10.8: Testing the Effectiveness of Antimicrobial Chemicals and Drugs
- Describe why the phenol coefficient is used
- Describe how the Kirby-Bauer disk diffusion test determines the susceptibility of a microbe to an antibacterial drug.
- Explain the significance of the minimal inhibitory concentration and the minimal bactericidal concentration relative to the effectiveness of an antimicrobial drug.
- Compare and contrast the disk-diffusion, use-dilution, and in-use methods for testing the effectiveness of antiseptics, disinfectants, and sterilants
The effectiveness of various chemical disinfectants is reflected in the terms used to describe them. Chemical disinfectants are grouped by the power of their activity, with each category reflecting the types of microbes and viruses its component disinfectants are effective against. High-level germicides have the ability to kill vegetative cells, fungi, viruses, and endospores, leading to sterilization, with extended use.
However, several environmental conditions influence the potency of an antimicrobial agent and its effectiveness. For example, length of exposure is particularly important, with longer exposure increasing efficacy. Similarly, the concentration of the chemical agent is also important, with higher concentrations being more effective than lower ones. Temperature, pH, and other factors can also affect the potency of a disinfecting agent.
One method to determine the effectiveness of a chemical agent includes swabbing surfaces before and after use to confirm whether a sterile field was maintained during use. Additional tests are described in the sections that follow. These tests allow for the maintenance of appropriate disinfection protocols in clinical settings, controlling microbial growth to protect patients, health-care workers, and the community.
Phenol Coefficient
The effectiveness of a disinfectant or antiseptic can be determined in a number of ways. Historically, a chemical agent’s effectiveness was often compared with that of phenol, the first chemical agent used by Joseph Lister. In 1903, British chemists Samuel Rideal (1863–1929) and J. T. Ainslie Walker (1868–1930) established a protocol to compare the effectiveness of a variety of chemicals with that of phenol, using as their test organisms Staphylococcus aureus (a gram-positive bacterium) and Salmonella enterica serovar Typhi (a gram-negative bacterium). They exposed the test bacteria to the antimicrobial chemical solutions diluted in water for 7.5 minutes. They then calculated a phenol coefficient for each chemical for each of the two bacteria tested. A phenol coefficient of 1.0 means that the chemical agent has about the same level of effectiveness as phenol. A chemical agent with a phenol coefficient of less than 1.0 is less effective than phenol. An example is formalin, with phenol coefficients of 0.3 ( S. aureus ) and 0.7 ( S. enterica serovar Typhi). A chemical agent with a phenol coefficient greater than 1.0 is more effective than phenol, such as chloramine, with phenol coefficients of 133 and 100, respectively. Although the phenol coefficient was once a useful measure of effectiveness, it is no longer commonly used because the conditions and organisms used were arbitrarily chosen.
What are the differences between the three levels of disinfectant effectiveness?
Disk-Diffusion (Kirby-Bauer) Method
The Kirby-Bauer disk diffusion test has long been used as a starting point for determining the susceptibility of specific microbes to various antimicrobial drugs or chemicals. The Kirby-Bauer assay starts with a Mueller-Hinton agar plate on which a confluent lawn (bacteria is spread across the entire surface of the plate) is inoculated with a sample or patient’s isolated bacterial pathogen. Filter paper disks impregnated with known amounts of antibacterial drugs or chemicals to be tested are then placed on the agar plate. As the bacterial inoculum grows, drug or chemical diffuses from the circular disk into the agar and interacts with the growing bacteria. Antimicrobial activity is observed as a clear circular zone of inhibition around the drug/chemical-impregnated disk. The diameter of the zone of inhibition, measured in millimeters and compared to a standardized chart, determines the susceptibility or resistance of the bacterial pathogen to the drug or chemical.(Figure \(\PageIndex{1}\))
There are multiple factors that determine the size of a zone of inhibition in this assay, including drug solubility (whether the agent is water soluble and able to diffuse in the agar), rate of drug diffusion through agar, the thickness of the agar medium, and the drug concentration impregnated into the disk. Due to a lack of standardization of these factors, interpretation of the Kirby-Bauer disk diffusion assay provides only limited information on susceptibility and resistance to the drugs tested. The assay cannot distinguish between bacteriostatic and bactericidal activities, and differences in zone sizes cannot be used to compare drug potencies or efficacies. Comparison of zone sizes to a standardized chart will only provide information on the antibacterials to which a bacterial pathogen is susceptible or resistant. However it is common to correlate larger zones to increased inhibition effectiveness of the chemical agent.
When comparing the activities of two disinfectants against the same microbe, using the disk-diffusion assay, and assuming both are water soluble and can easily diffuse in the agar, would a more effective disinfectant have a larger zone of inhibition or a smaller one?
How does one use the information from a Kirby-Bauer assay to predict the therapeutic effectiveness of an antimicrobial drug in a patient?
Unfortunately, infectious diseases don’t take a time-out for lab work. As a result, physicians rarely have the luxury of conducting susceptibility testing before they write a prescription. Instead, they rely primarily on the empirical evidence (i.e., the signs and symptoms of disease) and their professional experience to make an educated guess as to the diagnosis, causative agent(s), and drug most likely to be effective. This approach allows treatment to begin sooner so the patient does not have to wait for lab test results. In many cases, the prescription is effective; however, in an age of increased antimicrobial resistance, it is becoming increasingly more difficult to select the most appropriate empiric therapy. Selecting an inappropriate empiric therapy not only puts the patient at risk but may promote greater resistance to the drug prescribed.
Recently, studies have shown that antibiograms are useful tools in the decision-making process of selecting appropriate empiric therapy. An antibiogram is a compilation of local antibiotic susceptibility data broken down by bacterial pathogen. In a November 2014 study published in the journal Infection Control and Hospital Epidemiology , researchers determined that 85% of the prescriptions ordered in skilled nursing facilities were decided upon empirically, but only 35% of those prescriptions were deemed appropriate when compared with the eventual pathogen identification and susceptibility profile obtained from the clinical laboratory. However, in one nursing facility where use of antibiograms was implemented to direct selection of empiric therapy, appropriateness of empiric therapy increased from 32% before antibiogram implementation to 45% after implementation of antibiograms. 1 Although these data are preliminary, they do suggest that health-care facilities can reduce the number of inappropriate prescriptions by using antibiograms to select empiric therapy, thus benefiting patients and minimizing opportunities for antimicrobial resistance to develop.
Dilution Tests
As discussed, the limitations of the Kirby-Bauer disk diffusion test do not allow for a direct comparison of antibacterial potencies to guide selection of the best therapeutic choice. However, antibacterial dilution tests can be used to determine a particular drug’s minimal inhibitory concentration (MIC), the lowest concentration of drug that inhibits visible bacterial growth, and minimal bactericidal concentration (MBC), the lowest drug concentration that kills ≥99.9% of the starting inoculum. Determining these concentrations helps identify the correct drug for a particular pathogen. For the macrobroth dilution assay, a dilution series of the drug in broth is made in test tubes and the same number of cells of a test bacterial strain is added to each tube (Figure \(\PageIndex{2}\)). The MIC is determined by examining the tubes to find the lowest drug concentration that inhibits visible growth; this is observed as turbidity (cloudiness) in the broth. Tubes with no visible growth are then inoculated onto agar media without antibiotic to determine the MBC. Generally, serum levels of an antibacterial should be at least three to five times above the MIC for treatment of an infection.
The MIC assay can also be performed using 96-well microdilution trays, which allow for the use of small volumes and automated dispensing devices, as well as the testing of multiple antimicrobials and/or microorganisms in one tray (Figure \(\PageIndex{3}\)). MICs are interpreted as the lowest concentration that inhibits visible growth, the same as for the macrobroth dilution in test tubes. Growth may also be interpreted visually or by using a spectrophotometer or similar device to detect turbidity or a color change if an appropriate biochemical substrate that changes color in the presence of bacterial growth is also included in each well.
The Etest is an alternative method used to determine MIC, and is a combination of the Kirby-Bauer disk diffusion test and dilution methods. Similar to the Kirby-Bauer assay, a confluent lawn of a bacterial isolate is inoculated onto the surface of an agar plate. Rather than using circular disks impregnated with one concentration of drug, however, commercially available plastic strips that contain a gradient of an antibacterial are placed on the surface of the inoculated agar plate (Figure \(\PageIndex{4}\)). As the bacterial inoculum grows, antibiotic diffuses from the plastic strips into the agar and interacts with the bacterial cells. Because the rate of drug diffusion is directly related to concentration, an elliptical zone of inhibition is observed with the Etest drug gradient, rather than a circular zone of inhibition observed with the Kirby-Bauer assay. To interpret the results, the intersection of the elliptical zone with the gradient on the drug-containing strip indicates the MIC. Because multiple strips containing different antimicrobials can be placed on the same plate, the MIC of multiple antimicrobials can be determined concurrently and directly compared. However, unlike the macrobroth and microbroth dilution methods, the MBC cannot be determined with the Etest.
Compare and contrast MIC and MBC .
Use-Dilution Test
Other methods are also used for measuring the effectiveness of a chemical agent in clinical settings. The use-dilution test is commonly used to determine a chemical’s disinfection effectiveness on an inanimate surface. For this test, a cylinder of stainless steel is dipped in a culture of the targeted microorganism and then dried. The cylinder is then dipped in solutions of disinfectant at various concentrations for a specified amount of time. Finally, the cylinder is transferred to a new test tube containing fresh sterile medium that does not contain disinfectant, and this test tube is incubated. Bacterial survival is demonstrated by the presence of turbidity in the medium, whereas killing of the target organism on the cylinder by the disinfectant will produce no turbidity.
The Association of Official Agricultural Chemists International (AOAC), a nonprofit group that establishes many protocol standards, has determined that a minimum of 59 of 60 replicates must show no growth in such a test to achieve a passing result, and the results must be repeatable from different batches of disinfectant and when performed on different days. Disinfectant manufacturers perform use-dilution tests to validate the efficacy claims for their products, as designated by the EPA.
Is the use-dilution test performed in a clinical setting? Why?
In-Use Test
An in-use test can determine whether an actively used solution of disinfectant in a clinical setting is microbially contaminated (Figure \(\PageIndex{5}\)). A 1-mL sample of the used disinfectant is diluted into 9 mL of sterile broth medium that also contains a compound to inactivate the disinfectant. Ten drops, totaling approximately 0.2 mL of this mixture, are then inoculated onto each of two agar plates. One plate is incubated at 37 °C for 3 days and the other is incubated at room temperature for 7 days. The plates are monitored for growth of microbial colonies. Growth of five or more colonies on either plate suggests that viable microbial cells existed in the disinfectant solution and that it is contaminated. Such in-use tests monitor the effectiveness of disinfectants in the clinical setting.
What does a positive in-use test indicate?
Despite antibiotic treatment, Roberta’s symptoms worsened. She developed pyelonephritis, a severe kidney infection, and was rehospitalized in the intensive care unit (ICU). Her condition continued to deteriorate, and she developed symptoms of septic shock. At this point, her physician ordered a culture from her urine to determine the exact cause of her infection, as well as a drug sensitivity test to determine what antibiotics would be effective against the causative bacterium. The results of this test indicated resistance to a wide range of antibiotics, including the carbapenems, a class of antibiotics that are used as the last resort for many types of bacterial infections. This was an alarming outcome, suggesting that Roberta’s infection was caused by a so-called superbug: a bacterial strain that has developed resistance to the majority of commonly used antibiotics. In this case, the causative agent belonged to the carbapenem-resistant Enterobacteriaceae (CRE), a drug-resistant family of bacteria normally found in the digestive system . When CRE is introduced to other body systems, as might occur through improperly cleaned surgical instruments, catheters, or endoscopes, aggressive infections can occur.
CRE infections are notoriously difficult to treat, with a 40%–50% fatality rate. To treat her kidney infection and septic shock, Roberta was treated with dialysis, intravenous fluids, and medications to maintain blood pressure and prevent blood clotting. She was also started on aggressive treatment with intravenous administration of a new drug called tigecycline, which has been successful in treating infections caused by drug-resistant bacteria.
After several weeks in the ICU, Roberta recovered from her CRE infection. However, public health officials soon noticed that Roberta’s case was not isolated. Several patients who underwent similar procedures at the same hospital also developed CRE infections, some dying as a result. Ultimately, the source of the infection was traced to the duodenoscopes used in the procedures. Despite the hospital staff meticulously following manufacturer protocols for disinfection, bacteria, including CRE, remained within the instruments and were introduced to patients during procedures.
Carbapenem-resistant Enterobacteriaceae infections due to contaminated endoscopes have become a high-profile problem in recent years. Several CRE outbreaks have been traced to endoscopes, including a case at Ronald Reagan UCLA Medical Center in early 2015 in which 179 patients may have been exposed to a contaminated endoscope. Seven of the patients developed infections, and two later died. Several lawsuits have been filed against Olympus, the manufacturer of the endoscopes. Some claim that Olympus did not obtain FDA approval for design changes that may have led to contamination, and others claim that the manufacturer knowingly withheld information from hospitals concerning defects in the endoscopes.
Lawsuits like these raise difficult-to-answer questions about liability. Invasive procedures are inherently risky, but negative outcomes can be minimized by strict adherence to established protocols. Who is responsible, however, when negative outcomes occur due to flawed protocols or faulty equipment? Can hospitals or health-care workers be held liable if they have strictly followed a flawed procedure? Should manufacturers be held liable—and perhaps be driven out of business—if their lifesaving equipment fails or is found defective? What is the government’s role in ensuring that use and maintenance of medical equipment and protocols are fail-safe?
Protocols for cleaning or sterilizing medical equipment are often developed by government agencies like the FDA, and other groups, like the AOAC, a nonprofit scientific organization that establishes many protocols for standard use globally. These procedures and protocols are then adopted by medical device and equipment manufacturers. Ultimately, the end-users (hospitals and their staff) are responsible for following these procedures and can be held liable if a breach occurs and patients become ill from improperly cleaned equipment.
Unfortunately, protocols are not infallible, and sometimes it takes negative outcomes to reveal their flaws. In 2008, the FDA had approved a disinfection protocol for endoscopes, using glutaraldehyde (at a lower concentration when mixed with phenol), o-phthalaldehyde, hydrogen peroxide, peracetic acid, and a mix of hydrogen peroxide with peracetic acid. However, subsequent CRE outbreaks from endoscope use showed that this protocol alone was inadequate.
As a result of CRE outbreaks, hospitals, manufacturers, and the FDA are investigating solutions. Many hospitals are instituting more rigorous cleaning procedures than those mandated by the FDA. Manufacturers are looking for ways to redesign duodenoscopes to minimize hard-to-reach crevices where bacteria can escape disinfectants, and the FDA is updating its protocols. In February 2015, the FDA added new recommendations for careful hand cleaning of the duodenoscope elevator mechanism (the location where microbes are most likely to escape disinfection), and issued more careful documentation about quality control of disinfection protocols (Figure \(\PageIndex{7}\)).
There is no guarantee that new procedures, protocols, or equipment will completely eliminate the risk for infection associated with endoscopes. Yet these devices are used successfully in 500,000–650,000 procedures annually in the United States, many of them lifesaving. At what point do the risks outweigh the benefits of these devices, and who should be held responsible when negative outcomes occur?
Key Concepts and Summary
- Chemical disinfectants are grouped by the types of microbes and infectious agents they are effective against. High-level germicides kill vegetative cells, fungi, viruses, and endospores, and can ultimately lead to sterilization. Intermediate-level germicides cannot kill all viruses and are less effective against endospores. Low-level germicides kill vegetative cells and some enveloped viruses, but are ineffective against endospores.
- The effectiveness of a disinfectant is influenced by several factors, including length of exposure, concentration of disinfectant, temperature, and pH.
- Historically, the effectiveness of a chemical disinfectant was compared with that of phenol at killing Staphylococcus aureus and Salmonella enterica serovar Typhi, and a phenol coefficient was calculated.
- The disk-diffusion method is used to test the effectiveness of a chemical disinfectant against a particular microbe.
- The Kirby-Bauer disk diffusion test helps determine the susceptibility of a microorganism to various antimicrobial drugs. However, the zones of inhibition measured must be correlated to known standards to determine susceptibility and resistance, and do not provide information on bactericidal versus bacteriostatic activity, or allow for direct comparison of drug potencies.
- Antibiograms are useful for monitoring local trends in antimicrobial resistance/susceptibility and for directing appropriate selection of empiric antibacterial therapy.
- There are several laboratory methods available for determining the minimum inhibitory concentration (MIC) of an antimicrobial drug against a specific microbe. The minimal bactericidal concentration (MBC) can also be determined, typically as a follow-up experiment to MIC determination using the tube dilution method.
- The use-dilution test determines the effectiveness of a disinfectant on a surface. In-use tests can determine whether disinfectant solutions are being used correctly in clinical settings.
Footnotes
- J.P. Furuno et al. “Using Antibiograms to Improve Antibiotic Prescribing in Skilled Nursing Facilities.” Infection Control and Hospital Epidemiology 35 no. Suppl S3 (2014):S56–61.
Contributors and Attributions
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Nina Parker, (Shenandoah University), Mark Schneegurt (Wichita State University), Anh-Hue Thi Tu (Georgia Southwestern State University), Philip Lister (Central New Mexico Community College), and Brian M. Forster (Saint Joseph’s University) with many contributing authors. Original content via Openstax (CC BY 4.0; Access for free at https://openstax.org/books/microbiology/pages/1-introduction )


6.17: Problems
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1. For an oxygen molecule at 25 C, calculate (a) the most probable velocity, (b) the average velocity, (c) the root-mean-square velocity.
2. For a gas of oxygen molecules at 25 C and 1.00 bar, calculate (a) the collision frequency, (b) the mean time between collisions, (c) the mean free path. The diameter of an oxygen molecule, as estimated from gas-viscosity measurements, is 3.55 x 10\({}^{-10}\) m.
3. For oxygen molecules at 25 C, calculate (a) the fraction with speeds between 150 and 151 m s\({}^{-1}\), (b) the fraction with speeds between 400 and 401 m s\({}^{-1}\), (c) the fraction with speeds between 550 and 551 m s\({}^{-1}\).
4. For a hydrogen molecule at 100 C, calculate (a) the most probable velocity, (b) the average velocity, (c) the root-mean-square velocity.
5. For a gas of hydrogen molecules at 100 C and 1.00 bar, calculate (a) the collision frequency, (b) the mean time between collisions, (c) the mean free path. The diameter of a hydrogen molecule, as estimated from gas-viscosity measurements, is 2.71 x 10\({}^{-10}\) m.
6. For a uranium hexafluoride (UF\({}_{6}\)) molecule at 100 C, calculate (a) the most probable velocity, (b) the average velocity, (c) the root-mean-square velocity.
7. For a gas of uranium hexafluoride molecules at 100 C and 1.00 bar, calculate (a) the collision frequency, (b) the mean time between collisions, (c) the mean free path. Assume that the diameter of a uranium hexafluoride molecule is \(7.0\times {10}^{-10}\ \mathrm{m}\).
8. What is the average kinetic energy of hydrogen molecules at \(100\) C? What is the average kinetic energy of uranium hexafluoride \(\left(UF_6\right)\) molecules at \(100\) C?
9. Assuming the temperature in interstellar space is \(2.73\) K, calculate, for a hydrogen atom, (a) the most probable velocity, (b) the average velocity, (c) the root-mean-square velocity.
10. Assuming that interstellar space is occupied entirely by hydrogen atoms at a particle density of \({10}^2\) molecules \({\mathrm{m}}^{-3}\), calculate (a) the collision frequency, (b) the mean number of years between collisions, (c) the mean free path. Assume that the diameter of a hydrogen atom is \(2.40\times {10}^{-10}\ \mathrm{m}\).
11. Ignoring any effects attributable to its charge and assuming that the temperature is \(2.73\) K, calculate, for an electron in interstellar space, (a) the most probable velocity, (b) the average velocity, (c) the root-mean-square velocity.
12. If a wall of a gas-filled container contains a hole, gas molecules escape through the hole. If all of the molecules that hit the hole escape, but the hole is so small that the number escaping has no effect on the velocity distribution of the remaining gas molecules, we call the escaping process effusion . That is, we call the process effusion only if it satisfies three rather stringent criteria. First, the hole must be large enough (and the wall must be thin enough) so that most molecules passing through the hole do not hit the sides of the hole. Second, a molecule that passes through the hole must not collide with anything on the other side that can send it back through the hole into the original container. Third, the hole must be small enough so that the escaping molecules do not create a pressure gradient; the rate at which gas molecules hit the hole and escape must be determined entirely by the equilibrium distribution of gas velocities and, of course, the area of the hole. Show that the number of molecules effusing through a hole of area \(A\) in time \(t\) is
\[At\left(\frac{N}{V}\right) \left(\frac{kT}{2\pi m}\right)^{1/2} \nonumber \]
where \(\left({N}/{V}\right)\) is the number density of molecules in the container, and \(m\) is their molecular mass.
13. A vessel contains hydrogen and oxygen at \(350\) K and partial pressures of \(0.50\) bar and \(1.50\) bar, respectively. These gases effuse into a vacuum. What is the ratio of hydrogen to oxygen in the escaping gas?
14. How could we use effusion to estimate the molecular weight of an unknown substance?
15. An equimolar mixture of \({}^{235}UF_6\) and \({}^{238}UF_6\) is subjected to effusion. What is the ratio of \({}^{235}U\) to \({}^{238}U\) in the escaping gas?
16. Calculate the number of nitrogen molecules that collide with \({10}^{-6}\ \mathrm{m}^2\) of wall in \({10}^{-6}\ \mathrm{s}\), if the pressure is \(1.00\) bar and the temperature is \(300\) K.
17. Air is approximately \(20\%\) oxygen and \(80\%\) nitrogen by volume. Assume that oxygen and nitrogen molecules both have a radius of \(1.8\times {10}^{-8}\) m. For air at \(1.0\) bar and \(298\) K, calculate:
(a) The number of collisions that one oxygen molecule makes with nitrogen molecules every second.
(b) The number of collisions that occur between oxygen and nitrogen molecules in one cubic meter of air every second.
(c) The number of collisions that one oxygen molecule makes with other oxygen molecules every second.
(d) The number of collisions that occur between oxygen molecules in one cubic meter of air every second.
(e) The number of collisions that occur between oxygen and nitrogen molecules in one cubic meter each second in which the kinetic energy along the line of centers exceeds \(100\ \mathrm{kJ}\ \mathrm{mol}^{-1}\) or \(1.66\times {10}^{-19}\) J per collision.
(f) The number of oxygen-nitrogen collisions that occur in which the kinetic energy along the line of centers exceeds \(50\) kJ \(\mathrm{mol}^{-1}\).
18. Show that \(\int^{\infty }_0 {\rho }_v\left(v\right) dv\neq 1.\)
19. For what volume element, ʋ, is
\[P\left(\textrm{ʋ}\right)=f_{xyz}\left(v_x,v_y,v_z\right)? \nonumber \]
20. Using the model we develop in Section 2.10 :
(a) Show that the pressure, \(P_1\left(v\right)\), attributable to a single molecule of mass \(m\) and velocity \(v\) in a container of volume \(V\) is
\[P_1\left(v\right)=\frac{mv^2}{3V} \nonumber \]
(b) In Section 4.6 , we find that this pressure is
\[\delta P_1\left(v\right)=\frac{2mv^2 \mathrm{cos}^2 \theta}{V} \nonumber \]
for a molecule whose velocity vector lies between \(\theta\) and \(\theta +d\theta\) and between \(\varphi\) and \(\varphi +d\varphi\). This angular region comprises a solid angle whose magnitude is \(d\textrm{Ω}= \mathrm{sin} \theta d\theta d\varphi\). Since the solid angle surrounding a given point is \(4\pi\), the probability that a randomly oriented velocity vector lies between \(\theta\) and \(\theta +d\theta\) and between \(\varphi\) and \(\varphi +d\varphi\) is
\[\frac{d\textrm{Ω}}{4\pi }=\frac{\mathrm{sin} \theta d\theta d\varphi}{4\pi } \nonumber \]
Therefore, given that the scalar component of a molecule’s velocity is \(v\), its contribution to the pressure at \(A\) is
\[dP_1\left(v\right)=\left(\frac{mv^2}{2\pi V}\right) \mathrm{cos}^2 \theta ~ \mathrm{sin} \theta ~ d\theta d\varphi \nonumber \]
To find the pressure contribution made by this molecule irrespective of the values of \(\theta\) and \(\varphi\), we must integrate \(dP_1\left(v\right)\) over all values of \(\theta\) and \(\varphi\) that allow the molecule to impact the wall at \(A\). Recalling that these ranges are \(0\le \theta <{\pi }/{2}\) and \(0\le \varphi <2\pi\), show that
\[P_1\left(v\right)=\frac{mv^2}{3V} \nonumber \]
21. Taking \(P_1\left(v\right)={mv^2}/{3V}\) as the contribution made to the pressure by one molecule whose velocity is \(v\):
(a) Show that the expected value for the contribution made to the pressure by one molecule when the Maxwell–Boltzmann distribution function describes the distribution of molecular velocities is
\[\left\langle P_1\left(v\right)\right\rangle =\frac{kT}{V} \nonumber \]
(b) Show that the variance of the contribution made to the pressure by one molecule is
\[{\sigma }^2_{P_1\left(v\right)}=\frac{2k^2T^2}{3V^2} \nonumber \]
What is the standard deviation, \({\sigma }_{P_1\left(v\right)}\)?
(c) What is the value of the ratio
\[\frac{\sigma_{P_1\left(v\right)}}{ \left\langle P_1\left(v\right)\right\rangle} \nonumber \]
(d) Taking \(3\times {10}^{15}\) as the number of collisions of \(N_2\) molecules at \(1\) bar and \(300\) K with one one square millimeter per microsecond, what pressure, \(P_{avg}\), would we find if we could measure the individual contribution made by each collision and compute their average? What would be the variance, \({\sigma }^2_{avg}\), of this average? The standard deviation, \({\sigma }_{avg}\)? The ratio \({\sigma }_{avg}/P_{avg}\)?
22. Let \(\epsilon =mv^2/2\) be the translational kinetic energy of a gas molecule whose mass is \(m\). Show that the probability density function for \(\epsilon\) is
\[\frac{df}{d\epsilon }=2\pi \left(\frac{1}{\pi kT}\right)^{3/2} \epsilon ^{1/2}exp\left(\frac{-\epsilon }{kT}\right) \nonumber \]
Letting the translational kinetic energy per mole be \(E=\overline{N}\epsilon\), show that
\[\frac{df}{dE}=2\pi \left(\frac{1}{\pi RT}\right)^{3/2}E^{1/2}exp\left(\frac{-E}{RT}\right) \nonumber \]
Notes
\({}^{1}\) Our collision model and quantitative treatment of the role of activation energy in chemical reaction rates follow those given by Arthur A. Frost and Ralph G. Pearson, Kinetics and Mechanism , 2\({}^{nd}\) Ed., John Wiley and Sons, New York, 1961, pp 65-68. See also R. H. Fowler, Statistical Mechanics , Cambridge University Press, New York, 1936.


33.6: GUTs - The Unification of Forces
Learning Objectives
By the end of this section, you will be able to:
- State the grand unified theory.
- Explain the electroweak theory.
- Define gluons.
- Describe the principle of quantum chromodynamics.
- Define the standard model.
Present quests to show that the four basic forces are different manifestations of a single unified force follow a long tradition. In the 19th century, the distinct electric and magnetic forces were shown to be intimately connected and are now collectively called the electromagnetic force. More recently, the weak nuclear force has been shown to be connected to the electromagnetic force in a manner suggesting that a theory may be constructed in which all four forces are unified. Certainly, there are similarities in how forces are transmitted by the exchange of carrier particles, and the carrier particles themselves (the gauge bosons in [link]) are also similar in important ways. The analogy to the unification of electric and magnetic forces is quite good—the four forces are distinct under normal circumstances, but there are hints of connections even on the atomic scale, and there may be conditions under which the forces are intimately related and even indistinguishable. The search for a correct theory linking the forces, called the Grand Unified Theory (GUT) , is explored in this section in the realm of particle physics. Frontiers of Physics expands the story in making a connection with cosmology, on the opposite end of the distance scale.
Figure is a Feynman diagram showing how the weak nuclear force is transmitted by the carrier particle \(Z^0\), similar to the diagrams in [link] and [link] for the electromagnetic and strong nuclear forces. In the 1960s, a gauge theory, called electroweak theory , was developed by Steven Weinberg, Sheldon Glashow, and Abdus Salam and proposed that the electromagnetic and weak forces are identical at sufficiently high energies. One of its predictions, in addition to describing both electromagnetic and weak force phenomena, was the existence of the \(W^+,W^−\), and \(Z^0\) carrier particles. Not only were three particles having spin 1 predicted, the mass of the \(W^+\) and \(W^−\) was predicted to be 81 \(GeV/c^2\), and that of the \(Z^0\) was predicted to be 90 \(GeV/c^2\). (Their masses had to be about 1000 times that of the pion, or about 100 \(GeV/c^2\), since the range of the weak force is about 1000 times less than the strong force carried by virtual pions.) In 1983, these carrier particles were observed at CERN with the predicted characteristics, including masses having the predicted values as seen in [link]. This was another triumph of particle theory and experimental effort, resulting in the 1984 Nobel Prize to the experiment’s group leaders Carlo Rubbia and Simon van der Meer. Theorists Weinberg, Glashow, and Salam had already been honored with the 1979 Nobel Prize for other aspects of electroweak theory.
Although the weak nuclear force is very short ranged ( \(< 10^{–18} m\), its effects on atomic levels can be measured given the extreme precision of modern techniques. Since electrons spend some time in the nucleus, their energies are affected, and spectra can even indicate new aspects of the weak force, such as the possibility of other carrier particles. So systems many orders of magnitude larger than the range of the weak force supply evidence of electroweak unification in addition to evidence found at the particle scale.
Gluons (\(g\)) are the proposed carrier particles for the strong nuclear force, although they are not directly observed. Like quarks, gluons may be confined to systems having a total color of white. Less is known about gluons than the fact that they are the carriers of the weak and certainly of the electromagnetic force. QCD theory calls for eight gluons, all massless and all spin 1. Six of the gluons carry a color and an anticolor, while two do not carry color, as illustrated in Figure \(\PageIndex{2a}\). There is indirect evidence of the existence of gluons in nucleons. When high-energy electrons are scattered from nucleons and evidence of quarks is seen, the momenta of the quarks are smaller than they would be if there were no gluons. That means that the gluons carrying force between quarks also carry some momentum, inferred by the already indirect quark momentum measurements. At any rate, the gluons carry color charge and can change the colors of quarks when exchanged, as seen in Figure \(\PageIndex{2b}\). In the figure, a red down quark interacts with a green strange quark by sending it a gluon. That gluon carries red away from the down quark and leaves it green, because it is an \(R\bar{G}\) (red-antigreen) gluon. (Taking antigreen away leaves you green.) Its antigreenness kills the green in the strange quark, and its redness turns the quark red.
The strong force is complicated, since observable particles that feel the strong force (hadrons) contain multiple quarks. Figure \(\PageIndex{3}\) shows the quark and gluon details of pion exchange between a proton and a neutron as illustrated earlier in [link] and [link]. The quarks within the proton and neutron move along together exchanging gluons, until the proton and neutron get close together. As the u quark leaves the proton, a gluon creates a pair of virtual particles, a \(d\) quark and a \(d^−\) antiquark. The \(d\) quark stays behind and the proton turns into a neutron, while the \(u\)and \(\bar{d}\) move together as a \(π^+\) ([link] confirms the ud− composition for the π+.) The d− annihilates a d quark in the neutron, the u joins the neutron, and the neutron becomes a proton. A pion is exchanged and a force is transmitted.
It is beyond the scope of this text to go into more detail on the types of quark and gluon interactions that underlie the observable particles, but the theory ( quantum chromodynamics or QCD) is very self-consistent. So successful have QCD and the electroweak theory been that, taken together, they are called the Standard Model . Advances in knowledge are expected to modify, but not overthrow, the Standard Model of particle physics and forces.
MAKING CONNECTIONS: UNIFICATION OF FORCES
Grand Unified Theory (GUT) is successful in describing the four forces as distinct under normal circumstances, but connected in fundamental ways. Experiments have verified that the weak and electromagnetic force become identical at very small distances and provide the GUT description of the carrier particles for the forces. GUT predicts that the other forces become identical under conditions so extreme that they cannot be tested in the laboratory, although there may be lingering evidence of them in the evolution of the universe. GUT is also successful in describing a system of carrier particles for all four forces, but there is much to be done, particularly in the realm of gravity.
How can forces be unified? They are definitely distinct under most circumstances, for example, being carried by different particles and having greatly different strengths. But experiments show that at extremely small distances, the strengths of the forces begin to become more similar. In fact, electroweak theory’s prediction of the \(W^+, W^−\), and \(Z^0\) carrier particles was based on the strengths of the two forces being identical at extremely small distances as seen in Figure \(\PageIndex{4}\). As discussed in case of the creation of virtual particles for extremely short times, the small distances or short ranges correspond to the large masses of the carrier particles and the correspondingly large energies needed to create them. Thus, the energy scale on the horizontal axis of Figure \(\PageIndex{4}\) corresponds to smaller and smaller distances, with 100 GeV corresponding to approximately, \(10^{−18}m\) for example. At that distance, the strengths of the EM and weak forces are the same. To test physics at that distance, energies of about 100 GeV must be put into the system, and that is sufficient to create and release the \(W^+, W^−\), and \(Z^0\) carrier particles. At those and higher energies, the masses of the carrier particles becomes less and less relevant, and the \(Z^0\) in particular resembles the massless, chargeless, spin 1 photon. In fact, there is enough energy when things are pushed to even smaller distances to transform the, and \(Z^0\) into massless carrier particles more similar to photons and gluons. These have not been observed experimentally, but there is a prediction of an associated particle called the Higgs boson . The mass of this particle is not predicted with nearly the certainty with which the mass of the \(W^+,W^−\), and \(Z^0\) particles were predicted, but it was hoped that the Higgs boson could be observed at the now-canceled Superconducting Super Collider (SSC). Ongoing experiments at the Large Hadron Collider at CERN have presented some evidence for a Higgs boson with a mass of 125 GeV, and there is a possibility of a direct discovery during 2012. The existence of this more massive particle would give validity to the theory that the carrier particles are identical under certain circumstances.
The small distances and high energies at which the electroweak force becomes identical with the strong nuclear force are not reachable with any conceivable human-built accelerator. At energies of about \(10^{14}GeV\) (16,000 J per particle), distances of about \(10^{−30}m\) can be probed. Such energies are needed to test theory directly, but these are about \(10^{10}\) higher than the proposed giant SSC would have had, and the distances are about \(10^{−12}\) smaller than any structure we have direct knowledge of. This would be the realm of various GUTs, of which there are many since there is no constraining evidence at these energies and distances. Past experience has shown that any time you probe so many orders of magnitude further (here, about \(10^{12}\)), you find the unexpected. Even more extreme are the energies and distances at which gravity is thought to unify with the other forces in a TOE. Most speculative and least constrained by experiment are TOEs, one of which is called Superstring theory . Superstrings are entities that are \(10^{−35}m\) in scale and act like one-dimensional oscillating strings and are also proposed to underlie all particles, forces, and space itself.
At the energy of GUTs, the carrier particles of the weak force would become massless and identical to gluons. If that happens, then both lepton and baryon conservation would be violated. We do not see such violations, because we do not encounter such energies. However, there is a tiny probability that, at ordinary energies, the virtual particles that violate the conservation of baryon number may exist for extremely small amounts of time (corresponding to very small ranges). All GUTs thus predict that the proton should be unstable, but would decay with an extremely long lifetime of about \(10^{31}y\). The predicted decay mode is
\(p→π^0+e^+\), (proposed proton decay)
which violates both conservation of baryon number and electron family number. Although \(10^{31}y\) is an extremely long time (about \(10^{21}\) times the age of the universe), there are a lot of protons, and detectors have been constructed to look for the proposed decay mode as seen in Figure \(\PageIndex{5}\). It is somewhat comforting that proton decay has not been detected, and its experimental lifetime is now greater than \(5×10^{32}y\). This does not prove GUTs wrong, but it does place greater constraints on the theories, benefiting theorists in many ways.
From looking increasingly inward at smaller details for direct evidence of electroweak theory and GUTs, we turn around and look to the universe for evidence of the unification of forces. In the 1920s, the expansion of the universe was discovered. Thinking backward in time, the universe must once have been very small, dense, and extremely hot. At a tiny fraction of a second after the fabled Big Bang, forces would have been unified and may have left their fingerprint on the existing universe. This, one of the most exciting forefronts of physics, is the subject of Frontiers of Physics.
Summary
- Attempts to show unification of the four forces are called Grand Unified Theories (GUTs) and have been partially successful, with connections proven between EM and weak forces in electroweak theory.
- The strong force is carried by eight proposed particles called gluons, which are intimately connected to a quantum number called color—their governing theory is thus called quantum chromodynamics (QCD). Taken together, QCD and the electroweak theory are widely accepted as the Standard Model of particle physics.
- Unification of the strong force is expected at such high energies that it cannot be directly tested, but it may have observable consequences in the as-yet unobserved decay of the proton and topics to be discussed in the next chapter. Although unification of forces is generally anticipated, much remains to be done to prove its validity.
Glossary
- electroweak theory
- theory showing connections between EM and weak forces
- grand unified theory
- theory that shows unification of the strong and electroweak forces
- gluons
- eight proposed particles which carry the strong force
- Higgs boson
- a massive particle that, if observed, would give validity to the theory that carrier particles are identical under certain circumstances
- quantum chromodynamics
- the governing theory of connecting quantum number color to gluons
- standard model
- combination of quantum chromodynamics and electroweak theory
- superstring theory
- a theory of everything based on vibrating strings some \(10^{−35}m\) in length


9.2: Equilibrium Constants
By the end of this section, you will be able to:
- Derive reaction quotients from chemical equations
- Calculate values of reaction quotients and equilibrium constants
- Relate the magnitude of an equilibrium constant to properties of the chemical system
- Describe the ways in which an equilibrium system can be stressed
- Predict the response of a stressed equilibrium using Le Chatelier’s principle
The status of a reversible reaction is conveniently assessed by evaluating its reaction quotient ( Q ) . For a reversible reaction described by
\[ m \mathrm{A}+n \mathrm{B} \rightleftharpoons x \mathrm{C} + y \mathrm{D} \]
the reaction quotient is derived directly from the stoichiometry of the balanced equation where the subscript c denotes the use of molar concentrations in the expression as
\[ Q_c=\frac{[\mathrm{C}]^x[\mathrm{D}]^y}{[\mathrm{A}]^m[\mathrm{B}]^n} \]
Example
Writing Reaction Quotient Expressions
Write the concentration-based reaction quotient expression for each of the following reactions:(a)
(b)
(c)
Solution
(a) \( Q_c=\frac{[\mathrm{O_3}]^2}{[\mathrm{O_2}]^3} \)
(b) \( Q_c=\frac{[\mathrm{NH_3}]^2}{[\mathrm{N_2}] [\mathrm{H_2}]^3} \)
(c) \( Q_c=\frac{[\mathrm{NO_2}]^4 [\mathrm{H_2O}]^6}{[\mathrm{NH_3}]^4 [\mathrm{O_2}]^7} \)
Check Your Learning
Write the concentration-based reaction quotient expression for each of the following reactions:(a)
(b)
(c)
Answer:
(a) \(Q_c=\frac{[\mathrm{SO_3}]^2 }{[\mathrm{SO_2}]^2 [\mathrm{O_2}]} \)
(b) \( Q_c=\frac{[\mathrm{C_2H_4}]^2 }{[\mathrm{C_4H_8}]} \)
(c) \( Q_c=\frac{[\mathrm{CO_2}]^8 [\mathrm{H_2O}]^{10}}{[\mathrm{C_4H_10}]^2 [\mathrm{O_2}]^{13}} \)
The numerical value of Q varies as a reaction proceeds towards equilibrium; therefore, it can serve as a useful indicator of the reaction’s status. To illustrate this point, consider the oxidation of sulfur dioxide:
Two different experimental scenarios are depicted in Figure \(\PageIndex{1}\) , one in which this reaction is initiated with a mixture of reactants only, SO 2 and O 2 , and another that begins with only product, SO 3 . For the reaction that begins with a mixture of reactants only, Q is initially equal to zero:
\[ Q_c=\frac{\left[\mathrm{SO}_3\right]^2}{\left[\mathrm{SO}_2\right]^2\left[\mathrm{O}_2\right]}=\frac{0^2}{\left[\mathrm{SO}_2\right]^2\left[\mathrm{O}_2\right]}= 0 \]
As the reaction proceeds toward equilibrium in the forward direction, reactant concentrations decrease (as does the denominator of Q c ), product concentration increases (as does the numerator of Q c ), and the reaction quotient consequently increases. When equilibrium is achieved, the concentrations of reactants and product remain constant, as does the value of Q c .
If the reaction begins with only product present, the value of Q c is initially undefined (immeasurably large, or infinite):
\[ Q_c=\frac{\left[\mathrm{SO}_3\right]^2}{\left[\mathrm{SO}_2\right]^2\left[\mathrm{O}_2\right]}=\frac{[\mathrm{SO}_3]}{0}=\infty \] In this case, the reaction proceeds toward equilibrium in the reverse direction. The product concentration and the numerator of Q c decrease with time, the reactant concentrations and the denominator of Q c increase, and the reaction quotient consequently decreases until it becomes constant at equilibrium.
The constant value of Q exhibited by a system at equilibrium is called the equilibrium constant, K :
\[ K \equiv q ~ at ~ equlibrium \] Comparison of the data plots in Figure \(\PageIndex{1}\) shows that both experimental scenarios resulted in the same value for the equilibrium constant. This is a general observation for all equilibrium systems, known as the law of mass action : At a given temperature, the reaction quotient for a system at equilibrium is constant.
Example
Evaluating a Reaction Quotient
Gaseous nitrogen dioxide forms dinitrogen tetroxide according to this equation:When 0.10 mol NO 2 is added to a 1.0-L flask at 25 °C, the concentration changes so that at equilibrium, [NO 2 ] = 0.016 M and [N 2 O 4 ] = 0.042 M .
(a) What is the value of the reaction quotient before any reaction occurs?
(b) What is the value of the reaction quotent when the system reacheconstant for the reaction?
Solution
As for all equilibrium calculations in this text, use the simplified equations for Q and K and disregard any concentration or pressure units, as noted previously in this section.(a) Before any product is formed, and [N 2 O 4 ] = 0 M . Thus,
(b) At equilibrium, The reaction quotient is 1.6 10 2 .
Check Your Learning
For the reaction the concentrations at equilibrium are [SO 2 ] = 0.90 M , [O 2 ] = 0.35 M , and [SO 3 ] = 1.1 M . What is the value of the reaction quotient at equlibrium?Answer:
Q c = 4.3
By its definition, the reaction quotient at equilibrium is called the equilibrium constant. the magnitude of an equilibrium constant explicitly reflects the composition of a reaction mixture at equilibrium, and it may be interpreted with regard to the extent of the forward reaction. A reaction exhibiting a large equilibrium constant will reach equilibrium when most of the reactant has been converted to product, whereas a small equilibrium constant indicates the reaction achieves equilibrium after very little reactant has been converted. It’s important to keep in mind that the magnitude of the equilibrium constant does not indicate how rapidly or slowly equilibrium will be reached. Some equilibria are established so quickly as to be nearly instantaneous, and others so slowly that no perceptible change is observed over the course of days, years, or longer.
The equilibrium constant for a reaction can be used to predict the behavior of mixtures containing its reactants and/or products. As demonstrated by the sulfur dioxide oxidation process described above, a chemical reaction will proceed in whatever direction is necessary to achieve equilibrium. Comparing Q to the equilibrium constant for an equilibrium system of interest allows prediction of what reaction (forward or reverse), if any, will occur.
To further illustrate this important point, consider the reversible reaction shown below, K c is the symbol for the equilibrium constant:
The bar charts in Figure \(\PageIndex{2}\) represent changes in reactant and product concentrations for three different reaction mixtures. The reaction quotients for mixtures 1 and 3 are initially lesser than the reaction’s equilibrium constant, so each of these mixtures will experience a net forward reaction to achieve equilibrium. The reaction quotient for mixture 2 is initially greater than the equilibrium constant, so this mixture will proceed in the reverse direction until equilibrium is established.
Example
Predicting the Direction of Reaction
Given here are the starting concentrations of reactants and products for three experiments involving this reaction:Determine in which direction the reaction proceeds as it goes to equilibrium in each of the three experiments shown.
| Reactants/Products | Experiment 1 | Experiment 2 | Experiment 3 |
|---|---|---|---|
| [CO] i | 0.020 M | 0.011 M | 0.0094 M |
| [H 2 O] i | 0.020 M | 0.0011 M | 0.0025 M |
| [CO 2 ] i | 0.0040 M | 0.037 M | 0.0015 M |
| [H 2 ] i | 0.0040 M | 0.046 M | 0.0076 M |
Solution
Experiment 1:Q c < K c (0.040 < 0.64)
The reaction will proceed in the forward direction.
Experiment 2:
Q c > K c (140 > 0.64)
The reaction will proceed in the reverse direction.
Experiment 3:
Q c < K c (0.48 < 0.64)
The reaction will proceed in the forward direction.
Check Your Learning
Calculate the reaction quotient and determine the direction in which each of the following reactions will proceed to reach equilibrium.(a) A 1.00-L flask containing 0.0500 mol of NO(g), 0.0155 mol of Cl 2 (g), and 0.500 mol of NOCl:
(b) A 5.0-L flask containing 17 g of NH 3 , 14 g of N 2 , and 12 g of H 2 :
(c) A 2.00-L flask containing 230 g of SO 3 (g):
Answer:
(a) Q c = 6.45 10 3 , forward. (b) Q c = 0.23, reverse. (c) Q c = 0, forward.
Predicting the Direction of a Reversible Reaction
As we saw in the previous section, reactions proceed in both directions (reactants go to products and products go to reactants). We can tell a reaction is at equilibrium if the reaction quotient (\(Q\)) is equal to the equilibrium constant (K). We next address what happens when a system at equilibrium is disturbed so that \(Q\) is no longer equal to \(K\). If a system at equilibrium is subjected to a perturbance or stress (such as a change in concentration) the position of equilibrium changes. Since this stress affects the concentrations of the reactants and the products, the value of \(Q\) will no longer equal the value of \(K\). To re-establish equilibrium, the system will either shift toward the products (if \(Q < K\)) or the reactants (if \(Q > K\)) until \(Q\) returns to the same value as \(K\).
This process is described by Le Chatelier's principle: When a chemical system at equilibrium is disturbed, it returns to equilibrium by counteracting the disturbance. As described in the previous paragraph, the disturbance causes a change in \(Q\); the reaction will shift to re-establish \(Q = K\).
Le Chatelier's principle can be used to predict changes in equilibrium concentrations when a system that is at equilibrium is subjected to a stress. However, if we have a mixture of reactants and products that have not yet reached equilibrium, the changes necessary to reach equilibrium may not be so obvious. In such a case, we can compare the values of Q and K for the system to predict the changes.
A chemical system at equilibrium can be temporarily shifted out of equilibrium by adding or removing one or more of the reactants or products. The concentrations of both reactants and products then undergo additional changes to return the system to equilibrium.
The stress on the system in Figure \(\PageIndex{3}\) is the reduction of the equilibrium concentration of SCN − (lowering the concentration of one of the reactants would cause Q to be larger than K ). As a consequence, Le Chatelier's principle leads us to predict that the concentration of Fe(SCN) 2+ should decrease, increasing the concentration of SCN − part way back to its original concentration, and increasing the concentration of Fe 3 + above its initial equilibrium concentration.
The effect of a change in concentration on a system at equilibrium is illustrated further by the equilibrium of this chemical reaction:
\[\ce{H}_{2(g)}+\ce{I}_{2(g)} \rightleftharpoons \ce{2HI}_{(g)} \label{13.4.1a} \]
\[K_c=\mathrm{50.0 \; at\; 400°C} \label{13.4.1b} \]
The numeric values for this example have been determined experimentally. A mixture of gases at 400 °C with \(\mathrm{[H_2] = [I_2]} = 0.221\; M\) and \(\ce{[HI]} = 1.563 \;M\) is at equilibrium; for this mixture, \(Q_c = K_c = 50.0\). If \(\ce{H_2}\) is introduced into the system so quickly that its concentration doubles before it begins to react (new \(\ce{[H_2]} = 0.442\; M\)), the reaction will shift so that a new equilibrium is reached, at which
- \(\ce{[H_2]} = 0.374\; M\),
- \(\ce{[I_2]} = 0.153\; M\), and
- \(\ce{[HI]} = 1.692\; M\).
This gives:
\[Q_c=\mathrm{\dfrac{[HI]^2}{[H_2][I_2]}}=\dfrac{(1.692)^2}{(0.374)(0.153)}=50.0=K_c \label{13.4.2} \]
We have stressed this system by introducing additional \(\ce{H_2}\). The stress is relieved when the reaction shifts to the right, using up some (but not all) of the excess \(\ce{H_2}\), reducing the amount of uncombined \(\ce{I_2}\), and forming additional \(\ce{HI}\).
Effect of Change in Pressure on Equilibrium
Sometimes we can change the position of equilibrium by changing the pressure of a system. However, changes in pressure have a measurable effect only in systems in which gases are involved, and then only when the chemical reaction produces a change in the total number of gas molecules in the system. An easy way to recognize such a system is to look for different numbers of moles of gas on the reactant and product sides of the equilibrium. While evaluating pressure (as well as related factors like volume), it is important to remember that equilibrium constants are defined with regard to concentration (for \(K_c\)) or partial pressure (for \(K_P\)). Some changes to total pressure, like adding an inert gas that is not part of the equilibrium, will change the total pressure but not the partial pressures of the gases in the equilibrium constant expression. Thus, addition of a gas not involved in the equilibrium will not perturb the equilibrium.
As we increase the pressure of a gaseous system at equilibrium, either by decreasing the volume of the system or by adding more of one of the components of the equilibrium mixture, we introduce a stress by increasing the partial pressures of one or more of the components. In accordance with Le Chatelier's principle, a shift in the equilibrium that reduces the total number of molecules per unit of volume will be favored because this relieves the stress. The reverse reaction would be favored by a decrease in pressure.
Consider what happens when we increase the pressure on a system in which \(\ce{NO}\), \(\ce{O_2}\), and \(\ce{NO_2}\) are at equilibrium:
\[\ce{2NO (g) + O2(g) \rightleftharpoons 2NO2(g)} \label{13.4.3} \]
The formation of additional amounts of \(\ce{NO2}\) decreases the total number of molecules in the system because each time two molecules of \(\ce{NO_2}\) form, a total of three molecules of \(\ce{NO}\) and \(\ce{O_2}\) are consumed. This reduces the total pressure exerted by the system and reduces, but does not completely relieve, the stress of the increased pressure. On the other hand, a decrease in the pressure on the system favors decomposition of \(\ce{NO_2}\) into \(\ce{NO}\) and \(\ce{O_2}\), which tends to restore the pressure.
Now consider this reaction:
\[\ce{N2 (g) + O2 (g) \rightleftharpoons 2NO (g)} \label{13.4.4} \]
Because there is no change in the total number of molecules in the system during reaction, a change in pressure does not favor either formation or decomposition of gaseous nitrogen monoxide.
Effect of Change in Temperature on Equilibrium
Changing concentration or pressure perturbs an equilibrium because the reaction quotient is shifted away from the equilibrium value. Changing the temperature of a system at equilibrium has a different effect: A change in temperature actually changes the value of the equilibrium constant. However, we can qualitatively predict the effect of the temperature change by treating it as a stress on the system and applying Le Chatelier's principle.
When hydrogen reacts with gaseous iodine, heat is evolved.
\[\ce{H2(g) + I2(g) \rightleftharpoons 2HI(g) } \;\;\ ΔH=\mathrm{−9.4\;kJ\;(exothermic)} \label{13.4.5} \]
Because this reaction is exothermic, we can write it with heat as a product.
\[\ce{H2(g) + I2(g) \rightleftharpoons 2HI(g)} + \text{heat} \label{13.4.6} \]
Increasing the temperature of the reaction increases the internal energy of the system. Thus, increasing the temperature has the effect of increasing the amount of one of the products of this reaction. The reaction shifts to the left to relieve the stress, and there is an increase in the concentration of \(\ce{H2}\) and \(\ce{I2}\) and a reduction in the concentration of \(\ce{HI}\). Lowering the temperature of this system reduces the amount of energy present, favors the production of heat, and favors the formation of hydrogen iodide.
When we change the temperature of a system at equilibrium, the equilibrium constant for the reaction changes. Lowering the temperature in the \(\ce{HI}\) system increases the equilibrium constant: At the new equilibrium the concentration of \(\ce{HI}\) has increased and the concentrations of \(\ce{H2}\) and \(\ce{I2}\) decreased. Raising the temperature decreases the value of the equilibrium constant, from 67.5 at 357 °C to 50.0 at 400 °C.
Temperature affects the equilibrium between \(\ce{NO_2}\) and \(\ce{N_2O_4}\) in this reaction
\[\ce{N2O4(g) \rightleftharpoons 2NO2(g)}\;\;\; ΔH=\mathrm{57.20\; kJ} \label{13.4.7} \]
The positive ΔH value tells us that the reaction is endothermic and could be written
\[\text{heat}+\ce{N2O4(g) \rightleftharpoons 2NO2(g)} \label{13.4.8} \]
At higher temperatures, the gas mixture has a deep brown color, indicative of a significant amount of brown \(\ce{NO_2}\) molecules. If, however, we put a stress on the system by cooling the mixture (withdrawing energy), the equilibrium shifts to the left to supply some of the energy lost by cooling. The concentration of colorless \(\ce{N_2O_4}\) increases, and the concentration of brown \(\ce{NO_2}\) decreases, causing the brown color to fade.
Catalysts Do Not Affect Equilibrium
As we learned during our study of kinetics, a catalyst can speed up the rate of a reaction. Though this increase in reaction rate may cause a system to reach equilibrium more quickly (by speeding up the forward and reverse reactions), a catalyst has no effect on the value of an equilibrium constant nor on equilibrium concentrations.
The interplay of changes in concentration or pressure, temperature, and the lack of an influence of a catalyst on a chemical equilibrium is illustrated in the industrial synthesis of ammonia from nitrogen and hydrogen according to the equation
\[\ce{N2(g) + 3H2(g) \rightleftharpoons 2NH3(g)} \label{13.4.9} \]
A large quantity of ammonia is manufactured by this reaction. Each year, ammonia is among the top 10 chemicals, by mass, manufactured in the world. About 2 billion pounds are manufactured in the United States each year. Ammonia plays a vital role in our global economy. It is used in the production of fertilizers and is, itself, an important fertilizer for the growth of corn, cotton, and other crops. Large quantities of ammonia are converted to nitric acid, which plays an important role in the production of fertilizers, explosives, plastics, dyes, and fibers, and is also used in the steel industry.
Haber was born in Breslau, Prussia (presently Wroclaw, Poland) in December 1868. He went on to study chemistry and, while at the University of Karlsruhe, he developed what would later be known as the Haber process: the catalytic formation of ammonia from hydrogen and atmospheric nitrogen under high temperatures and pressures.
For this work, Haber was awarded the 1918 Nobel Prize in Chemistry for synthesis of ammonia from its elements. The Haber process was a boon to agriculture, as it allowed the production of fertilizers to no longer be dependent on mined feed stocks such as sodium nitrate.
\[\ce{N2(g) + 3H2(g) \rightleftharpoons 2NH3(g)} \nonumber \]
Currently, the annual production of synthetic nitrogen fertilizers exceeds 100 million tons and synthetic fertilizer production has increased the number of humans that arable land can support from 1.9 persons per hectare in 1908 to 4.3 in 2008. The availability of nitrogen is a strong limiting factor to the growth of plants. Despite accounting for 78% of air, diatomic nitrogen (\(\ce{N_2}\)) is nutritionally unavailable due the tremendous stability of the nitrogen-nitrogen triple bond. For plants to use atmospheric nitrogen, the nitrogen must be converted to a more bioavailable form (this conversion is called nitrogen fixation).
In addition to his work in ammonia production, Haber is also remembered by history as one of the fathers of chemical warfare. During World War I, he played a major role in the development of poisonous gases used for trench warfare. Regarding his role in these developments, Haber said, “During peace time a scientist belongs to the World, but during war time he belongs to his country.” 1 Haber defended the use of gas warfare against accusations that it was inhumane, saying that death was death, by whatever means it was inflicted. He stands as an example of the ethical dilemmas that face scientists in times of war and the double-edged nature of the sword of science.
Like Haber, the products made from ammonia can be multifaceted. In addition to their value for agriculture, nitrogen compounds can also be used to achieve destructive ends. Ammonium nitrate has also been used in explosives, including improvised explosive devices. Ammonium nitrate was one of the components of the bomb used in the attack on the Alfred P. Murrah Federal Building in downtown Oklahoma City on April 19, 1995.
Summary
Systems at equilibrium can be disturbed by changes to temperature, concentration, and, in some cases, volume and pressure; volume and pressure changes will disturb equilibrium if the number of moles of gas is different on the reactant and product sides of the reaction. The system's response to these disturbances is described by Le Chatelier's principle: The system will respond in a way that counteracts the disturbance. Not all changes to the system result in a disturbance of the equilibrium. Adding a catalyst affects the rates of the reactions but does not alter the equilibrium, and changing pressure or volume will not significantly disturb systems with no gases or with equal numbers of moles of gas on the reactant and product side.
| Disturbance | Observed Change as Equilibrium is Restored | Direction of Shift | Effect on K |
|---|---|---|---|
| reactant added | added reactant is partially consumed | toward products | none |
| product added | added product is partially consumed | toward reactants | none |
| decrease in volume/increase in gas pressure | pressure decreases | toward side with fewer moles of gas | none |
| increase in volume/decrease in gas pressure | pressure increases | toward side with more moles of gas | none |
| temperature increase | heat is absorbed | toward products for endothermic, toward reactants for exothermic | changes |
| temperature decrease | heat is given off | toward reactants for endothermic, toward products for exothermic | changes |
Footnotes
- Herrlich, P. “The Responsibility of the Scientist: What Can History Teach Us About How Scientists Should Handle Research That Has the Potential to Create Harm?” EMBO Reports 14 (2013): 759–764.
Glossary
- equilibrium
- state of a reversible reaction in which the forward and reverse processes occur at equal rates
- equilibrium constant ( K )
- value of the reaction quotient for a system at equilibrium; may be expressed using concentrations ( K c ) or partial pressures ( K p )
- heterogeneous equilibria
- equilibria in which reactants and products occupy two or more different phases
- homogeneous equilibria
- equilibria in which all reactants and products occupy the same phase
- law of mass action
- when a reversible reaction has attained equilibrium at a given temperature, the reaction quotient remains constant
- Le Châtelier’s principle
- an equilibrium subjected to stress will shift in a way to counter the stress and re-establish equilibrium
- reaction quotient ( Q )
- mathematical function describing the relative amounts of reactants and products in a reaction mixture; may be expressed in terms of concentrations ( Q c ) or pressures ( Q p )
- reversible reaction
- chemical reaction that can proceed in both the forward and reverse directions under given conditions


9.4: Polar Covalence
Make sure you thoroughly understand the following essential ideas which have been presented below.
- Define electronegativity, and describe the general way in which electronegativities depend on the location of an element in the periodic table.
- Explain how electronegativity values relate to the polar nature of a chemical bond.
- What two factors determine the magnitude of an electric dipole moment ?
- Sketch structural diagrams that illustrate how the presence of polar bonds in a molecule can either increase or diminish the magnitude of the molecular dipole moment.
- Calculate the formal charge of each atom in a structure, and comment on its significance in relation to the polarity of that structure.
- Select the more likely of two or more electron-dot structures for a given species.
- Explain the difference between formal charge and oxidation number.
- Describe the limiting cases of covalent and ionic bonds. Explain what both categories of bonds have in common.
The electrons constituting a chemical bond are simultaneously attracted by the electrostatic fields of the nuclei of the two bonded atoms. In a homonuclear molecule such as O 2 the bonding electrons will be shared equally by the two atoms. In general, however, differences in the sizes and nuclear charges of the atoms will cause one of them to exert a greater attraction on the bonding pair, causing the electron cloud to be displaced toward the more strongly-attracting atom.
Electronegativity
The electronegativity of an atom denotes its relative electron-attracting power in a chemical bond. It is important to understand that electronegativity is not a measurable property of an atom in the sense that ionization energies and electron affinities are, although it can be correlated with both of these properties. The actual electron-attracting power of an atom depends in part on its chemical environment (that is, on what other atoms are bonded to it), so tabulated electronegativities should be regarded as no more than predictors of the behavior of electrons, especially in more complicated molecules. There are several ways of computing electronegativities, which are expressed on an arbitrary scale. The concept of electronegativity was introduced by Linus Pauling and his 0-4 scale continues to be the one most widely used.
The 0-4 electronegativity scale of Pauling is the best known of several arbitrary scales of this kind. Electronegativity values are not directly observable, but are derived from measurable atomic properties properties such as ionization energy and electron affinity. The place of any atom on this scale provides a good indication of its ability to compete with another atom in attracting a shared electron pair to it, but the presence of bonds to other atoms, and of multiple- or nonbonding electron pairs may make predictions about the nature a given bond less reliable.
An atom that has a small electronegativity is said to be electropositive . As the diagram shows, the metallic elements are generally electropositive. The position of hydrogen in this regard is worth noting; although physically a nonmetal, much of its chemistry is metal-like.
Dipole moments
When non-identical atoms are joined in a covalent bond, the electron pair will be attracted more strongly to the atom that has the higher electronegativity. As a consequence, the electrons will not be shared equally; the center of the negative charges in the molecule will be displaced from the center of positive charge. Such bonds are said to be polar and to possess partial ionic character , and they may confer a polar nature on the molecule as a whole.
A polar molecule acts as an electric dipole which can interact with electric fields that are created artificially or that arise from nearby ions or polar molecules. Dipoles are conventionally represented as arrows pointing in the direction of the negative end. The magnitude of interaction with the electric field is given by the permanent electric dipole moment of the molecule. The dipole moment corresponding to an individual bond (or to a diatomic molecule) is given by the product of the quantity of charge displaced q and the bond length r :
\[μ = q \times r\]
In SI units, q is expressed in coulombs and r in meters, so μ has the dimensions of \(C \cdot m\). If one entire electron charge is displaced by 100 pm (a typical bond length), then
\[μ = (1.6022 \times 10^{–19}\; C) \times (10^{–10}\; m) = 1.6 \times 10^{–29}\; C \cdot m = 4.8 \;D\]
The quantity denoted by D, the Debye unit, is still commonly used to express dipole moments. It was named after Peter Debye (1884-1966), the Dutch-American physicist who pioneered the study of dipole moments and of electrical interactions between particles; he won the Nobel Prize for Chemistry in 1936.
How dipole moments are measured
When a solution of polar molecules is placed between two oppositely-charged plates, they will tend to align themselves along the direction of the field. This process consumes energy which is returned to the electrical circuit when the field is switched off, an effect known as electrical capacitance .
Measurement of the capacitance of a gas or solution is easy to carry out and serves as a means of determining the magnitude of the dipole moment of a substance.
Estimate the percent ionic character of the bond in the hydrogen fluoride molecule from the experimental data shown at the right.
Solution
Dipole moments as vector sums
In molecules containing more than one polar bond, the molecular dipole moment is just the vector combination of what can be regarded as individual "bond dipole moments". Being vectors, these can reinforce or cancel each other, depending on the geometry of the molecule; it is therefore not uncommon for molecules containing polar bonds to be nonpolar overall, as in the example of carbon dioxide:
The zero dipole moment of CO 2 is one of the simplest experimental methods of demonstrating the linear shape of this molecule.
H 2 O, by contrast, has a very large dipole moment which results from the two polar H–O components oriented at an angle of 104.5°. The nonbonding pairs on oxygen are a contributing factor to the high polarity of the water molecule. In molecules containing nonbonding electrons or multiple bonds, the elecronegativity difference may not correctly predict the bond polarity. A good example of this is carbon monoxide, in which the partial negative charge resides on the carbon, as predicted by its negative formal charge (below.)
Electron densities in a molecule (and the dipole moments that unbalanced electron distributions can produce) are now easily calculated by molecular modeling programs. In this example, for methanol CH 3 OH, the blue area centered on hydrogen represents a positive charge, the red area centered where we expect the lone pairs to be located represents a negative charge, while the light green around methyl is approximately neutral. The manner in which the individual bonds contribute to the dipole moment of the molecule is nicely illustrated by the series of chloromethanes:
(Bear in mind that all four positions around the carbon atom are equivalent in this tetrahedral molecule, so there are only four chloromethanes.)
Formal charge and oxidation number
Although the total number of valence electrons in a molecule is easily calculated, there is not always a simple and unambiguous way of determining how many reside in a particular bond or as non-bonding pairs on a particular atom. For example, one can write valid Lewis octet structures for carbon monoxide showing either a double or triple bond between the two atoms, depending on how many nonbonding pairs are placed on each: C::O::: and :C:::O: (see Problem Example 3 below). The choice between structures such as these is usually easy to make on the principle that the more electronegative atom tends to surround itself with the greater number of electrons. In cases where the distinction between competing structures is not all that clear, an arbitrarily-calculated quantity known as the formal charge can often serve as a guide.
The formal charge on an atom is the electric charge it would have if all bonding electrons were shared equally with its bonded neighbors.
The formal charge on an atom is calculated by the following formula:
in which the core charge is the electric charge the atom would have if all its valence electrons were removed. In simple cases, the formal charge can be worked out visually directly from the Lewis structure, as is illustrated farther on.
Find the formal charges of all the atoms in the sulfuric acid structure shown here.
Solution
The atoms here are hydrogen, sulfur, and double- and single-bonded oxygens. Remember that a double bond is made up of two electron-pairs.
- hydrogen: FC = 1 – 0 – 1 = 0
- sulfur: FC = 6 – 0 – 6 = 0
- hydroxyl oxygen: FC = 6 – 4 – 2 = 0
- double-bonded oxygen: FC = 6 – 4 – 2 = 0
Using formal charge to select the best Lewis structure
The general rule for choosing between alternative structures is that the one involving the smallest formal charges is most favored, although the following example shows that this is not always the case.
Write out some structures for carbon monoxide CO, both those that do and do not obey the octet rule, and select the "best" on the basis of the formal charges.
Solution
Structure that obeys the octet rule:
a) For :C:::O: Carbon: 4 – 2 – 3 = –1 ; Oxygen: 6 – 2 – 3 = +1
Structures that do not obey the octet rule (for carbon):
b) For :C:O::: Carbon: 4 – 2 – 1 = +1 ; Oxygen: 6 – 6 – 1 = –1
c) For :C::O:: Carbon: 4 – 2 –2 = 0 ; Oxygen: 6 – 4 – 2 = 0
Comment : All three structures are acceptable (because the formal charges add up to zero for this neutral molecule) and contribute to the overall structure of carbon monoxide, although not equally. Both experiment and more advanced models show that the triple-bonded form (a) predominates. Formal charge, which is no more than a bookkeeping scheme for electrons, is by itself unable to predict this fact.
In a species such as the thiocyanate ion \(SCN^-\) in which two structures having the same minimal formal charges can be written, we would expect the one in which the negative charge is on the more electronegative atom to predominate.
The electrons in the structures of the top row are the valence electrons for each atom; an additional electron (purple) completes the nitrogen octet in this negative ion. The electrons in the bottom row are divided equally between the bonded atoms; the difference between these numbers and those above gives the formal charges.
Formal charge can also help answer the question “where is the charge located?” that is frequently asked about polyatomic ions. Thus by writing out the Lewis structure for the ammonium ion NH 4 + , you should be able to convince yourself that the nitrogen atom has a formal charge of +1 and each of the hydrogens has 0, so we can say that the positive charge is localized on the central atom.
Oxidation number
This is another arbitrary way of characterizing atoms in molecules. In contrast to formal charge, in which the electrons in a bond are assumed to be shared equally, oxidation number is the electric charge an atom would have if the bonding electrons were assigned exclusively to the more electronegative atom. Oxidation number serves mainly as a tool for keeping track of electrons in reactions in which they are exchanged between reactants, and for characterizing the “combining power” of an atom in a molecule or ion.
The following diagram compares the way electrons are assigned to atoms in calculating formal charge and oxidation number in carbon monoxide.
Ionic compounds
The shared-electron pair model introduced by G.N. Lewis showed how chemical bonds could form in the absence of electrostatic attraction between oppositely-charged ions. As such, it has become the most popular and generally useful model of bonding in all substances other than metals. A chemical bond occurs when electrons are simultaneously attracted to two nuclei, thus acting to bind them together in an energetically-stable arrangement. The covalent bond is formed when two atoms are able to share a pair of electrons:
In general, however, different kinds of atoms exert different degrees of attraction on their electrons, so in most cases the sharing will not be equal. One can even imagine an extreme case in which the sharing is so unequal that the resulting "molecule" is simply a pair of ions:
The resulting substance is sometimes said to contain an ionic bond . Indeed, the properties of a number of compounds can be adequately explained using the ionic model. But does this mean that there are really two kinds of chemical bonds, ionic and covalent? According to the ionic electrostatic model, solids such as NaCl consist of positive and negative ions arranged in a crystal lattice. Each ion is attracted to neighboring ions of opposite charge, and is repelled by ions of like charge; this combination of attractions and repulsions, acting in all directions, causes the ion to be tightly fixed in its own location in the crystal lattice.
Since electrostatic forces are nondirectional, the structure of an ionic solid is determined purely by geometry: two kinds of ions, each with its own radius, will fall into whatever repeating pattern will achieve the lowest possible potential energy. Surprisingly, there are only a small number of possible structures; one of the most common of these, the simple cubic lattice of NaCl, is shown here.
Is there such as thing as an ionic bond?
When two elements form an ionic compound, is an electron really lost by one atom and transferred to the other one? In order to deal with this question, consider the data on the ionic solid LiF. The average radius of the neutral Li atom is about 2.52Å. Now if this Li atom reacts with an atom of F to form LiF, what is the average distance between the Li nucleus and the electron it has “lost” to the fluorine atom? The answer is 1.56Å; the electron is now closer to the lithium nucleus than it was in neutral lithium!
So the answer to the above question is both yes and no: yes, the electron that was now in the 2 s orbital of Li is now within the grasp of a fluorine 2 p orbital, but no, the electron is now even closer to the Li nucleus than before, so how can it be “lost”? The one thing that is inarguably true about LiF is that there are more electrons closer to positive nuclei than there are in the separated Li and F atoms. But this is just the rule we stated at the beginning of this unit: chemical bonds form when electrons can be simultaneously near two or more nuclei.
It is obvious that the electron-pair bond brings about this situation, and this is the reason for the stability of the covalent bond. What is not so obvious (until you look at the numbers such as are quoted for LiF above) is that the “ionic” bond results in the same condition; even in the most highly ionic compounds, both electrons are close to both nuclei, and the resulting mutual attractions bind the nuclei together. This being the case, is there really any fundamental difference between the ionic and covalent bond?
The answer, according to modern chemical thinking is probably “no”; in fact, there is some question as to whether it is realistic to consider that these solids consist of “ions” in the usual sense. The preferred picture that seems to be emerging is one in which the electron orbitals of adjacent atom pairs are simply skewed so as to place more electron density around the “negative” element than around the “positive” one.
This being said, it must be reiterated that the ionic model of bonding is a useful one for many purposes, and there is nothing wrong with using the term “ionic bond” to describe the interactions between the atoms in the very small class of “ionic solids” such as LiF and NaCl.
"Covalent, ionic or metallic" is an oversimplification!
If there is no such thing as a “completely ionic” bond, can we have one that is completely covalent? The answer is yes, if the two nuclei have equal electron attracting powers . This situation is guaranteed to be the case with homonuclear diatomic molecules-- molecules consisting of two identical atoms. Thus in Cl 2 , O 2 , and H 2 , electron sharing between the two identical atoms must be exactly even; in such molecules, the center of positive charge corresponds exactly to the center of negative charge: halfway between the two nuclei.
Categorizing all chemical bonds as either ionic, covalent, or metallic is a gross oversimplification; as this diagram shows, there are examples of substances that exhibit varying degrees of all three bonding characteristics.
Dative (Coordinate) covalent Bonds
In most covalent bonds, we think of the electron pair as having a dual parentage, one electron being contributed by each atom. There are, however, many cases in which both electrons come from only one atom. This can happen if the donor atom has a non-bonding pair of electrons and the acceptor atom has a completely empty orbital that can accommodate them. These are called dative or coordinate covalent bonds.
This is the case, for example, with boron trifluoride and ammonia. In BF 3 , one the 2p orbitals is unoccupied and can accommodate the lone pair on the nitrogen atom of ammonia. The electron acceptor, BF 3 , acts as a Lewis acid here, and NH 3 is the Lewis base . Bonds of this type (sometimes known as coordinate covalent or dative bonds) tend to be rather weak (usually 50-200kJ/mol); in many cases the two joined units retain sufficient individuality to justify writing the formula as a molecular complex or adduct .


20.31: Parental Recognition of Child Obesity
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Learning Objectives
- To study the parents' perception of their children's weight status
Research conducted by
Debra Etelson, Donald A. Brand, Patricia A. Patrick, and Anushree Shirali
Case study prepared by
Robert F. Houser and Georgette Baghdady
Overview
With increasing public awareness of child obesity as a major public health problem, studies are showing that it has not translated into an increased awareness of obesity in one’s own child. Dietary patterns and weight status in childhood tend to carry into adolescence and adulthood, promoting the onset of chronic and other diseases. A key ingredient for combating childhood obesity is parental involvement and commitment. However, this is predicated on whether or not parents can recognize overweight and obesity in their children.
This study examined parents’ perceptions of their children’s weight status, their understanding of the health risks of obesity relative to other conditions they may perceive as health risks, and their knowledge of some healthy eating practices. Children’s actual weight status was expressed as their body mass index ( BMI ) percentile, as determined by the CDC growth charts based on age and sex. According to the CDC growth charts for children, a child with a BMI percentile less than the \(5^{th}\) percentile is underweight; from the \(5^{th}\) to less than the \(85^{th}\), a child is at a healthy weight; from the \(85^{th}\) to less than the \(95^{th}\) percentile, a child is overweight; and a BMI percentile equal to or greater than the \(95^{th}\) percentile, a child is considered to be obese.
A visual analog scale was used to measure parents’ perceptions of their child’s weight. The visual analog scale consisted simply of a \(10\)-cm straight line anchored at the left end by the label “extremely underweight” and at the right end by the label “extremely overweight.” A parent placed a mark along the line to indicate where they perceived their child’s weight to be. The researchers interpreted the marks as percentiles in their analysis.
Questions to Answer
Do parents recognize when their children are overweight or obese? Do parents who make incorrect judgments about healthy food practices also make incorrect judgments about their child’s weight status?
Design Issues
This study defines a parent’s perception of their child’s BMI percentile as “accurate” if their score on a visual analog scale fell within \(30\) points of the child’s true BMI percentile. This wide range defining accuracy potentially allows for misclassification of a child’s weight status among normal, overweight, and obese categories. For example, a parent who perceives their child’s weight status as being at the \(80^{th}\) percentile, i.e., in the normal range, when in reality the child is obese with a BMI percentile of \(98\), the parent’s assessment would be considered accurate by the operational definition used in this study. The authors explain that they chose this definition to give parents as much leeway as possible in assessing their child’s weight on the visual analog scale.
Descriptions of Variables
| Variable | Description |
| Sex | The sex of the participating parent’s child |
|
Overwt_Obese |
Whether or not a child’s body mass index (BMI) is equal to or greater than the 85th percentile for the child’s age and sex, which means that the child is either overweight (85th to less than 95th percentile) or obese (95th percentile or above) |
| PA_overwt |
Parental attitude expressing level of concern if their child were overweight, measured on a 4-point Likert Scale. In data analysis, the four categories were condensed into two categories:
0 = “not at all” or “a little” concerned 1 = “quite” or “extremely” concerned |
| PA-TV |
Parental attitude expressing level of concern if their child watched >20 hours of TV per week, measured on a 4-point Likert Scale. In data analysis, the four categories were condensed into two categories:
0 = "not at all" or "a little" concerned 1 = "quite" or "extremely" concerned |
| Accurate | Whether or not the parent's perception of their child's weight status was accurate. Parent's perception was considered accurate if the BMI percentile it corresponded to fell within 30 points of the child's actual BMI percentile |
| Juice_boxes |
The amount of juice that a parent thinks is healthy for their child to drink each day (a juice box contains eight ounces). We condensed the original four response categories into two categories:
0 = "1 or 2 juice boxes per day" 1 = "3 to 8 juice boxes per day" |
| Fast_food_meals | How often a parent feels it is okay to eat at fast-food restaurants. We condensed the original four response categories into two categories: 0 = "once a month" |
Links
Etelson et al. article
BMI percentiles for children
References
- Debra Etelson, D., Brand, D. A., Patrick, P. A., Shirali, A. (2003). Childhood obesity: Do parents recognize this health risk? Obesity Research, 11, 1362-1368