Name: estimating equilibrium with small changes
Uploaded: 2020-10-19T09:47:37.000Z
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PS.Chm.5.2

The Small x Assumption

הנחת ה-x הקטן היא קירוב שבנסיבות מסוימות יכול לשמש כדי לפשט פתרון של ביטוי שיווי משקל ולהימנע משימוש בנוסחת השורשים הריבועית. אם קבוע שיווי המשקל של תגובה הוא קטן וריכוז המגיבים הראשוני גבוה במידה מספקת, ייתכן שרק כמות קטנה של מגיבים תומר לתוצרים. כשהתנאים הללו מתקיימים, ניתן להעריך את ריכוזי שיווי המשקל באמצעות ההנחה שהשינוי בריכוזים הראשוניים זניח.אך יש לאשר שההבדל בריכוז, X, נמוך מ-5 אחוז מהריכוזים ההתחלתיים כדי להראות שההנחה הייתה תקפה. הפירוק של 0.66 מולר גז קרבוניל דיכלוריד מייצר פחמן חד חמצני וגז כלור וקבוע שיווי המשקל של התגובה הזו הוא 2.2 10 בחזקת מינוס 10. כדי לחשב את ריכוזי שיווי המשקל, הערכים מתוארים בטבלת ICE יחד עם השינוי וריכוזי שיווי המשקל.בהצבת הריכוזים בשיווי המשקל בביטוי, K שווה x בריבוע חלקי 0.66 מינוס x. כיוון ש-K הוא קטן מאוד, השינוי בריכוז ההתחלתי, x, של קרבוניל דיכלוריד צפוי להיות זניח. לכן ניתן לאמוד את 0.66 פחות x ב-0.66.הפתרון מגלה ש-x שווה 1.2 כפול 10 בחזקת מינוס 5 מולר. כאן x הוא רק 0.0018 אחוז מריכוז קרבוניל דיכלוריד המקורי של 0.66 מולר, נמוך בהרבה מהמקסימום המותר, חמישה אחוז. לכן הנחת ה-x הקטן תקפה כאן.כשמשתמשים בערך של x בטבלת ICE, ריכוז שיווי המשקל של קרבוניל דיכלוריד עדיין עומד על 0.66 מולר עם ספרות משמעותיות, בעוד שהריכוז של פחמן חד חמצני וכלור עומדים שניהם על 1.2 כפול 10 בחזקת מינוס 5 מולר.

<ol>
	<li>לים, קירן פ. "שימוש במחשבונים גרפיים וגיליונות אלקטרוניים בכימיה: פתרון בעיות שיווי משקל". כתב עת לחינוך כימי 85, מס '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>

הנחת ה-x הקטן

<div style="direction:rtl">

 כאשר לתגובה יש קבוע שיווי משקל קטן, מיקום שיווי המשקל מעדיף את המגיבים. בתגובות כאלה, שינוי זניח בריכוז יכול להתרחש אם הריכוזים ההתחלתיים של המגיבים גבוהים וערך Kc קטן. בנסיבות כאלה, ריכוז שיווי המשקל שווה בערך לריכוז ההתחלתי שלו. ניתן להשתמש באומדן זה כדי לפשט את חישובי שיווי המשקל על ידי הנחה שכמה ריכוזי שיווי משקל שווים לריכוזים ההתחלתיים. עם זאת, כדי להניח הנחה זו, השינוי בריכוז של חומצה חלשה או בסיס, כלומר, x , .i.e פחות מ-5% מהריכוז ההתחלתי שלהם. אם x הוא יותר מ-5%, יש להשתמש בנוסחה הריבועית כדי לפתור את משוואת שיווי המשקל.

 חישוב ריכוזי שיווי משקל באמצעות הנחת אלגברה-מפשטת 

 מהם ריכוזי שיוויי המשקל של תמיסה של 0.15 M HCN? <img alt="Eq1" src="/files/ftp_upload/11477/11477_Equation_1.png" style="text-align: center; height: 50px;"> 

 שימוש ב-x כדי לייצג את הריכוז של כל תוצר בשיווי משקל מניב את טבלת ה-ICE הזו. <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;">ריכוז התחלתי (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;">שנה (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;">ריכוז שיווי משקל (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> 

 החלף את מונחי ריכוז שיווי המשקל בביטוי Kc <img alt="Eq2" src="/files/ftp_upload/11477/11477_Equation_2.png" style="text-align: center; height: 95px;"> 

 מסדרים אותו מחדש לצורת הריבוע ופותרים עבור x <img alt="Eq3" src="/files/ftp_upload/11477/11477_Equation_3.png" style="text-align: center; height: 95px;"> 

 לפיכך,[H+] = [CN–] = x = 8.6 ×10–6 M ו-[HCN] = 0.15 – x = 0.15 M. 

 במקרה זה, שימו לב שהשינוי בריכוז קטן משמעותית מהריכוז הראשוני (תוצאה של ה-K הקטן) ולכן הריכוז הראשוני עובר שינוי זניח: <img alt="Eq4" src="/files/ftp_upload/11477/11477_Equation_4.png" style="text-align: center; height: 50px;"> 

 קירוב זה מאפשר גישה מתמטית נוחה יותר לחישוב הנמנעת מחיפוש של שורשים של המשוואה הריבועית: <img alt="Eq5" src="/files/ftp_upload/11477/11477_Equation_5.png" style="text-align: center; height: 135px;"> 

 הערך המחושב של x למעשה קטן בהרבה מהריכוז ההתחלתי <img alt="Eq6" src="/files/ftp_upload/11477/11477_Equation_6.png" style="text-align: center; height: 50px;"> 

 ולכן הגישה הייתה מוצדקת. אם גישה מפושטת זו מייצרת ערך עבור x שאינו מצדיק את הקירוב, יהיה צורך לחזור על החישוב מבלי לבצע את הקירוב. 

 טקסט זה מותאם מ<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></div>

אם לתגובה יש קבוע שיווי משקל קטן, מיקום שיווי המשקל מעדיף את המגיבים. בתגובות כאלה, שינוי זניח בריכוז עלול ...

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Chemical Equilibrium

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KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

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...

homogeneous equilibria for gaseous reactions

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Relating Kc and Kp in Gas Equilibria

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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...

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...

dynamic equilibrium

Chemical Equilibrium in Reversible Reactions

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calculating equilibrium concentrations

Equilibrium Concentrations from an ICE Table

Equilibrium concentrations can be found from initial concentrations and an equilibrium constant. This skill matters in many areas of science and technology, including pharmaceutical product formulation and dosing. After a drug is ingested or injected, it may take part in several chemical equilibria that change its final concentration in the body system of interest. Scientists use these quantitative equilibrium relationships to help calculate a dosage that will produce the desired therapeutic...

le chatelier's principle: changing volume (pressure)

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Gas Equilibrium Under Volume Changes

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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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