Download Introduction to Physical Chemistry

Introduction to Physical Chemistry
Worksheet 1
Name _________________
Instructor's initials ______
Calculus, thermodynamics, and you.
I. Derivatives and differentials for thermodynamics. Since thermodynamics is
concerned with physical and chemical changes, derivatives and differentials are useful for
determining how one parameter (for example the energy of a system) will change when
another parameter (for example the temperature) is changed.
A. Derivatives. Derivatives show how a function changes when the variable is changed.
Evaluate the following derivatives (NOTE: A, B, C are constants.)
y = Ax4 + B
dy/dx=
y = (x2 +2)-2
dy/dx=
y= (x + 4)/(x4+2)3
dy/dx=
y=A ln(x2)
dy/dx=
y = B ln(x + A)
dy/dx=
B. Partial derivatives. Partial derivatives show how a function of more than one variable
changes when one of the variables is changed and the other variables are held constant.
Evaluate the following partial derivatives.
z = Ax2 + By3
(z/x)y =
(z/y)x =
P = nRT/V (n, R are constants)
(P/T)V=
(P/V)T=
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C. Differentials. A differential shows how a function changes as its variables change. It
can be written generally as shown below for functions of one and two variables. Write
the expression for the differential of the last function in the table below.
Function
y = f(x)
Differential
dy = (dy/dx)dx
z = f(x,y)
dz = (z/x)ydx + (z/y)xdy
w= f(x,y,z)
The pressure (P) of a gas depends on its temperature (T) and volume (V), so P=f(T,V).
Write the general differential for P.
dP =
Assuming that the gas is ideal (that is, there are not intermolecular forces and the
individual gas particles occupy no volume), the ideal gas law (PV=nRT) can be used to
evaluate the derivatives that appear in the differential of P. Insert the expressions you
found for the ideal gas law on page 1 into the differential of P.
dP =
II. State functions. A function that depends only on the state of a system (and not on the
manner in which the system was attained) is called a state function. The change in a state
function depends only on the changes in its variables and not on the order in which the
variables were changed. Physically this means that the change in a state function
depends only on the initial and final states and not on the path taken between the states.
A. State functions in thermodynamics.
1. P (pressure) is a state function. Show this by calculating P for the following two
paths.
Path a: One mole of gas occupies 2 liters at 200 K. First, the volume is doubled
to 4 liters. Second, the temperature is decreased to 150 K.
Path b: One mole of gas occupies 2 liters at 300 K. First, the temperature is
decreased to 150 K. Second, the volume is doubled to 4 liters.
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2. Other state functions that we will encounter are G (Gibbs Free Energy), A (Helmholtz
Free Energy), V (volume), U (energy), S (entropy) and H (enthalpy). Write the general
form of the differentials for the state functions below using the variables indicated.
Gibbs Free
Energy
G = f(T,P)
dG =
Helmholtz Free
Energy
A =f(T,V)
dA =
Internal energy
U = f(S,V) dU =
Enthalpy
H = f(S,P)
dH =
B. State functions and exact differentials. The differential of a state function is an exact
differential. An exact differential is one for which the second mixed partial derivatives
are equal. Demonstrate this statement for the equation of state for a van der Waals gas,
a gas made up of molecules or atoms that have attractive forces and occupy volume.
(Hint: solve the van der Waals equation for P, then evaluate the second mixed partial
derivatives.) The van der Waals equation of state is (P+an2/V2)(V-nb) =nRT
P=
(2P/VT) =
(2P/TV) =
Does the value of dP depend on whether the T or the V is changed first? (Use the
partial derivative expressions above to answer this question.)
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B. Functions and inexact differentials Inexact differentials are ones for which the
second mixed partial derivatives are NOT equal. For these, the order in which the
variables are changed affects the total change in the function. Such functions are NOT
state functions and in thermodynamics, describe quantities that are transferred during a
process, for example work (w) and heat (q).
1. Can a system contain work? Why or why not?
2. Can a system contain heat? Why or why not?
C. Integrated forms.
If X is a state function, then dX = X2-X1 = X (integration limits from X1 to X2)
If X is NOT a state function, then dX = X
Fill in the table below.
Function
G(T,P)
State
function?
(yes/no)
Differential
dG
Exact differential
(yes/no)
Integrated form
 dG = G2-G1 = G
H(S,P)
A(V,T)
w
dw
q
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