radioactivity pt 2

Radioactive Decay

In discussing isotopes of elements, we have touched on how the atomic number is the same for a given element, but the mass number can change depending on the isotope. Different isotopes can also be termed nuclides and have the notation AZX (e.g., 13C = carbon-13). Nucleons are protons and neutrons present in the nucleus. Think about heavier nuclei (more protons in the nucleus) and their stability: the nucleus contains both positively charged protons and neutral neutrons. Having a higher number of positively charged species in a small volume (the nucleus) can lead to nuclear instability, often manifested as radioactivity. In fact, all isotopes of elements with atomic numbers greater than 83 are unstable and are radioactive (bismuth and beyond in the periodic table).

Radioactive decay can occur in radioactive isotopes. The end result of the process, or a series of radioactive decay processes, is a stable nucleus with a lower atomic number and various particles, which may include (alpha), (beta), (gamma) particles, and smaller, also stable, nuclei.

The spontaneous change of an unstable nuclide into another is radioactive decay. The unstable nuclide is called the parent nuclide; the nuclide that results from the decay is known as the daughter nuclide. Three common types of natural radioactive decay are alpha, beta, and gamma decay.

Beta decay can be thought of as the conversion of a neutron into a proton and a β particle. The beta particle (electron) emitted is from the atomic nucleus and is not one of the electrons surrounding the nucleus.  Emission of an electron does not change the mass number of the nuclide but does increase the number of its protons and decrease the number of its neutrons.

Several different devices are used to detect and measure radiation, including Geiger counters, scintillation counters (scintillators), and radiation dosimeters. In this lab, you will be using a Vernier radiation monitor, which consists of a Geiger-Muller tube mounted in a small, rugged black case. A thin window at the bottom of the radiation monitor allows alpha, beta, and gamma emissions to be detected. Radiation causes the ionization of the gas in a Geiger-Müller tube; therefore, the rate of ionization is proportional to the amount of radiation. Each ionization event is measured and recorded as a "count." Geiger counters display these counts numerically and as an optional audio signal. 

A variety of units are used to measure various aspects of radiation. The SI unit for the rate of radioactive decay is the becquerel (Bq), with 1 Bq = 1 disintegration per second. The curie (Ci) and millicurie (mCi) are much larger units and are frequently used in medicine (1 curie = 1 Ci = 3.7×1010 disintegrations per second). The SI unit measuring tissue damage caused by radiation is the sievert (Sv). This takes into account both the energy and the biological effects of the type of radiation involved in the radiation dose. The roentgen equivalent for man (rem) is the unit for radiation damage that is used most frequently in medicine (100 rem = 1 Sv).  

Radiation in Everyday Life & Shielding

Different types of radiation have differing abilities to pass through a material. A very thin barrier, such as a sheet or two of paper or the top layer of skin cells, usually stops alpha particles.

Because of this, alpha particle sources are usually not dangerous if outside the body but are quite hazardous if ingested or inhaled.

Beta particles will pass through a hand or a thin layer of material like paper or wood but are stopped by a thin layer of metal.

Gamma radiation is very penetrating and can pass through a thick layer of most materials. Some high-energy gamma radiation is able to pass through a few feet of concrete. Certain dense, high atomic number elements (such as lead) can effectively attenuate gamma radiation with thinner material and are used for shielding.

The ability of various kinds of emissions to cause ionization varies greatly, and some particles have almost no tendency to produce ionization. Alpha particles have about twice the ionizing power of fast-moving neutrons, about 10 times that of β particles, and about 20 times that of γ rays and X-rays.

Chemical Reaction Rate

What is the rate of a reaction? It is the change in the amount of reactant or product per unit of time. Rate can be determined experimentally by measuring a property (such as absorbance, concentration, etc.) as it changes over time during the reaction process. The rate expression for a reaction is a mathematical one that indicates how the concentration of each involved species changes over time.

There are a number of factors that affect reaction rate. The identities of the reactants are one factor, as is the physical state. You can imagine that trying to react two solids together may be difficult, while reactions in solution (liquid or gas) may more readily occur. Other factors to consider are temperature, concentration, and catalysis.

Integrated Rate Laws

We are not concerned with the derivation of the rate laws in this particular course, but we will provide the background information for you to put it in context. The important part is to recognize there are three different equations for straight lines in y=mx+b format and that these inform us as to the order of the reaction. Different quantities need to be analyzed graphically vs. time, namely concentration, the natural logarithm of concentration, and the inverse concentration, to determine the order of the reaction.

In order to be able to determine the order and rate constant of a reaction, it is necessary to understand the integrated forms of the rate laws. Integrated rate laws come from combining and integrating two definitions of rate. From this, we get three integrated rate laws.

For zero-order reactions: 

 

For first-order reactions: 

 

For second-order reactions: 

 

In all equations, [A]0 is the initial concentration of the reactant, k is the rate constant, t is the time elapsed, and [A]t is the concentration of the reactant at any time t.

Notice that all of the equations are written in the form of y = mx + b. Therefore, if we plot the appropriate form of concentration on the y-axis and t on the x-axis, then we should get a line for all three orders.

For zero-order, the integrated rate law is [A]t = −kt + [A]0, so plotting [A]t versus time will yield a straight line:

For first-order, the integrated rate law is ln[A]t = −kt + ln[A]0, so plotting ln[A]t versus time will yield a straight line:

For second-order, the integrated rate law is 1/[A]t = kt + 1/[A]0, so plotting 1/[A]t versus time will yield a straight line:

You will measure the counts per minute of the radioactive decay of Ba-137m using a radiation monitor. In this experiment, you can use counts instead of concentration to determine the rate law.

You will use integrated rate law methods to determine the order of the reaction and the value of the rate constant k by graphing the counts and time data collected.

  • if a plot of A versus time is linear then x = 0, the reaction is zero order in Ba-137m, and the slope of the line is equal to -k1

  • if a plot of lnA versus time is linear then x = 1, the reaction is first order in Ba-137m, and the slope of the line is equal to -k1

  • if a plot of 1/A versus time is linear then n = 2, the reaction is second order in Ba-137m, and the slope of the line is equal to k1

You will generate all three plots. The most linear plot type will reveal the order of the reaction. Comparisons of the types of plot you will generate can be found in your text as well as various online sources.

Half-Life

Half-life is perhaps the facet of kinetics that is most applicable to a student's understanding of everyday life, specifically pharmaceuticals. 

Nearly every practical use of chemistry involves knowing about the kinetics of the reaction. It's not just the speed of the reaction that we care about, we are often more interested in being able to find the amount of material left at any point. If we have a mathematical way to determine the concentration, we don't have to do any invasive procedures to find it. Think about it for a second. Which would you rather do: get out the calculator to find the amount of a medicine left in your body or have your blood drawn for lab tests to find the same information?

How it works in pharmacology

Say you want to give a patient the drug "A" for the next 24 hours. You know that in order to be effective, the concentration in the blood serum must be between 15–30 mg/L. The metabolism of this drug follows first-order kinetic rate equations, and the half-life of the drug is 5 hours.

To begin, you give the patient enough of the drug so that the concentration in the blood is at the high end of the range (a "loading dose"). Blood makes up about 7 percent of the body's weight, so a 150-pound (68 kg) man would have ~5 L of blood. That calculation is: (30 mg/L) * (5 L) = 150 mg. So you give them 150 mg.

After 5 hours, the patient has metabolized half the drug, so the blood concentration falls by half. If it started at 30 mg/L, it's now only at 15 mg/L, right at the bottom of the effective dose. You have to give more of the drug, enough to bring the level back up to 30 mg/L. This time, you only give half the amount you gave to start because the patient is not starting from zero; they already have 15 mg/L in their blood. So, you give them 75 mg, which is half of the full dose (150 mg). That explains why the doctor so often tells you to "Take two pills right now, and then 1 every xx hours after that." This cycle repeats for the entire 24-hour treatment cycle.

The half-life of a reaction, t1/2, is the amount of time needed for the concentration of a reactant to decrease by half. This plays a key role in the administration of drugs, where half-life is used to determine the concentration within the body at any given time.

The half-life for a reaction depends on the rate law for the reaction.

 For zero-order and second-order reactions, you can see that the half-life depends on the concentration of the reactant [A]. For first-order reactions, the half-life does not depend on concentration and will stay constant no matter how much reactant is present. You can see that in the figure below:

Accuracy and Precision

We will begin with important definitions:

  • Measurement: The act or process of measuring. This can be simple (reading a thermometer) or complex (all operations required for analysis).

  • Precision (Reproducibility): A precise measurement is close to other values obtained in the same way.

  • Accuracy (Correctness): An accurate measurement is close to the true value or the accepted value if the true value is unknown.

  • Error: Anything causing a measurement to differ from the true value. The amount by which the measured value differs from the true value is also called the error.

Accuracy and precision can be expressed mathematically in a variety of ways. One simple expression of the precision of a series of measurements is range:

 

It should be intuitive that a small range suggests the measurements are precise, while a large one indicates they are not.

A statistically meaningful expression of precision is the standard deviation of the series. We will be using Excel to calculate standard deviation, but it is valuable to be familiar with the formula and its meaning.

 

Standard deviation measures how closely the individual values are clustered around the mean. If the measurements follow a normal distribution curve (shown below), 68% of all values will fall within the interval () +/- 1 s. A small standard deviation indicates greater precision or a more closely clustered data set.

Accuracy can be simply expressed as error:

 

 The percent error between your experimental value and the true value is also a measure of accuracy. It can be found by dividing the difference between your experimental value and the actual (or accepted) value by the actual value and expressing it as a percentage. Percent error is typically provided as a positive value, thus the absolute value bars below.