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An excerpt from a university chemistry course, specifically unit i of chem 3510 from fall 2006. It introduces the main subdivisions of physical chemistry, focusing on waves and vibrations in chemistry. Topics covered include the history of classical physics, the wave equation, and the solutions of standing waves. Useful for students studying physical chemistry, particularly those in the fields of quantum mechanics, spectroscopy, and thermodynamics.
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th
th
c. Frequency ν (Greek letter nu):
π
ω ν λ
ν
○ λ is
○ ν
○ ω is
2
2
2 2
2 () ( ) v
dt
d T t X x dx
d X x T t =
dt
d T t
dx Tt
d X x
X x
2
2
2 2
2 ()
v
constant K called the separation constant.
solving the following linear differential equations with constant coefficient (they are
ordinary differential equations):
u ( x , t )
2
2
− KX x = dx
d X x
v () 0
2
2
− K Tt = dt
d Tt
π
2
2
2
d X x β
i x i x X x ce c e
ix =cos ± sin
±
⇒ X ( x )= A cos β x + B sin β x
X ( 0 )= 0 ⇒ A = 0
X ( l )= 0 ⇒ B sin β l = 0 ⇒ β l = n π where n = 1, 2, ...
λ =
λ
π 2 π
2 ω
π
λ
π ω πν n
π ω = ) are called
π φ
π
ν =
π φ
π 2
ν =
π φ
π 3
ν =
π φ
π 4
π π ω
π ω
4
π ω 1 t =
2
π
4
3 π (^0) π 4
π ω 1 t =
2
π
4
3 π (^0) π
n
l n
d
l dn
l n n
2
increases.
d
dn 4
= − but v
c
c
d
c dv 2
= − ⇒ dv c
d
2
v dv c
dv c
dn
2 3
2
4
π ω
π ω ω
ω = ω is