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

La suposición de la x pequeña es una aproximación que se puede usar bajo ciertas condiciones para simplificar la resolución de la expresión de equilibrio y evitar usar la fórmula cuadrática. Si la constante de equilibrio para una reacción es pequeña y las concentraciones iniciales de reactivos son suficientemente altas, es posible que solo una pequeña cantidad de reactivos se convierta en productos. Cuando se cumplen estas condiciones, las concentraciones de equilibrio se pueden aproximar asumiendo que el cambio en las concentraciones iniciales es insignificante.Sin embargo, debe confirmarse que el cambio en la concentración, x, es menos del 5 por ciento de las concentraciones iniciales para demostrar que la suposición era válida. La descomposición de gas dicloruro de carbonilo 0, 66 molar produce monóxido de carbono y gas cloro, y la constante de equilibrio para esta reacción es 2, 2 x 10⁻¹⁰. Para calcular las concentraciones de equilibrio, los valores se tabulan en la tabla ICE junto con el cambio y las concentraciones de equilibrio.La sustitución de las concentraciones de equilibrio K es igual a x al cuadrado dividido por 0, 66 menos x. Como K es muy pequeño, se espera que el cambio en la concentración inicial, x, de dicloruro de carbonilo sea insignificante. Por lo tanto, 0, 66 menos x se puede aproximar a 0, 66.Cuando se resuelve, x es igual a 1, 2 10⁻⁵ molar. Aquí, x es sólo el 0, 0018 por ciento de la concentración inicial de 0, 66 molar de dicloruro de carbonilo, que es mucho menor que el máximo permitido del 5 porciento. Por tanto, la suposición de la x pequeña es válida aquí.Utilizando el valor de la x en la tabla ICE, la concentración de equilibrio de dicloruro de carbonilo sigue siendo 0, 66 molar con cifras significativas, mientras que las concentraciones de monóxido de carbono y cloro son 1, 2 10⁻⁵ molar.

<ol>
	<li>Lim, Kieran F. "Uso de calculadoras gráficas y hojas de cálculo en química: resolución de problemas de equilibrio". Revista de Educación 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>

el supuesto x es pequeña

Si una reacción tiene una constante de equilibrio pequeña, la posición de equilibrio favorece a los reactivos. En tales reacciones, puede producirse un cambio insignificante en la concentración si las concentraciones iniciales de reactivos son altas y el valor de Kc es pequeño. En dichas circunstancias, la concentración en equilibrio es aproximadamente igual a su concentración inicial.   Esta estimación se puede utilizar para simplificar los cálculos en equilibrio asumiendo que algunas concentraciones en equilibrio son iguales a las concentraciones iniciales. Sin embargo, para hacer esta suposición, el cambio en la concentración de un ácido o base débil, es decir, x, debe ser inferior al 5% de su concentración inicial. Si x es más del 5%, entonces la fórmula cuadrática necesita ser utilizada para resolver la ecuación de equilibrio.   
 Cálculo de concentraciones de equilibrio utilizando un supuesto de simplificación algebraica
 ¿Cuáles son las concentraciones en equilibrio de una solución 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 la concentración de cada producto en equilibrio se obtiene esta tabla 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 (ac)           </td>
 <td style="color: black; border: 1px solid black; text-align: center;">H+ (ac)</td>
 <td style="color: black; border: 1px solid black; text-align: center;">CN− (ac)</td>
 </tr>
 <tr>
 <td style="color: black; border: 1px solid black; text-align: center;">Concentración 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;">Cambiar (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;">Concentración en equilibrio (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>
 Sustituya los términos de concentración en equilibrio en la expresión Kc
 <img alt="Eq2" src="/files/ftp_upload/11477/11477_Equation_2.png" style="text-align: center; height: 95px;" />
 reorganice a la forma cuadrática y resuelva para hallar x
 <img alt="Eq3" src="/files/ftp_upload/11477/11477_Equation_3.png" style="text-align: center; height: 95px;" />
 Así, [H+] = [CN−] = x = 8,6 × 10−6 M y [HCN] = 0,15 − x = 0,15 M.
 Observe en este caso que el cambio en la concentración es significativamente menor que la concentración inicial (una consecuencia de la K pequeña), y por lo tanto la concentración inicial experimenta un cambio insignificante:
 <img alt="Eq4" src="/files/ftp_upload/11477/11477_Equation_4.png" style="text-align: center; height: 50px;" />
 Esta aproximación permite un enfoque matemático más conveniente para el cálculo que evita la necesidad de resolver las raíces de una ecuación cuadrática:
 <img alt="Eq5" src="/files/ftp_upload/11477/11477_Equation_5.png" style="text-align: center; height: 135px;" />
 El valor calculado de x es, de hecho, mucho menor que la concentración inicial
 <img alt="Eq6" src="/files/ftp_upload/11477/11477_Equation_6.png" style="text-align: center; height: 50px;" />
 y así la aproximación estaba justificada. Si este enfoque simplificado tuviera arrojara un valor para x que no justificara la aproximación, el cálculo tendría que repetirse sin hacer la aproximación.
 Este texto ha sido adaptado de<a href="https://openstax.org/books/chemistry-2e/pages/13-4-equilibrium-calculations"> Openstax, Química 2e, Sección 13.4 Cálculos de Equilibrio</a><a href="https://openstax.org/books/chemistry-2e/pages/13-4-equilibrium-calculations">.</a>

El Supuesto x es Pequeña

Si una reacción tiene una constante de equilibrio pequeña, la posición de equilibrio favorece a los reactivos. En tales reacciones, un cambio insignificante en la concentración puede ...

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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

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Shifting Equilibrium by Changing Concentration

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le chatelier's principle: changing temperature

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Le Chatelier's Principle in Heating and Cooling

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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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