Category: Intermediate Chemistry

Radioactive decay is the spontaneous process by which an unstable atomic nucleus transforms into a more stable one by emitting particles or energy.

It follows first-order kinetics, meaning the rate at which a radioactive nucleus transforms is directly proportional to the number of un-decayed nuclei  at any given time . Because each nucleus decays independently, the probability that a nucleus will decay in a small interval of time is constant, regardless of how long the atom has existed. Mathematically, the rate law is given by:

where , the decay constant, represents the probability of decay per unit time.

An example of radioactive decay is the transformation of uranium-238 into thorium-234 with the emission of an alpha particle (a helium nucleus):

Thorium-234, which is also radioactive, then undergoes beta decay (a beta particle is an electron) to produce protactinium-234, and continues through a chain until a stable lead-206 nucleus is formed:

 

 

Solving eq60 by integration gives:

where is the number of un-decayed nuclei at .

Therefore, the probabilistic decay behaviour leads to an exponential decrease in the number of radioactive nuclei as time passes.

For a first order reaction, the half life is constant and given by:

Because is specific to each radioactive isotope, every radionuclide has a characteristic half-life, which can be exploited to gain insight into the age of a material. For example, the analysis of radioactive elements within meteorites allow scientists to estimate the age of the earth. One commonly used method involves the decay of uranium-238 to lead-206. Uranium-238 has a half-life of about 4.47 billion years, meaning that after this time, half of the original ²³⁸U atoms in a sample will have decayed to ²⁰⁶Pb. By measuring the present-day ratio of parent isotope (²³⁸U) to daughter isotope (²⁰⁶Pb) in ancient minerals, such as zircon crystals found in some of the oldest meteorites, scientists can determine how many half-lives have elapsed.

 

 

Suppose zircon samples from the oldest meteorite on Earth are measured by TIMS to have values very close to:

From eq61, the number of ²³⁸U atoms remaining at time is:

Zircon (ZrSiO₄), during its initial formation, incorporates U⁴⁺ easily but not Pb²⁺ because U⁴⁺ closely matches Zr⁴⁺ in both charge and ionic radius, allowing it to substitute easily, whereas Pb²⁺ has a lower charge and a much larger radius, making substitution energetically unfavourable. Therefore, lead is typically below detection levels initially, and the number of ²⁰⁶Pb atoms formed is:

It follows that the uranium-to-lead ratio can be expressed as:

Solving for yields:

Substituting eq62, where billion years, into eq64 gives billion years. Since represents the time required for the original ²³⁸U to produce the measured uranium-to-lead ratio of 0.99, the earth is therefore approximately 4.5 billion years old.

 

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Radiocarbon dating is a method of determining the age of carbonaceous materials by measuring the amount of radioactive carbon-14 remaining in them.

Developed by the chemist Willard F. Libby (see above picture) in 1949, radiocarbon dating, also known as carbon-14 dating, is one of the most widely used techniques in archaeology, paleontology and environmental science because it allows scientists to estimate when an organism died. The method is based on the radioactive decay of ¹⁴C, an unstable isotope of carbon that forms naturally in the upper atmosphere.

When cosmic rays strike atmospheric ¹⁴N, a small but fairly constant amount of ¹⁴C is produced:

This radioactive carbon combines with oxygen to form carbon dioxide, which plants absorb through photosynthesis. Animals then consume the plants and incorporate ¹⁴C into their bodies. As long as an organism is alive, it constantly exchanges carbon with the environment, keeping the ratio of stable ¹²C to radioactive ¹⁴C roughly constant.

When an organism dies, however, it stops taking in carbon. From that moment on, the ¹⁴C in its tissues begins to decay at a known rate, transforming back into ¹⁴N through beta decay:

The decay rate is characterised by ¹⁴C’s half-life of 5,730 years. By comparing the amount of ¹⁴C remaining in a dead organism to the amount it would have contained while alive, scientists can determine how long it has been since the organism died.

The fundamental principle behind the method is expressed mathematically using the first-order radioactive decay equation:

where

is the amount of radioactive carbon remaining after time .
is the original amount at the time of death.
is the decay constant, and is related to the half-life of the decaying species by .
is the time that has passed since death.

To illustrate how radiocarbon dating calculations work, consider the example of an archaeological team that discovers a wooden tool handle at an ancient settlement. The ratios of ¹⁴C to ¹²C in a sample of the tool and in modern wood of the same type are analysed using mass spectrometry to be and respectively.

Because all living organisms share essentially the same carbon isotopic composition as the contemporary atmosphere, any decrease in ¹⁴C relative to ¹²C reflects radioactive decay occurring after death. Assuming that the concentration of the stable isotope ¹²C is the same in the sample and in modern wood, the fraction of radiocarbon remaining in the sample is:

Combining the radioactive decay equation with the half-life equation gives:

Substituting eq70 and ¹⁴C’s half-life of 5,730 years into eq71 yields years, which is the uncalibrated radiocarbon age of the wooden tool.

Although the raw radiocarbon age can be calculated precisely, it does not necessarily correspond directly to calendar years. The amount of ¹⁴C in the atmosphere has not been constant over time; it has fluctuated with solar activity, geomagnetic field strength, ocean circulation changes, and most recently human activities such as fossil fuel burning and atmospheric nuclear tests. To correct for these variations, radiocarbon ages must be calibrated using independent chronological records, most importantly tree-ring sequences (dendrochronology). Calibration curves allow raw radiocarbon ages to be converted into more accurate calendar ages.

Finally, radiocarbon dating is reliable for materials up to about 50,000–60,000 years old. Beyond that point, so little ¹⁴C remains that it becomes difficult to distinguish from background radiation and measurement noise. Despite these complications, radiocarbon dating remains extraordinarily valuable. It has been used to date archaeological sites, reconstruct past climates, verify the authenticity of historical artifacts, and study ecological changes. For instance, it helped determine the age of the Dead Sea Scrolls, revealed the timing of glacial advances and retreats, and allowed scientists to track how ancient cultures expanded or declined.

 

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The Clausius-Clapeyron equation describes how the vapour pressure of a pure substance changes with temperature during a phase change. It has the form

where

is the vapour pressure,
is the enthalpy of vaporisation or enthalpy of sublimation,
is the universal gas constant,
is the temperature in Kelvin,

We can derive the Clausius-Clapeyron equation from the van’t Hoff equation , where for phase transitions such as or , the activity of the pure liquid or solid phase in the equilibrium constant is taken as 1. This simplifies the expression and leads to eq 26.

 

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Infrared (IR) spectroscopy measures the interaction of infrared radiation with vibrating molecules, allowing the identification of chemical substances or functional groups.

Molecules consists of atoms connected by chemical bonds. These bonds can vibrate – stretching, bending and twisting – when the molecules absorb specific wavelengths of IR radiation. For instance, when absorbs a specific frequency of IR radiation, it undergoes a vibrational mode known as an asymmetric stretch (see diagram above). During this motion, the two oxygen atoms vibrate periodically in a direction opposite to that of the carbon atom. However, If it absorbs a different IR frequency, the carbon and oxygen atoms move perpendicularly in and out of the screen. Notably, the carbon atom moves in opposite direction to the oxygen atoms. We refer to this type of vibrational mode as bending.

 

 

Each molecule has a unique set of vibrational modes. When IR radiation interacts with a molecule, it selectively excites these vibrational modes. The IR spectrometer measures the absorbance of IR radiation by a molecule and provides an absorption spectrum with peaks at different frequencies. By comparing a sample’s absorption peaks to a database of known spectra, scientists can identify the molecule.

There are two commonly used instruments that measure the interaction of IR radiation with molecules – dispersive IR spectrometer and Fourier-transform IR spectrometer. The diagram below outlines a typical double-beam dispersive IR spectrometer, which utilises a heated rod, wire or coil as the radiation source. The materials used for fabricating these components can be rare-earth oxides (for the Nernst Glower), silicon carbide (for the Globar) or nichrome (for the Nichrome Coil). All of these materials produce continuous radiations when heated to specific temperatures within the range of 1000 ^oC to 1800^oC.

A beam of broad spectrum infrared light produced is first split into two separate beams by mirrors. One beam passes through the sample cell, and the other through a reference cell. For an aqueous sample, the reference cell is filled with the corresponding pure solvent; otherwise it is empty. A rotating chopper periodically blocks one of the beams, allowing the other to reach the diffraction grating, which disperses the radiation into its component frequencies. With the help of a rotating mirror, the component frequencies of the dispersed spectrum are then scanned across the exit, beyond which lies the photon detector.

The photodetector operates based on the photoelectric effect. This is where the interaction of IR radiation with a semiconductor promotes electrons to the conduction band, generating a small current that is proportional to the intensity of the radiation. When the sample absorbs at a specific component frequency, the reference and sample signals reach the detector periodically. Since the detector generates a larger current every time it receives a reference signal and a smaller current when it receives a sample signal, it produces an alternating current (AC) that has the same frequency as the rotating chopper.

The AC signal is proportional to the absorption intensity at a specific component frequency. If the sample does not absorb at that frequency, the intensities of the sample and reference beams reaching the detector are equal. In this case, the AC operational amplifier, which is designed to pass only AC signals, provides no output. Finally, the motor that rotates the mirror also moves the chart paper. This movement allows the spectrum to be plotted as a function of frequency or wavenumber.

The dispersive IR spectrometer is particularly useful for identifying functional groups of organic compounds. Some characteristic absorption ranges are as follows:

Bond	Functional group	Absorption wavenumber/ cm-1	Appearance of peak
	Alcohols, ethers, esters	1040-1300	Strong
	Aromatic compounds, alkenes	1500-1680	Weak unless conjugated
	Amides Ketones & aldehydes Esters	1640-1690 1670-1740 1715-1750	Strong Strong Strong
	Alkynes	2150-2250	Weak unless conjugated
	Nitriles	2200-2250	Weak
	Alkanes Alkenes & arenes	2850-2950 3000-3100	Strong Weak
	Amines, amides	3300-3500	Weak
	Carboxylic acids H-bonded alcohols Non H-bonded alcohols	2500-3000 3200-3600 3580-3650	Strong and broad Strong and broad Strong

For example, the spectra of butanone and butan-2-ol are:

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Ultraviolet-visible (UV-Vis) spectroscopy is an analytical technique that measures the amount of discrete ultraviolet and visible light that is absorbed by a sample, allowing for the identification and quantification of various compounds.

A photon of UV-Vis radiation possesses an energy  given by the equation , where represents the Planck constant, denotes the speed of light and signifies a wavelength in the range of 200 to 800nm. This quantum of energy corresponds to the energy difference between the highest occupied orbitals and lowest unoccupied orbitals in many atoms, ions and molecules. When these chemical species absorb UV-Vis photons, they undergo electronic transitions between these orbitals. As different chemical species absorb at characteristic wavelengths, they can be identified through UV-Vis spectroscopy.

A typical double-beam UV-Vis spectrometer comprises a light source that emits a beam of broad-spectrum electromagnetic radiation with a wavelength between 200 and 800nm (see above diagram). Three light sources, namely the deuterium arc lamp, the tungsten-halogen lamp and the Xenon flash lamp, are commonly utilised. The generated light beam initially passes through a monochromator, where it is split into its component wavelengths. A beam with a very narrow range of wavelengths is then selected and divided into two using mirrors. These beams are separately directed towards the reference cell and the sample cell. For an aqueous sample, the reference cell is filled with the corresponding solvent. Otherwise, it remains empty.

The photodetector operates based on the photoelectric effect. This is where the interaction of UV-Vis radiation with a semiconductor promotes electrons to the conduction band, thereby generating a small current that is proportional to the intensity of the radiation. The magnitude of the current is separately recorded for sample cell and the reference cell, and the final spectrum is generated by the computer using a ratio of the sample spectrum against the reference spectrum.

In the visible electromagnetic range, UV-Vis spectroscopy proves useful in determining the concentration of coloured aqueous compounds. This is largely due to the Beer-Lambert law, which is applicable for most diluted aqueous compounds. For instance, if a 0.10M solution of copper(II) sulphate yields an absorbance  of 0.55, the absorbance when the concentration is doubled can be calculated as follows:

Beyond a single measurement, UV-Vis spectroscopy can also be employed to monitor changes in concentration of a coloured compound during a reaction.

In the realm of organic chemistry, it is possible to predict the extent of UV-Vis absorption by organic compounds using the principles of UV-Vis spectroscopy. For example, a saturated organic compound is likely to only absorb short wavelengths of UV light, while a conjugated compound and an aromatic compound may absorb light with wavelengths in the visible range.

 

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The Beer-Lambert Law states that the absorbance of a solution is directly proportional to the path length of the sample and the concentration of the absorbing species in the solution.

Consider a beam of electromagnetic radiation with a narrow range of wavelengths passing through a sample of molar concentration and length . Let  be the number of photons striking on the sample per unit time, and be the number of photons absorbed by the sample. The probability of the number of photons absorbed by the sample is dependent on the length of sample  and the concentration of the sample. Therefore, we have

where is a proportionality constant.

As spectroscopy involves the absorption of radiation intensity at different wavelengths, we need to express the above equation in terms of intensity.

The intensity of the radiation is proportional to the number of photons  striking per unit area of the sample per unit time. Let the intensity of radiation that is incident on the sample and the outgoing intensity of the radiation be and  respectively. It follows that, the greater the number of incident photons, the higher the outgoing intensity will be, i.e. . Furthermore, if is the change in intensity after the radiation passes through the sample, then , which implies that . So, we have

where is another proportionality constant.

Integrating the above equation throughout the length of the sample, we have , which gives or equivalently,

where and .

Eq1 is the Beer-Lambert law. is the sample’s absorbance and is known as the molar absorption coefficient, which represents the nature of the sample. Conventionally, the units of and are and respectively. Moreover, is expressed in , and so, is unitless. The Beer-Lambert law is applicable to UV, visible and IR spectroscopy.

 

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Spectroscopy is the scientific study of the absorption and emission of electromagnetic radiation by matter.

The instrument used for analysing these interactions is known as a spectrometer. A typical spectrometer consists of a light source that produces a beam with a range of electromagnetic frequencies (see diagram below). This beam passes through the sample, where it may be absorbed. The remaining transmitted radiation is detected and recorded, forming an absorption spectrum.

The absorption of a photon of energy causes a chemical species to undergo a transition from a lower energy state to a higher energy state . Different magnitudes of elicit various transitions. These transitions can occur between rotational states, vibrational states, electronic states, and even nuclear spin states of a chemical species (see diagram below).

A variety of spectroscopic methods are used to analyse these transitions. For example, infrared (IR) spectroscopy focuses on transitions between vibrational states, rotational spectroscopy analyses transitions between rotational states, and nuclear magnetic resonance (NMR) spectroscopy studies transitions between different nuclear spin states.

 

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The root mean square speed of a gas provides a measure of the average molecular speed in the gas.

Consider a gas particle of mass moving in a container to the right with velocity (see diagram below). The change in momentum of the particle in the -coordinate before and after colliding elastically with a surface of the container is

with the particle colliding the same surface every

The force acting on the particle at collision is

By Newton’s third law, the force acting on the surface of the container by the particle is

The total force exerted on the surface by particles is

where .

Comparing and the definition of root mean square speed, we have , i.e. the square of the root mean square speed of the gas in the -direction.

         

If the container is three-dimensional (see above diagram), the velocity vector for a particle in Cartesian coordinates in three dimensions is:

Even though a particle has three velocity components in space, the force exerted on the shaded wall is dependent only on the -component velocity of the particle.

To express the root mean square speed for particles, let’s begin by summing the squares of the velocity vectors of all particles:

Next, group the velocity components and divide the equation throughout by ,

The velocity components of the particles in the container are assumed to be random, with the particles evenly distributed in the container at any time. , like and , can be positive or negative, but their squares,, and , are always positive. If the velocity components of the particles in the container are random, and the number of particles is very large,

Otherwise, there is a preference of direction in a particular axis and the particles’ movement is no longer random (you can convince yourself the validity of eq8 by summing the squares of three sets of randomly generated numbers in a spreadsheet and finding their averages; the larger the number of elements in each set, the closer the average values are to each other). Hence, we can rewrite eq7 as:

Substituting eq9 in eq5 gives

The pressure (not to be confused with momentum) exerted by particles on the shaded wall is

where is the volume of the container.

Hence,

Substituting the ideal gas law in eq11 yields

Since (see this article for derivation ) and ,

Since and the root mean square speed is , we have . Therefore,

where is the molar mass of the gas.

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The root mean square speed of a gas is a statistical measure of the average speed of each particle in the gas.

As mentioned in an earlier article, a gas in a container is made up of point masses that are moving randomly and colliding elastically with each other. The velocity of a gas particle is a vector with both magnitude and direction. With every elastic collision, the magnitude of a gas particle’s velocity is unchanged but its direction changes. The particles remain evenly distributed because the sum of the velocity vectors of all particles in the container at any time is zero (a non-zero net velocity implies that the particles are moving in a particular direction and are no longer in random motion).

We cannot determine the individual velocity of a gas particle and have to rely on an average value, which is zero as explained above. This poses a problem and we need to devise a different way to average the velocities of the particles. This new averaging method, which is described in the previous article, is called the root mean square speed and is given by

In the ideal case where the magnitudes of the velocities of all the particles are the same, is the same as taking the average of the absolute values of the velocities of the particles.

 

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The standard deviation of a set of numbers is the measure of the dispersion of the numbers with respect to the mean.

Consider the following sets of numbers:

-2, -2, 2, 2 with a mean of 0

-3, -1, 0, 4 with a mean of 0

The two sets of numbers have the same mean, , or average value. However, the numbers in the first set are clearly less dispersed from the central value of 0 than those in the second set. Therefore, we need an averaging indicator that can tell us the deviation or spread of the numbers about the mean. Suppose we find the average of the absolute values of the numbers:

We end up with the same value with no information regarding the respective spreads. If we try this approach,

we have a reasonable distinction between the spreads.

What we have done for each set in the above computation is that we have:

1. Taken the difference between an element in the set and the mean.
2. Squared the difference and repeat step 1 and step 2 for the rest of the elements in the set.
3. Summed all the squared differences.
4. Divided the sum with the number of elements in the set.
5. Computed the square root of the result in step 4.

This method of measure is called standard deviation . In general,

If the mean of the set of numbers is zero,

We call the standard deviation for this special case, the root mean square value of the set. Notice that the results obtained from the absolute value method and the standard deviation method (or root mean square) are the same if the magnitudes of all the elements in the set are the same.

 

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