Chemical Equilibrium Discussion Examples | How to Determine the Equilibrium Constant and Causes of Error

What Is Chemical Equilibrium?

Chemical equilibrium is a state in a reversible reaction in which the rates of the forward and reverse reactions become equal and the concentrations of reactants and products no longer appear to change.
At equilibrium, the reaction has not stopped; rather, the forward and reverse reactions continue at the same rate.
Therefore, even though the concentrations appear constant, the system is in a state of dynamic equilibrium in which reactions continue at the molecular level.

In chemical equilibrium experiments, the proportions of reactants and products present at equilibrium are investigated.
The value representing these proportions is the equilibrium constant.
When the equilibrium constant is large, the equilibrium is considered to be shifted toward the products, while when it is small, the equilibrium is considered to be shifted toward the reactants.

Main Items to Include in the Results

In a chemical equilibrium report, organize the initial concentrations, equilibrium concentrations, measured values, calculation equations, equilibrium constant, and comparison with literature values.
Because the equilibrium constant depends greatly on the concentration-calculation process, it is important to clearly state which measured values were used to determine the concentration of each component.

Main Items to Include in the Results

• Reaction equation
• Initial concentrations
• Measured values at equilibrium
• Equilibrium concentrations
• Changes in concentration
• Equation for the equilibrium constant
• Calculated equilibrium constant
• Measurement temperature
• Literature or theoretical value
• Error rate
• If a graph or calibration curve was used, its fitted equation
• Presence or absence of outliers or variation

Reference Experimental Values and Analysis Examples for Chemical Equilibrium

Here, equilibrium concentrations, equilibrium constants, concentration changes, temperature changes, and measurement errors obtained in chemical equilibrium experiments are organized as reference experimental values that are easy to discuss in a report.
The initial concentration, change, and equilibrium concentration are summarized in tables, and the method for determining the equilibrium constant K and the sources of error are examined.

In chemical equilibrium, the forward and reverse reactions proceed simultaneously until the concentrations appear to stop changing.
At equilibrium, the concentration ratio between reactants and products becomes constant, and this value is expressed as the equilibrium constant.
The equilibrium constant depends on temperature, and even if concentration or volume is changed, it is expected to approach the same value at the same temperature.

Reference Experimental Conditions

Basic Equation for the Equilibrium Constant

For the reaction aA + bB ⇄ cC + dD, the equilibrium constant Kc is expressed as follows.

Kc = [C]^c[D]^d / [A]^a[B]^b

Here, square brackets represent concentrations at equilibrium.
The coefficients of the reactants and products are treated as exponents in the equilibrium-constant equation.

Example Measurement of the FeSCN2+ Formation Reaction

In the reaction between iron(III) ions and thiocyanate ions, red FeSCN²⁺ is formed.
The equilibrium concentration of FeSCN²⁺ can be determined from absorbance, and the equilibrium constant can then be calculated.

Fe³⁺ + SCN⁻ ⇄ FeSCN²⁺

Example of Determining the Equilibrium Concentration From a Calibration Curve

Suppose the calibration curve for FeSCN²⁺ is represented by A = 5000 × [FeSCN²⁺].
If the absorbance of Sample B is 0.375,

[FeSCN²⁺] = 0.375 ÷ 5000 = 7.50×10⁻⁵ mol/L

This value is treated as the equilibrium concentration of FeSCN²⁺, and the concentration changes before and after the reaction are then considered.

Example of Organizing Initial Concentrations, Changes, and Equilibrium Concentrations

For Sample B, let the initial [Fe³⁺] = 1.00×10⁻³ mol/L, initial [SCN⁻] = 2.00×10⁻⁴ mol/L, and equilibrium [FeSCN²⁺] = 7.50×10⁻⁵ mol/L.

Example Calculation of the Equilibrium Constant K

The equilibrium constant for Fe³⁺ + SCN⁻ ⇄ FeSCN²⁺ is expressed as follows.

K = [FeSCN²⁺] / ([Fe³⁺][SCN⁻])

Substituting the equilibrium concentrations for Sample B gives

K = 7.50×10⁻⁵ / {(9.25×10⁻⁴) × (1.25×10⁻⁴)}

K = 7.50×10⁻⁵ / 1.16×10⁻⁷ = 647

Under these conditions, the equilibrium constant is approximately 6.5×10².

Comparison of Equilibrium Constants Determined From Multiple Samples

At the same temperature, the equilibrium constant should be close to the same value even if the initial concentrations differ.
However, variation occurs among samples because of errors in absorbance measurement and concentration calculations.

Example of Reaction Quotient Q and Equilibrium Shift

The reaction quotient Q is the value obtained by substituting concentrations at any point in time into the equilibrium-constant expression.
Comparing Q with K makes it possible to determine the direction in which the reaction will proceed.

Example Observation of Equilibrium Shift Caused by Concentration Changes

Change in the Equilibrium Constant With Temperature

In this example, the FeSCN²⁺ concentration and K decrease as the temperature increases.
This can be interpreted as indicating that the FeSCN²⁺ formation reaction is exothermic and that increasing the temperature shifts the equilibrium toward the reverse reaction.

Linear Range and Error in Absorbance Measurements

Effect of Concentration Error on the Equilibrium Constant

Example of Checking the Time Required to Reach Equilibrium

If absorbance before equilibrium is used, the product concentration may be underestimated and the equilibrium constant may also be calculated as too small.

Main Sources of Error

Example of How to Write the Results

Red FeSCN²⁺ was formed by the reaction of Fe³⁺ and SCN⁻.
In the absorbance measurement, Sample B showed an absorbance of 0.375, and substitution into the calibration curve A = 5000[FeSCN²⁺] gave an equilibrium [FeSCN²⁺] of 7.50×10⁻⁵ mol/L.

Because Fe³⁺ and SCN⁻ are consumed by the same amount that FeSCN²⁺ is formed, the equilibrium [Fe³⁺] was 9.25×10⁻⁴ mol/L and the equilibrium [SCN⁻] was 1.25×10⁻⁴ mol/L.
Substituting these values into the equilibrium-constant expression gave K = 647.

When K was determined for multiple samples, the values generally fell within the range of 5.6×10² to 7.6×10².
Because the equilibrium constant should be constant at the same temperature, this variation is considered to originate from errors in absorbance measurement, dilution operations, and the calibration curve.
In particular, because the product concentration is also involved in calculating the equilibrium concentrations of the reactants, even a small error in absorbance can have a large effect on K.

Points for Connecting the Results to the Discussion

In a discussion of chemical equilibrium, it is important not only to calculate the equilibrium constant but also to explain why equilibrium shifts when concentration changes, why K changes when temperature changes, and why K varies among samples.

• Can you write the correct equilibrium-constant equation from the reaction equation?
• Can you determine equilibrium concentrations from absorbance or titration values?
• Can you organize initial concentration, change, and equilibrium concentration in a table?
• Do you understand that reactants decrease by the amount of product formed?
• Can you explain that K should be similar even when initial concentrations differ at the same temperature?
• Can you compare the reaction quotient Q with the equilibrium constant K and explain the direction of the reaction?
• Can you explain equilibrium shifts caused by concentration changes using Le Chatelier’s principle?
• Can you discuss why the equilibrium constant changes with temperature in relation to exothermic and endothermic reactions?
• Can you explain how absorbance errors, calibration curves, dilution factors, and time required to reach equilibrium affect K?

Example Discussion

In this experiment, the absorbance of FeSCN²⁺ formed by the reaction of Fe³⁺ and SCN⁻ was measured, and the equilibrium constant was determined.
Because FeSCN²⁺ is red, a larger absorbance indicates a higher product concentration.
For Sample B, the absorbance of 0.375 gave [FeSCN²⁺] = 7.50×10⁻⁵ mol/L.

When the equilibrium concentrations were organized, Fe³⁺ and SCN⁻ had been consumed by the same amount of FeSCN²⁺ formed.
Using these values, the equilibrium constant was calculated as K = 647.
The K values obtained under multiple initial-concentration conditions did not agree perfectly, but at the same temperature they generally fell within a similar range.
This indicates that the equilibrium constant is determined by temperature rather than by the initial concentrations of reactants and products.

When Fe³⁺ or SCN⁻ was added, the red color became darker.
This is because increasing the reactant concentration temporarily made the reaction quotient Q smaller than K, causing the equilibrium to shift in the direction of forming FeSCN²⁺.
This result can be explained by Le Chatelier’s principle, according to which equilibrium shifts in the direction that counteracts an externally applied change.

When the temperature was increased, the absorbance decreased and the apparent equilibrium constant also became smaller.
This suggests that the FeSCN²⁺ formation reaction is exothermic and that increasing the temperature shifted the equilibrium toward the reverse reaction.
Because the equilibrium constant depends on temperature, K values obtained at different temperatures cannot be compared as values under the same conditions.

Possible sources of error include errors in absorbance readings, contamination or bubbles in the cell, errors in constructing the calibration curve, mistakes in dilution factors, and measurement before equilibrium was reached.
In particular, an error in the FeSCN²⁺ concentration affects not only the product concentration but also the equilibrium concentrations of Fe³⁺ and SCN⁻.
Therefore, even a small absorbance error may appear as a large variation in the equilibrium constant K.

Summary

In chemical equilibrium, K can be determined by substituting the equilibrium concentrations into the equilibrium-constant expression.
At the same temperature, K should be close to the same value even if the initial concentrations differ, but variation occurs because of measurement errors and differences in the time required to reach equilibrium.

This reference example covered concentration calculation from a calibration curve, organization of initial concentration, change, and equilibrium concentration, the equilibrium constant K, reaction quotient Q, equilibrium shifts caused by concentration changes, temperature changes, absorbance errors, and time required to reach equilibrium using the FeSCN²⁺ formation reaction as an example.
In a report, it is useful to discuss the calculation results in relation to Le Chatelier’s principle, temperature dependence, and measurement errors.

What Is the Equilibrium Constant?

The equilibrium constant is a value representing the ratio of product concentrations to reactant concentrations at equilibrium.
If a general reaction is expressed as follows, the equilibrium constant is written using the equilibrium concentration of each component.

aA + bB ⇄ cC + dD

K = [C]^c[D]^d ÷ [A]^a[B]^b

Here, [A], [B], [C], and [D] are the concentrations at equilibrium.
The important point is to use equilibrium concentrations rather than initial concentrations.
The coefficients in the reaction equation are reflected as exponents in the equilibrium-constant expression.

Meaning of a Large or Small Equilibrium Constant

When the equilibrium constant is large, more product is present at equilibrium and the reaction is considered to be shifted toward the products.
Conversely, when the equilibrium constant is small, more reactant remains and the equilibrium is considered to be shifted toward the reactants.

However, reaction rate cannot be determined from the magnitude of the equilibrium constant alone.
The equilibrium constant indicates “which side is favored at the end,” not “how quickly equilibrium is reached.”
Reaction rate is considered separately using factors such as rate constants and activation energy.

How to Determine Equilibrium Concentrations

To determine the equilibrium constant, the concentration of each component at equilibrium is required.
However, it is not always possible to measure the concentration of every component directly.
Therefore, in experiments, the concentration of one component may be determined by titration or absorbance, and the concentrations of the other components calculated from the stoichiometric relationships in the reaction equation.

When organizing equilibrium concentrations, it is useful to prepare a table of initial concentration, change, and equilibrium concentration.
Organizing them separately as Initial, Change, and Equilibrium, as in an ICE table, helps prevent calculation errors.

Concentration Calculations Using an ICE Table

An ICE table is used to organize concentration calculations in chemical equilibrium.
I represents initial concentration, C represents change, and E represents equilibrium concentration.
A common source of error when calculating equilibrium constants is confusing the initial concentration with the equilibrium concentration.
Using an ICE table makes it easier to organize how much each component increases or decreases.

When the Equilibrium Constant Is Determined From Absorbance

In chemical equilibrium experiments, the absorbance of a colored substance produced at equilibrium may be measured to determine its concentration.
Within the range where absorbance is proportional to concentration, a calibration curve can be used to determine the product concentration at equilibrium.
From this product concentration, the reactant concentrations are calculated using the stoichiometric relationships in the reaction equation, and the equilibrium constant is determined.

When absorbance is used, the linearity of the calibration curve, blank correction, contamination of the cell, wavelength setting, and measurement range become sources of error.
In particular, if the absorbance is too high or too low, the reliability of the concentration conversion may decrease.

When the Equilibrium Constant Is Determined From Titration

The concentration of a component in an equilibrium mixture may also be quantified by titration, and the equilibrium constant determined from that concentration.
For example, the extent of reaction can be calculated by titrating the acid remaining at equilibrium or a component formed during the reaction.
The concentration is determined from the titration volume, and the other component concentrations are then calculated from the stoichiometric relationships in the reaction equation.

When titration is used, endpoint determination, concentration of the standard solution, burette readings, sample volume, and timing of stopping the reaction become sources of error.
Whether the operation could be performed without changing the equilibrium state is also an important point for discussion.

Handling Units of the Equilibrium Constant

In high-school chemistry and basic student experiments, the equilibrium constant is often treated as a ratio of concentrations.
In such cases, an apparent unit may arise depending on the reaction equation.
On the other hand, in thermodynamics, activities are used, so the equilibrium constant may be treated as dimensionless.

In university laboratory reports, it is important to follow the instructions of the instructor or laboratory manual.
When treating it as a concentration equilibrium constant, clearly state the concentration units and the calculation equation used.

Effect of Temperature on the Equilibrium Constant

The equilibrium constant changes with temperature.
Therefore, it is important to keep the temperature constant in experiments where the equilibrium constant is determined.
If the measurement temperature differs from that used for the literature value, the experimental and literature values may differ.

In general, the temperature dependence of equilibrium is related to whether the reaction is endothermic or exothermic.
When the temperature changes, the position of equilibrium changes and the product and reactant concentrations change, so the calculated equilibrium constant also changes.

Discussion When Equilibrium Has Not Been Reached

To determine the equilibrium constant, the reaction must have sufficiently reached equilibrium.
If measurement is performed before equilibrium is reached, the concentration of each component is still changing and does not represent the true equilibrium concentration.
Therefore, the calculated equilibrium constant deviates from its true value.

Possible causes of insufficient equilibrium attainment include a reaction time that is too short, insufficient stirring, unstable temperature, and a slow reaction rate.
Measuring at different times and checking whether the measured values have become constant is useful for determining whether equilibrium has been reached.

Error Caused by Concentration Preparation

Initial concentrations and equilibrium concentrations are important in calculating the equilibrium constant.
Therefore, errors in weighing reagents, dilution, or the use of volumetric flasks and pipettes directly affect the equilibrium constant.
If the initial concentration is incorrect, the calculated extent of reaction and equilibrium concentrations also become incorrect.

In particular, because concentrations may appear as exponents in the equilibrium-constant equation, even a small concentration error can have a large effect on the equilibrium constant.

Error in Reading Measured Values

Experiments used to determine equilibrium constants rely on measured values such as absorbance, titration volume, pH, temperature, mass, and volume.
If these measured values contain reading errors, the calculation of equilibrium concentrations is affected.
For example, even a small deviation in absorbance is converted into a concentration through the calibration curve, so it causes error in the equilibrium constant.

Error in the Calibration Curve

When equilibrium concentrations are determined using absorbance, the accuracy of the calibration curve is important.
If the fitted line of the calibration curve shows large scatter, errors are also introduced into the concentration conversion of unknown samples.
Errors in preparing standard-solution concentrations, absorbance measurements, blank correction, and wavelength setting are sources of calibration-curve error.

In addition, if the absorbance of the unknown sample lies outside the calibration-curve range, concentration estimation becomes an extrapolation and reliability decreases.
In that case, measures such as diluting the sample so that it falls within the calibration-curve range are necessary.

Error in Blank Correction

In absorbance measurements, a blank measurement is performed to correct for absorption originating from the solvent or reagents themselves.
If blank correction is insufficient, the measured absorbance includes absorption from components other than the target substance.
As a result, the product concentration may be overestimated or underestimated.

Discussion When Side Reactions Occur

In experiments for determining an equilibrium constant, calculations are often performed assuming that only the target reversible reaction is occurring.
However, if a side reaction occurs, some of the reactants or products are converted into other substances, causing errors in the calculated equilibrium concentrations.
As a result, the equilibrium constant may deviate from its true value.

Possible side reactions can be discussed from observations such as changes in color, formation of precipitates, decomposition, pH changes, extra absorption in spectra, and inconsistencies in titration results.

Discussion Using Le Chatelier’s Principle

Le Chatelier’s principle is the idea that when a change in concentration, temperature, pressure, or another condition is applied to a system at equilibrium, the equilibrium shifts in the direction that counteracts that change.
In chemical equilibrium experiments, it can be used to explain how the equilibrium position changes when concentration or temperature is changed.

However, Le Chatelier’s principle explains the direction of equilibrium shift, while the equilibrium constant itself does not change when concentration is changed at constant temperature.
Changing the concentration changes the equilibrium concentrations, but at the same temperature the calculated equilibrium constant should approach the same value.

Discussion When K Varies Under Multiple Conditions

When the equilibrium constant is determined under multiple initial-concentration conditions, it should theoretically approach the same value at the same temperature.
In practice, however, K may vary.
This variation can be caused by concentration-preparation errors, measurement errors, insufficient equilibrium attainment, temperature changes, and calibration-curve errors.

When comparing multiple K values, the mean and standard deviation may be used to evaluate reproducibility.
For conditions showing large variation, check whether there were problems with the measured values or experimental operations.

When the Equilibrium Constant Is Larger Than the Literature Value

If the experimentally determined equilibrium constant is larger than the literature value, the product concentration may have been overestimated or the reactant concentrations underestimated.
If the product concentration is calculated too high from absorbance measurements, or if the amount of reactant remaining is underestimated by titration, K tends to become large.

When the Equilibrium Constant Is Smaller Than the Literature Value

If the experimentally determined equilibrium constant is smaller than the literature value, the product concentration may have been underestimated or the reactant concentrations overestimated.
If measurement is performed before equilibrium is reached, the product has not been formed sufficiently and K may be calculated as too small.
K may also become small if the product decomposes or is lost during measurement.

Calculation of Error Rate

When comparing with a literature or theoretical value, the error rate can be calculated to show how far the experimental value deviates.
The error rate is obtained by dividing the difference between the experimental and literature values by the literature value and expressing it as a percentage.

Error rate (%) = |Experimental value − Literature value| ÷ Literature value × 100

When writing the error rate, do not end with the numerical value alone; discuss why the error occurred.
It is useful to relate it to temperature conditions, measured values, concentration preparation, attainment of equilibrium, and calibration curves.

When the Results Can Be Considered Good

Chemical equilibrium experiment results can be considered good when the equilibrium constants obtained under multiple conditions are close to one another and do not greatly contradict literature or theoretical values.
It is also important that the calibration curve show good linearity, that sufficient time has been allowed to reach equilibrium, and that the temperature is kept constant.

Example Discussion When the Experiment Did Not Go Well

If a chemical equilibrium experiment does not go well, possible causes are considered from results such as large variation in K, large differences from the literature value, poor linearity of the calibration curve, failure to reach equilibrium, or inconsistent concentration calculations.
Organizing equilibrium concentrations, temperature, measurement method, concentration preparation, and insufficient equilibrium attainment separately makes the discussion easier.

How to Write Points for Improvement

In a discussion of chemical equilibrium, including points for improvement as well as sources of error makes the report easier to organize.
Improvements are easier to consider when divided into concentration preparation, attainment of equilibrium, measurement, temperature control, and calculations.

Improvements to Concentration Preparation

• Use volumetric flasks and volumetric pipettes correctly
• Prepare standard-solution concentrations accurately
• Perform dilution operations carefully
• Reduce errors when measuring reagents
• Do not confuse initial concentrations with equilibrium concentrations

Improvements to Equilibrium Attainment and Temperature Control

• Allow sufficient reaction time
• Stir sufficiently
• Confirm that equilibrium has been reached before measurement
• Keep the temperature constant using a thermostatic bath or similar equipment
• Compare with literature values at the same temperature

Improvements to Measurement and Analysis

• Perform blank correction in absorbance measurements
• Measure within the linear range of the calibration curve
• Determine the endpoint carefully in titration
• Take multiple measurements and use the average
• Organize concentration changes using an ICE table
• Substitute equilibrium concentrations into the equilibrium-constant equation

Difference Between a Superficial Discussion and a Good Discussion

In a discussion of chemical equilibrium, simply writing that “the equilibrium constant was determined” or “there was an error” results in a superficial discussion.
Relating equilibrium concentrations, the reaction equation, concentration changes, temperature, measurement errors, and differences from literature values produces a more persuasive discussion.

Examples of Expressions That Can Be Used in Reports

The following expressions can be used when writing the results and discussion of chemical equilibrium experiments.
Adjust the necessary parts according to your own experimental results.

• At equilibrium, the rates of the forward and reverse reactions become equal and the concentration of each component becomes constant.
• The equilibrium constant is expressed as the ratio of product concentrations to reactant concentrations at equilibrium.
• Equilibrium concentrations rather than initial concentrations must be used in calculating the equilibrium constant.
• The concentration changes were organized according to the coefficients in the reaction equation, and the equilibrium concentrations were determined.
• Because the obtained equilibrium constant was large, the equilibrium is considered to be shifted toward the products.
• Because the equilibrium constant depends on temperature, differences in measurement temperature may have affected the difference from the literature value.
• If measurement is performed before equilibrium is reached, the equilibrium concentrations are not correctly reflected and error occurs in K.
• Error in the calibration curve affects the conversion of product concentration and leads to error in the equilibrium constant.
• Possible causes of variation in K under multiple conditions include concentration-preparation errors and measurement errors.
• At a constant temperature, the equilibrium constant should approach the same value even if the initial concentrations differ.

Points to Check When Discussing Chemical Equilibrium

Checking the following points before writing the report makes the discussion easier to write.

• Have you written the reaction equation correctly?
• Does the equilibrium-constant equation correspond to the coefficients in the reaction equation?
• Have you distinguished between initial and equilibrium concentrations?
• Have you organized concentration changes using an ICE table or similar method?
• Have you correctly determined the equilibrium concentrations from the measured values?
• Have you related the magnitude of the equilibrium constant to the direction in which the equilibrium is shifted?
• Have you compared the result with a literature or theoretical value?
• Have you calculated the error rate?
• Have you considered temperature dependence?
• Have you discussed insufficient attainment of equilibrium?
• Have you considered errors in calibration curves, titration, and absorbance measurements?
• Do the points for improvement correspond to the sources of error?

Summary

In chemical equilibrium experiments, the equilibrium constant is determined from the concentration of each component at equilibrium.
The important point in calculating the equilibrium constant is to use equilibrium concentrations rather than initial concentrations.
Organizing concentration changes based on the stoichiometric relationships in the reaction equation and using an ICE table or similar method to determine the equilibrium concentration of each component helps prevent calculation errors.

When the equilibrium constant is large, the equilibrium is considered to be shifted toward the products, while when it is small, it is considered to be shifted toward the reactants.
However, the equilibrium constant does not represent reaction rate.
In addition, because the equilibrium constant depends on temperature, the temperature conditions must be checked when comparing with literature values.

In a report, discuss not only the value of the equilibrium constant but also the method used to determine the equilibrium concentrations, the measurement method, concentration preparation, attainment of equilibrium, temperature control, and errors in calibration curves and titration.
A more persuasive chemical equilibrium report can be produced by explaining why K varied among multiple conditions and the causes of differences from literature values.