Name: estimating equilibrium with small changes
Uploaded: 2020-10-19T09:47:37.000Z
Duration: 2 min 20 s
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The Small x Assumption

The small x assumption is an approximation that can be used under certain conditions to simplify solving the equilibrium expression and to avoid using the quadratic formula. 
If the equilibrium constant for a reaction is small and the initial concentrations of reactants are sufficiently high, it is possible that only a small amount of reactants will be converted into products.
When these conditions are met, the equilibrium concentrations can be approximated using the assumption that the change in the initial concentrations is negligible. However, it must be confirmed that the change in concentration, x, is less than 5% of the initial concentrations to show the the assumption was valid.
The decomposition of 0.66 molar carbonyl dichloride gas produces carbon monoxide and chlorine gas, and the equilibrium constant for this reaction is 2.2 &#215; 10&#8722;10.
To calculate the equilibrium concentrations, the values are tabulated in the ICE table along with the change and equilibrium concentrations.
Substituting the equilibrium concentrations in the expression, K equals x squared divided by 0.66 minus x. As K is very small, the change in the initial concentration, x, of carbonyl dichloride is expected to be negligible.
Therefore, 0.66 minus x can be approximated to 0.66. When solved, x equals 1.2 &#215; 10&#8722;5 molar.
Here, x is only 0.0018 percent of the initial 0.66 molar concentration of carbonyl dichloride, which is far less than the 5% maximum allowed. Thus, the small x assumption is valid here.
Using the value for x in the ICE table, the equilibrium concentration of carbonyl dichloride is still 0.66 molar with significant figures, while the concentrations of carbon monoxide and chlorine are both 1.2 &#215; 10&#8722;5 molar.

the small x assumption

If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration may occur if the initial concentrations of reactants are high and the Kc value is small. In such circumstances, the equilibrium concentration is approximately equal to its initial concentration. This estimation can be used to simplify the equilibrium calculations by assuming that some equilibrium concentrations are equal to the initial concentrations. However, to make this assumption, the change in the concentration of a weak acid or base, i.e., x, must be less than 5% of its initial concentration. If x is more than 5%, then the quadratic formula needs to be used to solve the equilibrium equation. 
Calculation of Equilibrium Concentrations Using an Algebra-Simplifying Assumption
What are the concentrations at equilibrium of a 0.15 M solution of HCN?
<img alt="Quantum entanglement circuit diagram with Hadamard and CNOT gates, illustrating qubit operations." src="/files/ftp_upload/11477/11477_Equation_1.png" style="text-align: center; height: 50px;" data-altupdatedat="1754904545000" data-alt="Polymerase chain reaction diagram showing DNA amplification process with thermal cycles." />
Using x to represent the concentration of each product at equilibrium gives this ICE table.
<table style="margin-left: auto; margin-right: auto;">
	<tbody>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;"></td>
			<td style="color: black; border: 1px solid black; text-align: center;"> HCN (aq) </td>
			<td style="color: black; border: 1px solid black; text-align: center;">H+ (aq)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">CN&#8722; (aq)</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Initial Concentration (M)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.15</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Change (M)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">&#8722;x</td>
			<td style="color: black; border: 1px solid black; text-align: center;">+x</td>
			<td style="color: black; border: 1px solid black; text-align: center;">+x</td>
		</tr>
		<tr>
			<td style="color: black; border: 1px solid black; text-align: center;">Equilibrium Concentration (M)</td>
			<td style="color: black; border: 1px solid black; text-align: center;">0.15 &#8722; x</td>
			<td style="color: black; border: 1px solid black; text-align: center;">x</td>
			<td style="color: black; border: 1px solid black; text-align: center;">x</td>
		</tr>
	</tbody>
</table>
Substitute the equilibrium concentration terms into the Kc expression
<img alt="Static equilibrium formula &#931;Fx=0, &#931;Fy=0 vector diagram with force components for physics analysis." src="/files/ftp_upload/11477/11477_Equation_2.png" style="text-align: center; height: 95px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium; &#931;Fx=0, &#931;Fy=0, &#931;M=0; diagram with applied forces and moments; vector analysis." />
rearrange to the quadratic form and solve for x
<img alt="static equilibrium formula diagram, force vectors &#931;Fx=0, illustrated for physics education and research" src="/files/ftp_upload/11477/11477_Equation_3.png" style="text-align: center; height: 95px;" data-altupdatedat="1754904545000" data-alt="Static equilibrium, &#931;Fx=0, MA=0, diagram, illustrating force balance on an object in mechanical stability." />
Thus, [H+] = [CN&#8211;] = x = 8.6 &#215; 10&#8211;6 M and [HCN] = 0.15 &#8211; x = 0.15 M.
Note in this case that the change in concentration is significantly less than the initial concentration (a consequence of the small K), and so the initial concentration experiences a negligible change:
<img alt="Static equilibrium diagram, &#931;Fx=0, &#931;Fy=0, beam balance, force vectors, equilibrium analysis." src="/files/ftp_upload/11477/11477_Equation_4.png" style="text-align: center; height: 50px;" data-altupdatedat="1754904545000" data-alt="Nernst equation diagram illustrating electrochemical cell potential with ion concentration." />
This approximation allows for a more expedient mathematical approach to the calculation that avoids the need to solve for the roots of a quadratic equation:
<img alt="Static equilibrium equations diagram, &#931;Fx=0, &#931;Fy=0, moment MA=0, truss analysis setup." src="/files/ftp_upload/11477/11477_Equation_5.png" style="text-align: center; height: 135px;" data-altupdatedat="1754904545000" data-alt="Chemical equilibrium equation CH3COOH &#8652; CH3COO&#8722; + H+. Acid ionization concept diagram." />
The value of x calculated is, indeed, much less than the initial concentration
<img alt="DNA transcription process diagram, showing DNA to RNA conversion, nucleotides, and base pairing." src="/files/ftp_upload/11477/11477_Equation_6.png" style="text-align: center; height: 50px;" data-altupdatedat="1754904545000" data-alt="Spectroscope diagram illustrating light dispersion and measurement in optical studies." />
and so the approximation was justified. If this simplified approach were to yield a value for x that did not justify the approximation, the calculation would need to be repeated without making the approximation.
This text has been adapted from <a href="https://openstax.org/books/chemistry-2e/pages/13-4-equilibrium-calculations">Openstax, Chemistry 2e, Section 13.4 Equilibrium Calculations</a><a href="https://openstax.org/books/chemistry-2e/pages/13-4-equilibrium-calculations">.</a>

If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration may ...

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estimating equilibrium with small changes

A small equilibrium constant can make equilibrium calculations easier. In these reactions, the equilibrium position favors the reactants. When the initial reactant concentration is high and the Kc value is small, the change in concentration may be negligible. In that case, the equilibrium concentration is close to the initial concentration.

This idea helps simplify weak acid and weak base problems. Chemists can assume that some equilibrium concentrations are equal to the initial concentrations. The change in concentration is written as x. To use the shortcut, x must be less than 5% of the initial concentration. If x is greater than 5%, the quadratic formula is needed instead.

A solution of HCN shows how the approximation works. For a 0.15 M HCN solution, an ICE table gives HCN, H+, and CN− with initial concentrations of 0.15, 0, and 0 M. The change is −x for HCN and +x for each product, so the equilibrium concentrations are 0.15 − x, x, and x. Substituting these terms into the Kc expression and solving gives [H+] = [CN−] = 8.6 × 10−6 M and [HCN] = 0.15 M.

In this example, the calculated x is much smaller than the initial concentration. That means the initial HCN concentration changes very little. The shortcut is justified, and the calculation avoids solving a quadratic equation. If the result had not met the 5% rule, the calculation would need to be repeated without the approximation.

Estimating Equilibrium with Small Changes

estimating-equilibrium-with-small-changes

the equilibrium constant

Chemical Equilibrium and Kc Rules

Chemical equilibrium and the equilibrium constant Kc describe how reactants and products are balanced in a reaction. In the oxidation of sulfur dioxide, a mixture that starts with reactants only has no product at first. As the reaction moves forward toward equilibrium, reactant concentrations decrease and product concentration increases. When equilibrium is reached, the concentrations of both stay constant. A reaction can also begin with only products present. In that case, the reaction moves...

le chatelier's principle: changing concentration

le-chatelier-s-principle-changing-concentration

Shifting Equilibrium by Changing Concentration

Chemical equilibrium is a dynamic balance. At equilibrium, the forward and reverse reactions happen at equal rates. If the system is stressed in a way that changes one rate more than the other, it is no longer at equilibrium and will shift until balance is restored. This behavior is described by Le Châtelier’s principle. Changes in concentration are a major stress on an equilibrium system. Reaction rates depend mainly on concentration and temperature. For that reason, changing the amount of a...

homogeneous equilibria for gaseous reactions

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Relating Kc and Kp in Gas Equilibria

Gas-phase equilibrium can be written using either concentration or pressure. The equilibrium constant may be shown as Kc, which uses molar concentration, or as Kp, which uses partial pressure of the reactants and products. The link between Kc and Kp comes from the ideal gas equation and the definition of molarity. Molarity is the number of moles divided by volume. In the ideal gas equation, P stands for partial pressure, V for volume, n for number of moles, R for the gas constant, and T for...

calculating the equilibrium constant

Finding Kc from Equilibrium Data

The equilibrium constant, Kc, can be found from the equilibrium concentrations of reactants and products. When those concentrations are known, the calculation is straightforward. The values are placed into the Kc expression for the reaction. One example uses nitrogen dioxide and dinitrogen tetroxide. When 0.10 mol NO2 is added to a 1.0-L flask at 25 °C, the concentrations change until equilibrium is reached. At equilibrium, [NO2] is 0.016 M and [N2O4] is 0.042 M, and these values are used to...

reaction quotient

Using Q to Track Equilibrium Changes

Reaction quotient Q helps track how far a reversible reaction has moved toward equilibrium. For a balanced reaction such as m A + n B ⇌ x C + y D, Q comes directly from the stoichiometric coefficients in the equation. When concentrations are used, the expression is written as Qc. For gases, a similar form uses partial pressures. These expressions use relative concentration and pressure values, not absolute values. That means reaction quotients are dimensionless and have no units. As a reaction...

calculating equilibrium concentrations

Equilibrium Concentrations from an ICE Table

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dynamic equilibrium

Chemical Equilibrium in Reversible Reactions

Chemical equilibrium in reversible reactions happens when a reaction moves in both directions at the same time. A reversible reaction uses a double arrow to show that products can turn back into reactants. At equilibrium, the concentrations of reactants and products stay constant over time because the forward and reverse reaction rates are equal. The video uses the reversible decomposition of colorless dinitrogen tetroxide to brown nitrogen dioxide as an example. At time zero, there is finite...

le chatelier's principle: changing volume (pressure)

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Gas Equilibrium Under Volume Changes

Gas-phase equilibrium can shift when the volume of the system changes. In these reactions, a change in pressure changes concentration because the partial pressure of an ideal gas is proportional to its molar concentration. That means pressure changes affect equilibrium in the same way as concentration changes. When the volume of a gas mixture changes, the concentrations of reactants and products change together. Since every gas in the equilibrium occupies the same space, the same volume change...

le chatelier's principle: changing temperature

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Le Chatelier's Principle in Heating and Cooling

Le Chatelier's principle helps explain how chemical equilibrium responds to heating and cooling. A system at equilibrium will shift to restore balance after a temperature change, but the equilibrium constant, K, will not stay the same. Instead, the mixture reaches a new equilibrium with a different composition. This temperature effect is different from a concentration change, which shifts equilibrium without changing K. To understand why temperature matters, the reaction can be written with...

A small equilibrium constant can make equilibrium calculations easier. In these reactions, the equilibrium position favors the reactants. When the initial reactant concentration is high and the Kc value is small, the change in concentration may be negligible. In that case, the equilibrium concentration is close to the initial concentration. This idea helps simplify weak acid and weak base problems. Chemists can assume that some equilibrium concentrations are equal to the initial concentrations.

key terms and concepts

Learn how a small equilibrium constant lets students use the 5% rule and an ICE table to estimate equilibrium concentrations without solving a quadratic.

Estimating Equilibrium with Small Changes | JoVE K12

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Estimating Equilibrium with Small Changes

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