How to Derive Half-Life Equations for Chemical Reactions

TL;DR

To derive half-life equations, start with the formulas for zero-order, first-order, and second-order reactions. For first-order reactions, the half-life is given by ln(2)/k; for zero-order, it's A_initial/2k; and for second-order, it's 1/(A_initial * k). The half-life is inversely related to the rate constant for all reaction types.

Transcript

in this video we're going to talk about how we can derive the equation for half-life so we're going to start with this equation a is equal to p e raised to the RT so this equation is associated with exponential growth particularly when that growth is compounded continuously p is the principle a is the future value of the account R is the interest r... Read More

Key Insights

• 😀 The equation A = P * e^(RT) describes exponential growth in continuous compounding.
• 🛟 The equation A final = A initial * e^(-KT) is used for exponential decay in half-life problems.
• 🧑‍💻 The natural log of both sides and log properties help derive the integrated rate law expressions for different reaction orders.
• 🛟 The half-life equation for a first-order reaction is ln2/K.
• 🛟 The half-life equation for a zero-order reaction is A initial / 2K.
• 😉 The half-life equation for a second-order reaction is 1 / (A initial * K).
• ☠️ The half-life is inversely related to the rate constant for all three reaction orders.

Questions & Answers

Q: What does the equation A = P * e^(RT) represent?

This equation represents exponential growth in continuous compounding, where A is the future value, P is the principle, R is the interest rate or growth rate, T is the time, and e is the number 2.71828.

Q: How can you derive the equation for half-life in exponential decay?

Start with the equation A final = A initial * e^(-KT) and take the natural log of both sides to get Ln A final = Ln A initial - KT Ln e. Simplify further to Ln A final = -KT + Ln A initial, which is the integrated rate law expression for a first-order reaction.

Q: How is the half-life equation different for zero-order reactions?

The half-life equation for a zero-order reaction is t1/2 = A initial / 2K. It is different because the half-life is dependent on the initial concentration, whereas for a first-order reaction, it only depends on the rate constant.

Q: What is the effect of increasing the initial concentration on the half-life of a second-order reaction?

Increasing the initial concentration in a second-order reaction decreases the half-life. Conversely, decreasing the initial concentration increases the half-life. This relationship is opposite to that of a zero-order reaction.

Summary & Key Takeaways

• The equation A = P * e^(RT) describes exponential growth in continuous compounding.

• The equation A final = A initial * e^(-KT) is used for exponential decay in half-life problems.

• Taking the natural log of both sides and using log properties helps derive the integrated rate law expressions for different reaction orders.