Theoretical background

1. Introduction

In this experiment the rate of reaction between sucrose and water catalyzed by a hydrogen ion is followed by measuring the angle of rotation of polarized light passing through the solution. The angle of rotation of polarized light passing through the solution is measured using a polarized beam of He-Ne laser and a polarizer. The reaction is:

C12H22O11(sucrose) + H2O + H+ ⇒ C6H12O6(fructose) + C6H 12O6(glucose) + H+

Sucrose is dextrorotatory , but the resulting mixture of glucose and fructose is slightly levorotatory because the levorotatory fructose has a greater molar rotation than the dextrorotatory glucose. As the sucrose is used up and the glucose-fructose mixture is formed, the angle of rotation to the right (as the observer looks in the direction opposite to that of the light propagation) reduces until the light is rotated to the left.
The reaction of sucrose hydrolysis can be catalyzed not only by hydrogen ions but also by enzymes (for example by ß-fructofuranidase). The basic mechanism for enzyme catalyzed reactions was first proposed Michaelis and Menten in 1913 and was confirmed by a study of the kinetics of the sucrose hydrolysis.
The experiment includes two parts. In the first part you will learn the properties of polarized light from a He-Ne laser and in the second, you will study the chemical kinetics of the hydrogen ion catalyzed inversion of sucrose.

1.1. What do you need to know

  1. Malus Law.
  2. Polarized light (linear and circular).
  3. Optical activity.
  4. Chemical Kinetics.
  5. Specific rotation.
  6. Operation principles of: He-Ne laser, photodiode and polarizer.

2. Classification of Polarization

Light in the form of a plane wave in space is said to be linearly polarized. Light is a transverse electromagnetic wave, but natural light is generally unpolarized, all planes of propagation being equally probable. If light is composed of two plane waves of equal amplitude but differing in phase by 90°, then the light is said to be circularly polarized. If two plane waves of differing amplitude are related in phase by 90°, or if the relative phase is other than 90° then the light is said to be elliptically polarized.

Figure 1: Linear, Circular and Elliptical Light sources.

2.1. Linear Polarization

A plane electromagnetic wave is said to be linearly polarized. The transverse electric field wave is accompanied by a magnetic field wave as illustrated.

Figure 2:The magnetic and electric components in linear light source.

2.2. Circular Polarization

Circularly polarized light consists of two perpendicular electromagnetic plane waves of equal amplitude and 90° difference in phase. The light illustrated is right- circularly polarized.

Figure 3: The electric components in circular light source changes with the propagation of light. Watch an animated gif illustration in wikipedia.

If light is composed of two plane waves of equal amplitude by differing in phase by 90°, then the light is said to be circularly polarized. If you could see the tip of the electric field vector, it would appear to be moving in a circle as it approached you. If while looking at the source, the electric vector of the light coming toward you appears to be rotating clockwise, the light is said to be right-circularly polarized. If counterclockwise, then left-circularly polarized light. The electric field vector makes one complete revolution as the light advances one wavelength toward you. Circularly polarized light may be produced by passing linearly polarized light through a quarter- wave plate at an angle of 45° to the optic axis of the plate.

2.3. Malus Law

Let A be the amplitude of plane polarized light incident on a polarizer. Electrical field of the polarized light is inclined at angle θ to axis of polarization.

Figure 4: Malus law.

The component of A, Acosθ is transmitted and Asinθ is blocked out. Recall that the intensity is proportional to the square of amplitude, so that the intensity transmitted is:

(1) I = I0·cos2(θ)

where I0 is the intensity of light incident on polarizer. If unpolarized light Iu falls on the polarizer, ideally only half of the light (i.e. Iu/2) is transmitted (the field of the incident wave has components parallel and perpendicular to the polarizing axis. The incident light is a random mix of both so each component is equal). If a second polarizer is now placed in the beam, with vibration direction θ relative to the first polarizer, the intensity transmitted is thus:

(2)
Iθ = Iucos2θ
2

Note that the polarization direction of the light after passing the polarizer is identical to the polarizer orientation, however the intensity is reduced according to Malus law. When a polarized light passes an optically active substance the polarization direction changes, however the intensity remains unchanged.

Refer to phy.hk web site where you will find a Java applet that illustrates how the intensity and polarization angle of light (circular, linear or unpolarized) changes while passing through (up to three) polarizers. You may change the angle of the polarizers, eliminate some of them, change the light polarization and more.

3. Optically active compounds

Two compounds are called isomers if they have the same molecular formula but different chemical structures. Optical isomers are those in which two compounds have not only the same molecular formula but also identical bonding connections between the various atoms. A pair of optical isomers remain distinct from each other, however, because they are nonsuperimposable mirror images of each other. One optical isomer cannot be superimposed on the other, just as your left hand cannot be superimposed on your right hand. Compounds that exist as optical isomers are frequently referred to as chiral compounds, and each member of a pair of optical isomers is named an enantiomer. Molecules such as H2O and CH4, which do not exist as nonsuperimposable pairs, are called achiral. A few examples of chiral compounds are presented in Figure 5.

Figure 5: Two examples for chiral compounds.

Enantiomers have identical atoms and bonds, different optical properties. If plane polarized light is passed through a solution of a chiral compound, the plane of polarization of the light is rotated either clockwise or counterclockwise. The extent of this rotation depends on the nature of the compound, the path length of the solution, and environmental factors such as temperature. Under the same conditions, enantiomers rotate light to the same extent, except that one rotates the plane of polarized light to the left and the other structure rotates it to the right.
Only chiral compounds rotate light in this way, and for this reason chiral compounds are often called optically active compounds (and achiral compounds are referred to as optically inactive). The reason chiral compounds rotate polarized light is complex and is fully explained only by a quantum mechanical treatment of the interaction of electromagnetic radiation with chiral compounds. One way of thinking about optical rotation, however, is to consider linearly polarized light as a superposition of right-handed and left-handed circularly polarized light. Because circularly polarized light has a handedness like chiral molecules, it is not surprising that right- and left-handed circularly polarized light interact differently with chiral molecules. Specifically, the index of refraction of a chiral substance is different for left- and right-handed circularly polarized light. Thus, the right- and left-handed components of linearly polarized light travel through a chiral medium with different velocities, and one handedness of light is retarded with respect to the other. The net effect of this retardation is to rotate the direction of the polarization of the linearly polarized light.
No easy method can predict whether a particular chiral compound will rotate light clockwise or counterclockwise. Distinguishing between two enantiomers by the direction in which they rotate the polarization of light is often convenient. By convention left-rotating enantiomers are labeled (-) and right-rotating ones (+). Some chiral compounds rotate light more than others, and chiral compounds can be assigned specific rotation values (often designated in tables by the symbol [α]), which reflects the extent to which the compound rotates light. The specific rotation is defined as:

(3)
[α] = α
ℓ·c

where α refers to the rotation (in degrees) of the electric field vector in the light undergoes in traveling a distance ℓ (in decimeter) through a solution with a concentration, c, (in g/mL). Note that g/mL means gram of compound per mL of the total solution! The procedure of preparation of a given concentration is as follows: First dilute a necessary amount of substance in small amount of a solvent and then add a solvent up to desired total volume. The specific rotation of a solution should be independent of the concentration of the solution, because implicit in the definition of specific rotation is the assumption that the rotation of the light is directly proportional to the concentration of the solution. The specific rotation of a solution does depend, however, on the temperature of the solution and the wavelength of light traveling through it. Thus, specific rotations are often labeled with a superscript that indicates the temperature (in °C) and a subscript that indicates the wavelength of light (in nm). The specific rotation of the solutions in this experiment might be reported as:

(4)
[α]25632.8

if a He-Ne laser is used, because the wavelength of light produced by a red He-Ne laser is 632.8 nm. The bottle that you will get in the lab will be marked by [α]25D, while the subscript D means 589 nm Na- D-line standard.

3.1. Chirality in Organic and Biological Chemistry

Chirality is an especially important concept in organic chemistry, because a vast number of organic compounds are chiral. The chiral nature of organic compounds usually results from an asymmetric carbon atom, which is one that is bonded to four different substituents. A simple example of a chiral organic molecule is CHClBrI. The two enantiomers can be thought of as having two of the attached atoms reversed, causing the two forms to be nonsuperimposable. Many organic compounds have more than one asymmetric carbon, each of which is called a chiral center. In these compounds each chiral center has 2, or 2n total possible configurations. where n is the number of chiral centers. Some of these configurations are mirror images of each other and are called enantiomeric pairs. Any given pair of these molecules are not necessary mirror images, however, and in general are called diasteriomers.

Figure 6: Diasteriomers and two enantiomers.

The vast majority of important biological compounds, such as sucrose, have at least one chiral center. The structures of sucrose and quinine are shown in Figure 7.

Figure 7: Quinine and Sucrose Molecules.

Amino acids and sugars have a special labeling system to distinguish between different diasteriomers; The letters D and L, from the Latin dexter (right) and laevus (left), are used to indicate how the -H and -OH groups are attached to a particular carbon atom. Despite their names, the labels do not indicate which way the light is rotated. Interestingly, in most life forms, only L forms of amino acids are produced, whereas the D forms of sugars predominate. Although there is no intuitive reason for one configuration to predominate over the other, biological systems have evolved such that for many chiral compounds, only one isomer is observed in nature. This fact has profound implications for the pharmaceutical industry. Because many drugs are chiral. Different isomers of the same drug can have entirely different effects on the body. One isomer of a drug may have the desired healing effect, while another may have no effect or even be harmful. Thus, controlling the chirality of biological compounds when they are being synthesized is almost always necessary, and chiral selectivity presents a key challenge for the pharmaceutical industry in developing safe and effective drugs.

4. Subject of Chemical Kinetics

Chemical kinetics, a topic in several chemistry courses, illustrates the connection between mathematics and chemistry. Chemical kinetics deals with chemistry experiments and interprets them in terms of a mathematical model. The experiments are performed on chemical reactions as they proceed with time. The models are differential equations for the rates at which reactants are consumed and products are produced. By combining models with experiments, chemists are able to understand how chemical reactions take place at the molecular level.

4.1. First Order Reactions

These are characterized by the property that their rate is proportional to the amount of reactant. It follows that the differential rate law contains the amount (or concentration) of reactant and a proportionality constant (the rate constant):

(5)
Differential Rate Law:           d A  = -κA
dt

Mathematicians call equations that contain the first derivative but no higher derivatives first order differential equations. Chemists call the equation d[A]/dt = -k[A] a first order rate law because the rate is proportional to the first power of [A]. Integration of this ordinary differential equation is elementary, giving:

(6)
Integrated Rate Law:           [A] = [A]0 exp(-κ t)

A common way for a chemist to discover that a reaction follows first order kinetics is to plot the measured concentration versus the time on a semi-log plot. Namely, the concentration versus time data are fit to the following equation:

(7)
Data Analysis:           ln([A]) = ln([A]0) - κ·t.

A plot of ln([A]) versus t is a straight line with slope -κ. Alternatively, a plot of rate versus [A] is a straight line with slope -κ. From experimental data the rate constant can be found from the slope of the appropriate plot.

4.2. Second Order Reactions

Second Order Reactions are characterized by the property that their rate is proportional to the product of two reactant concentrations (or the square of one concentration). Suppose that A ⟹ products is second order in A, or suppose that A + B ⟹ products is first order in A and also first order in B. Then the differential rate laws in these two cases are given by differential rate laws:

(8)
Differential Rate Law:           d A  = -κA2 (for 2A ⟶ Products)
dt
(9)
Differential Rate Law:           d x  = -κA·B (for A + B ⟶ Products; x∈{A,B})
dt

In mathematical language, these are first order differential equations because they contain the first derivative and no higher derivatives. A chemist calls them second order rate laws because the rate is proportional to the product of two concentrations. By elementary integration of these differential equations integrated rate laws can be obtained:

(10)
Rate Law:           1  -  1  = κt (for 2A ⟶ Products)
[A] [A]0

or

(11)
Rate Law:           1  · (ln [A]0-x  — ln [A]0 ) = κt (for A + B ⟶ Products)
[A]0-[B]0 [B]0-x [B]0

where [A]0 and [B]0 are the initial concentrations of A and B (assuming [A]0 ≠ [B]0), and x is the extent of reaction at time t. Note that function (11) can also be written as:

(12)
Rate Law:           [A]0-x  =  [A]0  exp {([A]0-[B]0)·κT}
[B]0-x [B]0

A common way for a chemist to discover that a reaction follows second order kinetics is to plot 1/[A] versus the time in the former case, or ln([B]0([A]0-x)/[A]0([B]0-x) versus t in the latter case.

(13)
Data Analysis:           1  =  1  + κt
[A] [A]0

A plot of 1/[A] versus t is a straight line with slope k.

5. Specific Rotation

Determining the Specific Rotation

The observed rotation is dependent upon the path length of the light passing through the sample compartment and is also dependent upon the number of molecules of the isomer. The observed rotation is converted to a specific rotation by using formula presented in equation (3). Let's take an example the following case. Suppose that the observed rotation of an optically active isomer produced an observed rotation of +13.00 degrees. The sample had a concentration of 1000g/liter and the length of the sample tube was 20 cm in length. What would be the reported specific rotation of this dextrorotatory isomer?

  1. Convert the concentration to g/ml.
    1000g / liter X 1 liter / 1000 ml = 1.0 g/ml
  2. Convert the length of the tube to decimeters knowing that 10 cm = 1 decimeter.
    20 cm X 1 dm / 10 cm = 2 dm
  3. Using the above formula plug in the observed rotation, the length and concentration.
    Specific Rotation = +13.00 / (1.0 g/ml) (2 dm) = +6.50 degrees

Let's see if you can do one of these. Suppose that the observed rotation was -45.5 degrees of rotation. The concentration was 3.00 grams/ml and the length of the tube was 20 cm. Identify whether this is a dextrorotatory or levorotatory isomer and determine the specific rotation.
Answer, The following procedure is followed:

  1. 1. Convert the length of the tube to dm
    20 cm X 1 dm / 10 cm = 2 dm
  2. Plug in the concentration in g/ml and the length of the tube into the formula.
    Specific Rotation = —45.5 degrees / (3.0 g/ml) (2 dm)= —7.58 degrees rotation.
  3. Since the rotation is negative it will be the levorotatory isomer.

5.1 Using the Specific Rotation to Find Rate Constants

Equation (3) discribes the relation between the specific rotation of a substance and its rotation in a given concentration [gr/ml] and length of the beam path.

(14a)
α =  [α] ·ℓ·c

In any reaction, c, the concentration of a substance is dependent on time ,t, so we can write:

(14b)
α(t) =  [α] ·ℓ·c(t)

From the stoichiometric reaction:

C12H22O11(sucrose) + H2O + H+ ⇒ C6H12O6(fructose) + C6H 12O6(glucose) + H+

We can conclude that the concentration [in molar] of the fructose and glucose in any time, t, is equal to:

(14c)
cf(t) = cg(t) = cs(0)-cs(t)

cs(t), cf(t), cg(t) are the concentrations of sucrose, fructose and glucose, respectivly, in time. cs(0) is the initial concentrations of sucrose. Fructose and glucose have the same molecular weights (MW) so in order to transform units of molar concentration to [gr/ml] we can write in:

(14d)
cf(t)=cg(t) = MW·(cS(0)-cS(t))

Let's place eq (14d) into eq (14b) and we shell get a set of equations describing the rotation angle as a function of time for each substance:

(14e)
αs(t) =  [αs] ·ℓ·cs(t)
(14f)
αf(t) =  [αf] ·ℓ·cf(t) =  [αf] ·ℓ·MW(cs(0)-cs(t))
(14g)
αg(t) =  [αg] ·ℓ·cg(t) = [αg] ·ℓ·MW(cs(0)-cs(t))

Bear in mind that at any given time of the reaction, the solution's rotational angle is the sum of its compounds' rotational angle so If one substitute αs(t), αf(t), αg(t), in the expression {α(t)—α(∞)}/{α(0)—α(∞)}:

(14h)
α(t)—α(∞)  =  αs(t)+αf(t)+αg(t)—αs(∞)—αf(∞)—αg(∞)
α(0)—α(∞) αs(0)+αf(0)+αg(0)—αs(∞)—αf(∞)—αg(∞)

Placing equations (14e)-(14g) in equation (14h), get you to (after some algebra to):

(14i)
α(t)—α(∞)  =  cs(t)
α(0)—α(∞) cs(0)

Since the reaction of the sucrose is a first order reaction:

(14j)
α(t)—α(∞)  =  cs(t)  = exp(-κ·t)
α(0)—α(∞) cs(0)

Now we have the relation between the time, t, and the rotational angle of the solution in any given time t and from that we can calculate the rate constant, κ, of the reaction.

6. Principles of Lasers

Let us consider a collection of atoms or molecules interacting with electromagnetic wave that is propagating along the z axis. If the wave is of appropriate frequency to cause stimulated transitions in the system, each stimulated emission generates a photon while each absorption removes a photon. The change in radiant flux with the distance dz due to absorption is given by dΦ=—Φniσ·dz, where σ is the transition cross-section in cm2. The change in flux due to stimulated emission in distance dz is dΦ=Φnjσ·dz. Thus we can write the total change in radiant flux in distance dz is:

(15)
dΦ = σ·Φ·(nj-ni)·dz

This equation shows that there is a net gain in a flux when (nj>ni). When the system is at thermodynamic equilibrium, (nj)eq<(ni)eq the medium is absorbent. However, if the upper level population can be made to exceed that of the lower level nj>ni, the system behaves like an amplifier at frequency νij. Under this conditions, the atomic or molecular system is called an active medium, and it has undergone population inversion. In the IR-to-visible frequency region, such an amplifier is called a LASER for Light Amplification by Stimulated Emission of Radiation. The population inversion can be achieved by excitation of the active medium by external source. The excitation process, called pumping, can be by optical methods with other sources, by electrical methods using an electrical discharge, by chemical reaction or by rapid adiabatic expansion. In a laser device scheme the amplifier is located in 2 (or more) mirror resonator, which provides a positive feedback if a wave performing a roundtrip inside the resonator undergoes constructive interference. The next figure shows a principle scheme of laser device:

Figure 8: Laser scematics.

6.1. He-Ne laser

The He-Ne laser is the most common of all lasers. It is a CW-laser pumped by electrical discharge. The lasing transitions occur between the Ne energy levels, while He is added to increase the pumping efficiency. The helium atoms are ionized to maintain the discharge. Metastable levels in He atoms transfer energy efficiently to Ne. The dominant laser transitions are at 632.8 nm, 1.15µm, and 3.39µm. The outputs of many He-Ne lasers as well as other noble gas lasers are linearly polarized. This can be understood from the scheme of He-Ne laser which is shown on the next figure:

Figure 9: He-Ne Laser scematics.

The figure shows polarized output He-Ne laser. The discharge tube is terminated by end windows titled at Brewster’s angle (this issue will be explained shortly). Thus no reflections occur at the windows for light polarized with the electric field vector in the plane of the figure. The orthogonal component is partially reflected on each pass and quickly becomes an insignificant part of the output beam. However, simple He-Ne lasers made for use as alignment aids are available without Brewster angle windows and thus with unpolarized outputs. The He-Ne laser is simple and inexpensive. The output power is limited, however, to the range 0.5-50 mW because at low currents few metastables form, whereas at high currents the He metastables ionize (take a look at the wikipedia to understand why this is important).

7. Plane of incidence

Before continuing on the subject of Brewster’s angle, the phrase "Plane of incidence" should be defined. The plane of incidence is the plane spanned by the surface normal and the propagation vector of the incoming radiation. Let the blue path be a light beam that hits a reflecting surface. The green plane is defined by two vectors, the first is the surface normal and the second is the propagation vector of the light towards the reflecting surface. The plane of incidence must contain a third vector, which is the propagation vector of the light from the reflecting surface.

Figure 10: Plane of incidence marked in green.

7.1. Brewster’s angle

If the incidence ray is polarized in such a way that its electric field oscillates only in the plane which is perpendicular to the refraction plane then the intensity of the reflected ray depends very strongly on the incidence angle. In the extreme case when directions of the reflected ray and polarization of the refracted ray are parallel (fig. 11), intensity of the reflected ray becomes equal to zero. The angle between the reflected ray and refracted ray in this case is equal to 90° and the incidence angle is called Brewster's angle. The refraction index can be calculated exclusively from Brewster's angle αB (using Snell's law) because for αB = 90° - β

(16)
n = tan(αB)

Figure 11: Polarized beam that has electric field oscillates perpendicular to the refraction plane is not reflected from the more dence medium.

Here are schematic illustrations explaining how the brewster window enables the laser beam to become polarized. This happens in a time scale of a second or so.

Figure 12(a): Before reaching steady state, the laser beam is unpolarized. The Brewster angle windows reflects the bean that contains only photons that have polarization (electrical component) which is perpendicular to the plane of incidence. The reflected beam is absorbed and rapidly decays, leaving the bean that has polarization parallel to the plane of incidence to propagate. Figure 12(b): After reaching steady state, the laser beam is polarized. The Brewster angle windows DO NOT reflect the beam that contains polarization parallel to the plane of incidence, allowing it to propagate back and forth in the amplifying medium.
Figure 12(c): An illustration of Brewster window "in action" taken from www.repairfaq.org/sam/. This figure shows a Brewster window reflects the unwanted polarization away from the path of the laser beam and by that the reflected beam does not amplify in the cavity of the laser.

7.2. Polarizers

Types of Polarizing Filters

Figure 13: Polarizer effect on an unpolarized beam.

LINEAR POLARIZERS: Synthetic linear polarizing filters (polarizers) possess special properties for selectively absorbing light vibrations in certain planes. When unpolarized light which is a complex mixture of vibrations lying in all possible directions transverse to the line of travel is passed through a linear polarizer, its vibrations become confined to a single linear plane and the light is considered " polarized ". This linearly polarized light can be modified to suppress unwanted reflections and to eliminate glare for a variety of applications.
RETARDERS: Optical retarders are polarization form converters. They are clear, oriented polyvinyl alcohol film laminated to a cellulose acetate butyrate substrate. The retarder has two principal axes, slow and fast, and it resolves a light beam into two polarized components (the one parallel to the slow axis lags the one parallel to the fast axis), and then recombines the two to form a single emerging beam of a specific polarization form. Used mainly to produce circular polarization when used with a linear polarizer.
CIRCULAR POLARIZERS: A circular polarizer comprises a linear polarizer and a 1/4 wave retarder whose slow and fast axes are at 45° to the axis of the polarizer. A ray of unpolarized light, passing through the linear polarizer, becomes polarized at 45° to the axis of the retarder. When this polarized light ray passes through the retarder its vibration direction is made to move in a helical pattern. After the light ray is reflected from a specular surface the sense of rotation of the vibration reverses. This rotation is stopped in the return through the retarder. The light ray is now linearly polarized in a plane 90° to its original polarization plane, and is absorbed by the linearly polarized component of the circular polarizer.

Figure 14: Circular polarizers effect.

8. Photodiodes

Photodiodes are semiconductor light sensors that generate a current or voltage when P-N junction in the semiconductor is illuminated by light. The term photodiode can be broadly defined to include even solar batteries, but it usually refers to sensors used to detect the intensity of light. Photodiodes can be classified by function and construction.

8.1. Operation Principle of Photodiodes

The following figure shows a cross section of a photodiode.

Figure 15: Operation principle of photodiodes.

The P—layer material at the active surface and the N material at the substrate form a PN junction which operates as a photoelectric converter. The usual P—layer for a silicon photodiode is formed by selective diffusion of boron, to a thickness of approximately 1 µm or less and the neutral region at the junction between the Pmdash; and N—layers is known as depletion layer. By varying and controlling the thickness of the outer P— layer, substrate N—layer and bottom N+—layer as well as the doping concentration, the spectral response and frequency response can be controlled.
When light strikes a photodiode, the electron within the crystal structure becomes stimulated. If the light energy is greater than the band gap energy Eg, the electrons are pulled up into the conduction band, leaving holes in their place in the valence band (see figure 13b). These electron- hole pairs occur throughout the P—layer, depletion layer and N&mdashlayer materials. In the depletion layer the electric field accelerates these electrons toward the N—layer and the holes toward the P—layer. Of the electron- hole pairs generated in the N— layer, the electrons, along with electrons that have arrived from the P—layer, are left in the N—layer conduction band. The holes at this time are being diffused through the N—layer up to the depletion layer while being accelerated, and collected in the P- layer valence band. In this manner, electron&mdashhole pairs which are generated in proportion to the amount of incident light are collected in the N— and P—layers. This results in a positive charge in the P—layer and a negative charge in the N—layer. If an external circuit is connected between the P— and N—layers, electrons will flow away from N—layer, and holes will flow away from the P—layer towards the opposite respective electrodes.