Name: estimating equilibrium with small changes
Uploaded: 2020-10-19T09:47:37.000Z
Duration: 2 min 20 s
Channel: JoVE

K12 Standards

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North Carolina K-12 Science Standard Course of Study

9-12

Chemistry

Matter and its Interactions

PS.Chm.5.2

The Small x Assumption

O pequeno x pressuposto é uma aproximação que pode ser utilizada em certas condições para simplificar a resolução da expressão de equilíbrio e para evitar a utilização da fórmula quadrática. Se a constante de equilíbrio para uma reação é pequena e as concentrações iniciais dos reagentes são suficientemente elevadas, é possível que apenas uma pequena quantidade de reagentes seja convertida em produtos. Quando estas condições forem cumpridas, as concentrações de equilíbrio podem ser aproximadas utilizando a hipótese de que a alteração nas concentrações iniciais é insignificante.No entanto, deve ser confirmada que a alteração na concentração, x, é inferior a 5 por cento das concentrações iniciais para mostrarem que a suposição era válida. A decomposição de 0, 66 de dicloreto de carbonilo de gás molar produz monóxido de carbono e gás cloro, e o equilíbrio constante para esta reação é de 2, 2 10-¹⁰. Para calcular as concentrações de equilíbrio, os valores são tabulados na Tabela ICE juntamente com a alteração e concentrações de equilíbrio.Substituindo as concentrações de equilíbrio na expressão, K igual a x ao quadrado dividido por 0, 66 menos x. Como K é muito pequeno, a alteração na concentração inicial, x, de dicloreto de carbonilo espera-se que seja insignificante. Portanto, 0, 66 menos x pode ser aproximado a 0, 66.Quando resolvido, x é igual a 1, 2 10-⁵ molar. Aqui, x é apenas 0, 0018 por cento da concentração 0, 66 molar inicial de dicloreto de carbonilo, que é muito menos do que do 5%máximo permitido. Assim, o pequeno x pressuposto é válido aqui.Utilizando o valor para x na tabela do ICE, a concentração de equilíbrio de dicloreto de carbonilo é ainda 0, 66 molar com números significativos, enquanto as concentrações molares de monóxido de carbono e cloro são ambas 1, 2 10-⁵.

<ol>
	<li>Lim, Kieran F. "Usando calculadoras gráficas e planilhas em química: resolvendo problemas de equilíbrio." Jornal de Educação Química 85, nº 10 (2008): 1347. <a href="https://pubs.acs.org/doi/pdf/10.1021/ed085p1347">https://pubs.acs.org/doi/pdf/10.1021/ed085p1347</a></li>
</ol>

a suposição do x pequeno

Se uma reação tiver uma constante de equilíbrio pequena, a posição de equilíbrio favorece os reagentes. Em tais reações, pode ocorrer uma alteração negligenciável da concentração se as concentrações iniciais de reagentes forem elevadas e o valor de Kc for pequeno. Nessas circunstâncias, a concentração de equilíbrio é aproximadamente igual à sua concentração inicial.  Esta estimativa pode ser utilizada para simplificar os cálculos de equilíbrio, partindo do princípio de que algumas concentrações de equilíbrio são iguais às concentrações iniciais. No entanto, para fazer esta suposição, a variação da concentração de ácido ou base fraco, ou seja, x, deve ser inferior a 5% da sua concentração inicial. Se x for superior a 5%, então a fórmula quadrática precisa ser usada para resolver a equação de equilíbrio.  
Cálculo das Concentrações de Equilíbrio Utilizando uma Suposição para Simplificação da Álgebra
Quais são as concentrações em equilíbrio de uma solução de 0,15 M de HCN?
<img alt="Eq1" src="/files/ftp_upload/11477/11477_Equation_1.png" style="text-align: center; height: 50px;" />
Usando x para representar a concentração de cada produto em equilíbrio obtemos esta tabela ICE.
<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− (aq)</td>
</tr>
<tr>
<td style="color: black; border: 1px solid black; text-align: center;">Concentração Inicial (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;">Alteração (M)</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>
<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;">Concentração em Equilíbrio (M)</td>
<td style="color: black; border: 1px solid black; text-align: center;">0,15 − 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>
Substituindo os termos de concentração de equilíbrio na expressão de Kc
<img alt="Eq2" src="/files/ftp_upload/11477/11477_Equation_2.png" style="text-align: center; height: 95px;" />
reorganizando para a forma quadrática e resolvendo para x
<img alt="Eq3" src="/files/ftp_upload/11477/11477_Equation_3.png" style="text-align: center; height: 95px;" />
Assim, [H+] = [CN–] = x = 8,6 × 10–6 M e [HCN] = 0,15 – x = 0,15 M.
Note, neste caso, que a alteração da concentração é significativamente inferior à concentração inicial (consequência do K pequeno), pelo que a concentração inicial sofre uma alteração negligenciável:
<img alt="Eq4" src="/files/ftp_upload/11477/11477_Equation_4.png" style="text-align: center; height: 50px;" />
Essa aproximação permite uma abordagem matemática mais rápida para o cálculo que evita a necessidade de resolver para as raízes de uma equação quadrática:
<img alt="Eq5" src="/files/ftp_upload/11477/11477_Equation_5.png" style="text-align: center; height: 135px;" />
O valor de x calculado é, de facto, muito inferior à concentração inicial
<img alt="Eq6" src="/files/ftp_upload/11477/11477_Equation_6.png" style="text-align: center; height: 50px;" />
e portanto a aproximação foi justificada. Se esta abordagem simplificada produzisse um valor para x que não justificasse a aproximação, o cálculo teria de ser repetido sem fazer a aproximação.
Este texto foi adaptado de <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>

A Suposição do x Pequeno

Se uma reação tem uma pequena constante de equilíbrio, a posição de equilíbrio favorece os reagentes. Em tais reações, uma mudança insignificante na concentração pode ...

Education

JoVE Core

Química

Chemical Equilibrium

Equilíbrio Químico

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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

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le chatelier's principle: changing concentration

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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...

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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...

the small x assumption

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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