
By comparing this form of dA with the first, we can see that p  
 and
 V T
 A 
S    . Since order of differention of A doesn't matter, we get:
 T V

 2 A  S 
2 A
 p 

 
 
VT  V  T  T V
TV
 S   p 
The inside relation, 
    , is the Maxwell relation for the Helmholtz energy.
 V T  T V
1. Develop the corresponding Maxwell relationships for total energy, E, Gibbs free
energy, G, and enthalpy, H.
Indefinite & Definite integrals
No justification is required; see if you can remember how to do these. Try to do them,
don't just look them up.
1.
2.
 (3x  1) dx
 2  5 y dy
234

3.
x
2
e  ax dx
4


4.
3  ax
 x e dx
4

Differential volume elements
Many problems in this course will require your analytical geometry skills. These
problems are designed to refresh them.
4
1. Prove that the volume of a sphere of radius R is R 3 .
3
2
2
2
2. The circle x + y = a is rotated about the x axis to generate a sphere. What is the
volume of the sphere? (You can solve this from integrating an appropriate volume
element or from the equation for the volume of a sphere that you just derived).
3. In your own words and pictures, explain why the differential volume element in the
standard spherical polar coordinate system shown below is given by r2sin()drdd.
Homework #1 / Math Guide -- Page 6

r

Generation of power series
The chemistry of polymerization, and also chain statistics for polymer dimensions,
require evaluation of sums like:
n 1
S0   xi
i 0
or
n 1
Sa   i a xi
i 0
where a is an integer. A sum like S0 can be evaluated by writing out a few terms. Then,
underneath this expression, write the sum S1. Subtract and simplify to obtain an
expression for S0.
1)
2)
Follow the above prescription to obtain expressions for S0 and S1
Simplify your expressions further for the case x << 1 and n >> 1.
The process gets tedious when a > 1. Then it becomes useful to apply a little trick, which
you will now prove.
3)
4)
 dS

Prove that S a  x a 1 
 dx 
Assuming x << 1 and n >> 1, write an expression for:
n 1
S3   i 3 x i
i 0
Homework #1 / Math Guide -- Page 7
Future topics:
Probability
Matrix algebra
Geometric series
Degeneracy
Undetermined multipliers
Contour integrals (near exact differentials)
Integral representations of del operator (Gauss' law, etc.)
Maxwell equations for E&M
Homework #1 / Math Guide -- Page 8