Many chemical phenomena can be explained by physical theories. The main theory for chemistry is quan — Physical Chemistry — Kinetics Chemistry Question
Schrödinger cat and chemistry
Many chemical phenomena can be explained by physical theories. The main theory for chemistry is quantum mechanics, which gives the solid foundation for the observed chemical periodicity. One of the cornerstones of quantum mechanics is the superposition principle that says:
“If a quantum system can be found in the states 1 and 2 described by wavefunctions Ψ1 and Ψ2, it can also be found in a mixed state with the wavefunction Ψ = c1Ψ1 + c2Ψ2, where factors c1 and c2 characterize the contributions of the pure states 1 and 2 to the mixed state”.
The sum or difference of some wave functions taken with certain factors is called a superposition (a linear combination) of these functions.
In a mixed state the quantum system exists in both pure states simultaneously. When you perform some measurement on the system being in the mixed state, this measurement transfers the system to one of the pure states. We can never predict the specific final state; it is determined by the probability laws. The probability of any of the final states after measurement is proportional to the square of the modulus of the corresponding factor: p1 ~ |c1| 2, p2 ~ |c2| 2. Of course, the probability to find the system in either of the states is unity: p1 + p2 = 1.
The superposition principle is applicable to quantum systems only and is not valid when applied to macrosystems. To illustrate this idea, E. Schrödinger proposed the following mental experiment. Consider the Geiger counter which detects the entering electrons. The counter is connected to a device which breaks the glass with the poison when the particle enters the counter. Near the glass is a live cat. If the particle enters the counter, the cat is poisoned. But if the counter did not perform the measurement and is in the mixed state between the detected and undetected particle then the state of the cat is a superposition of life and death. Evidently, this is nonsense: the cat can be either alive or dead.
In chemistry, the superposition principle is used in the theories of hybridization, resonance, and molecular orbitals.
The superposition principle in theory of hybridization.
2.1 An sp3-hybrid atomic orbital is a linear combination of one s and three p-orbitals:
3 x y z1 s 2 p 3 p 4 psp = + + +Ψ Ψ Ψ Ψ Ψc c c c .
i) If we assume that all the orbitals make an equal contribution to a hybrid orbital, what are the absolute values of the coefficients c1 – c4?
ii) Similarly, find the absolute values of the coefficients c1 – c3 for an sp2 hybrid orbital.
Model Answer
2.1 (i) All orbitals make equal contribution, hence |c1| 2 = |c2| 2 =|c3| 2 =|c4| 2 = 1/4, because the sum of squares of all modulus is unity. Therefore, |c1| = |c2| = |c3| = = |c4| = 1/2.
(ii) For the sp2-orbital |c1| 2 = |c2| 2 = |c3| 2 = 1/3, hence |c1| = |c2| = |c3| = 1 / 3 .
The superposition principle in molecular orbital theory.
2.2 The molecular orbital for the ground state of H2 + molecule ion has the form:
a b 1s 1s
1 1 = +
2 2 Ψ Ψ Ψ ,
where a and b denote hydrogen atoms. What is the probability to find an electron on the 1s-orbital of the atom a?
Model Answer
2.2 The probability of being found in a definite state is equal to the square of the modulus of the corresponding coefficient:
pa = 2
1 1 =
22 .
This result is obvious because both hydrogen atoms are indistinguishable in H2 +.
The superposition principle in theory of resonance
2.3 Covalent bonds have a partial ionic character. Thus the wavefunction of a hydrogen halide bond can be presented as a linear combination of two wavefunctions characterizing its ionic (ΨH + Hal –) and covalent (ΨH:Hal) states:
+ -HHal cov H:Hal ion H Hal = +Ψ Ψ Ψc c
L. Pauling in his famous book «The nature of the chemical bond» (1947) claimed that in the HCl molecule the chemical bond is 17 % ionic in character. Find the absolute values of ccov and cion for HCl.
Model Answer
2.3 The probability of ionic state is 17 %:
|cion| 2 = 0.17,
Whence |cion| = 0.17 ≈ 0.41. Similarly, |ccov| = 0.83 ≈ 0.91.
One of the benzene wavefunctions can be presented as a linear combination of wavefunctions that correspond to two Kekule and three Dewar structures:
+ Ψ +ΨC6H6 = Ψ + Ψ + Ψ Ψ
2.4 What is the total contribution of the Kekule structures to this electronic state of benzene?
Model Answer
2.4 The total contribution of two Kekule structures is equal to the sum of squares of the moduli of all the corresponding coefficients in the linear combination: 2 2
Kekule
2 2 4 = + =
5 5 5 p
.
It means that in a given state 80% of benzene molecules have one of the Kekule structures, and 20 % – one of the Dewar ones.
In chemical reactions molecular structure changes over time so that the electronic state of a molecule is a function of time. In some cases structure of a molecule can be presented by a superposition of the initial and final states with time-dependent coefficients.
Let’s assume that a molecule oscillates between two pure states, one with a wave function Ψ1, and another with a wavefunction Ψ2, with the frequency ω. Initially (t = 0) the molecule is in the pure first state and after a half-period (t = π / ω) – in the second pure state.
2.5 Find the time-dependent coefficients of the superposition of these states describing the electronic structure of the molecule. Write the total wave function at a quarter of a period.
Model Answer
2.5 1 2 1 2(x, ) = ( ) (x) + ( ) (x)Ψ Ψ Ψt c t c t
c1(t), c2(t) – are periodic functions of time with the boundary conditions c1(0) = 1, c1(π/ω) = 0, c2(0) = 0, c2(π/ω) = 1. It is natural to express these coefficients via the sine and cosine trigonometric functions:
1 2( ) = cos , ( ) = sin 2 2 ω t ω t
c t c t
After a quarter of a period, at t = π/(2ω), the total wave function is a superposition of both states with equal masses:
1 2 1 2
1 1 x, = cos (x) + sin (x) = (x) + (x)
2 2 2 2 2 2 2
ω ω
ω ω ω
π π π
Ψ Ψ Ψ Ψ Ψ