Name: reading reaction progress with integrated laws
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The Integrated Rate Law: The Dependence of Concentration on Time

משוואת קצב משמשת לקביעת קצב התגובה מריכוזי המגיזים ואת קבועי הקצב ניתן להמיר למשוואות קצב המתארות את התלות של קצב התגובה בריכוז המגיב ובזמן. משוואות קצב אלה יכולות לשמש כדי ללמוד כמה לאט או כמה מהר מגיב נצרך או כמה זמן נדרש כדי להגיע לחצי מהריכוז של מגיב. כדי להתחיל, בחנו את משוואת הקצב הדיפרנציאלית, המבטאת את קצב התגובה כשינוי בריכוז המגיב בפרק זמן מסוים.חישוב אינטגרל על משוואת הקצב נותן את הפיתרון למשוואת הקצב הדיפרנציאלית המבטא את קצב התגובה כקשר בין הריכוז הההתחלתי של מגיב לבין הריכוז שלו לאחר פרק זמן מסוים. אינטגרל משוואת הקצב תלוי בסדר התגובה הכוללת ולכן משתנה עבור כל סוג תגובה. יחד עם זאת, ללא תלות בסדר הכולל, כל פתרונות משוואות הקצב הם כמשוואות ליניאריות סטנדרטיות עם רכיבי y, m, x ו-b הניתנות לשרטוט לכדי יצירית קו ישר.באינטגרל משוואת קצב דיפרנציאלית מסדר אפס t הוא ריכוז המגיב בזמן _t k הוא קבוע הקצב, t הוא זמן ו-0_[A]הוא ריכוז המגיב ההתחלתי. בתגובה מסדר אפס, שרטוט של ריכוז המגיב כפונקציה של זמן תייצר קו ישר. השיפוע הוא הערך השלילי של קבוע הקצב והחיתוך עם ציר ה-y הוא ריכוז המגיב ההתחלתי.בתגובה מסדר ראשון, הלוגריתם הטבעי של ריכוז מגיב המשורטט כפונקציה של זמן נותן קו ישר. השיפוע תואם לערך השלילי של קבוע הקצב, בעוד שערך נקודת החיתוך עם ה-y נותן את הלוגריתם הטבעי של ריכוז המגיב ההתחלתי. לפי האינטגרל של משוואת הסדר השני, שרטוט הערך ההופכי של ריכוז המגיב לעומת הזמן נותן קו ישר.השיפוע שווה לקבוע הקצב, והערך בנקודת החיתוך עם ציר ה-y מייצג את הערך ההופכי של ריכוז המגיב ההתחלתי. את הסדר הכולל של התגובה ניתן לזהות באמצעות נתונים קינטיים נסיוניים על ידי שרטוט האינטגרלים של משוואות הקצב השונות. רק השרטוט שיניב גרף הליניארי תואם לסדר התגובה הכולל.ניתוח נוסף מאפשר לקבוע את קבוע הקצב ואת ריכוז המגיב בכל זמן נתון.

חוק הקצבים המשולבים: תלות הריכוז בזמן

<div style="direction:rtl">

בעוד שחוק הקצב הדיפרנציאלי מתייחס לקצב ולריכוזים של מגיבים, צורה שנייה של חוק קצב הנקראת חוק הקצב המשולב מתייחסת לריכוזים של מגיבים וזמן. ניתן להשתמש בחוקי קצב משולבים כדי לקבוע את כמות המגיב או התוצר הקיים לאחר פרק זמן או להעריך את הזמן הנדרש לתגובה להתקדם במידה מסוימת. לדוגמה, חוק קצב משולב עוזר לקבוע את משך הזמן שבו יש לאחסן חומר רדיואקטיבי כדי שהרדיואקטיביות שלו תתפרק לרמה בטוחה. 

באמצעות ניתוח מתמטי, ניתן לשלב את חוק הקצב הדיפרנציאלי עבור תגובה כימית ביחס לזמן כדי לתת משוואה המתייחסת בין כמות המגיב/תוצר לזמן שחלף של התגובה. 

תגובות מסדר ראשון 

שילוב של חוק הקצב לתגובה פשוטה מסדר ראשון (קצב = k[A]) מביאה למשוואה המתארת את השונות בריכוז המגיבים עם הזמן: 
<img alt="Eq1" src="/files/ftp_upload/11377/11377_Equation 1.png" style="height: 50px;"> 

כאן, [A]t הוא הריכוז של A בכל זמן t, [A]0 הוא הריכוז ההתחלתי של A, ו-k הוא קבוע הקצב מהסדר הראשון. מטעמי נוחות מתמטית, משוואה זו מסודרת מחדש לצורה המראה תלות לינארית של ריכוז בזמן שלובשת צורה של משוואת קו ישר (y = mx + b):
<img alt="Eq2" src="/files/ftp_upload/11377/11377_Equation 2.png" style="height: 50px;"> 

המשוואה מציעה שחלקה של ln[A]t מול t עבור תגובה מסדר ראשון היא ישר עם שיפוע של −k ו-y-חיזור של ln[A]0. אם קבוצה של נתוני קצב משורטטת בצורה זו אך אינה מביאה לקו ישר, התגובה אינה מסדר ראשון ב-A.

תגובות מסדר שני 

חוק הקצב הדיפרנציאלי לתגובה פשוטה מסדר שני הוא קצב =k[A]2, וחוק הקצב המשולב הוא:
<img alt="Eq3" src="/files/ftp_upload/11377/11377_Equation 3.png" style="text-align: center; height: 70px;"> 

חוק הקצב המשולב מסדר שני לובש גם את הצורה של המשוואה עבור קו ישר. לפי המשוואה, חלקה של 1/[A]t מול t לתגובה מסדר שני היא ישר עם שיפוע של k ו-y-חתך של 1/[A]0. אם הגרף אינו קו ישר, הרי שהתגובה אינה מסדר שני. 

תגובות מסדר אפס 

עבור תגובות מסדר אפס, חוק הקצב הדיפרנציאלי הוא קצב = k.. תגובה מסדר אפס מציגה קצב תגובה קבוע, ללא קשר לריכוז המגיבים שלה. קינטיקה מסדר אפס נצפית עבור תגובות מסוימות רק בתנאים ספציפיים מסוימים. אותן תגובות מפגינות התנהגויות קינטיות שונות כאשר התנאים הספציפיים אינם מתקיימים, ומסיבה זו, משתמשים לפעמים במונח הזהיר יותר מסדר אפס.

חוק הקצב המשולב לתגובה מסדר אפס הוא גם פונקציה לינארית, בצורת y = mx + b: 
<img alt="Eq4" src="/files/ftp_upload/11377/11377_Equation 4.png" style="height: 50px;"> 

גרף של [A] לעומת זמן t עבור תגובה מסדר אפס היא קו ישר עם שיפוע של −k וחתך y של [A]0. 

טקסט זה מותאם מ <a href="https://openstax.org/books/chemistry-2e/pages/12-4-integrated-rate-laws">Openstax, Chemistry 2e, Section 12.4: Integrated Rate Laws.</a></div>

בעוד חוק התעריף הדיפרנציאלי מתייחס לשיעור ולריכוזים של מגיבים, צורה שנייה של חוק התעריף הנקראת חוק השיעור המשולב מתייחסת ...

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reading reaction progress with integrated laws

Integrated rate laws connect reaction concentration to time. They help show how much reactant or product is present after a certain period. They can also help estimate how long a reaction must run to reach a given extent. A common use is finding how long radioactive material must be stored before its radioactivity drops to a safe level.

These equations come from the differential rate law after it is integrated with respect to time. For a first-order reaction, where rate = k[A], the integrated form relates the concentration of A at time t to the initial concentration [A]0 and the first-order rate constant k. The equation can be rearranged into the straight-line form y = mx + b. That means a plot of ln[A]t versus t should be linear. The slope is −k, and the y-intercept is ln[A]0. If the data do not make a straight line, the reaction is not first order in A.

Second-order reactions follow a different pattern. For a simple second-order reaction, the differential rate law is rate = k[A]2. Its integrated rate law also gives a straight-line relationship. A plot of 1/[A]t versus time t should be linear, with slope k and y-intercept 1/[A]0. If the plot is not straight, the reaction is not second order.

Zero-order reactions behave differently again. For these reactions, the rate is rate = k, so the reaction rate stays constant even when reactant concentration changes. Some reactions show zero-order kinetics only under certain conditions, so the term pseudo-zero-order is sometimes used. The integrated rate law is also linear here. A plot of [A] versus time t is a straight line with slope −k and y-intercept [A]0.

Reading Reaction Progress with Integrated Laws

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reaction rate

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the integrated rate law: the dependence of concentration on time

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half-life of a reaction

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key terms and concepts

Learn how integrated rate laws connect concentration and time for first-order, second-order, and zero-order reactions, including straight-line plots and slopes.

Reading Reaction Progress with Integrated Laws | JoVE K12

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Explain how atomic energy levels and emission spectra present evidence for the wave particle duality.

IPC.7.E

Science concepts. The student knows that relationships exist between the structure and properties of matter. The student is expected to:

Reading Reaction Progress with Integrated Laws

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