Decay Rate

Definition

Decay rate is a measure of how quickly a quantity, such as the amount of a radioactive substance or the amplitude of a vibrating system, decreases or diminishes over time. It is a fundamental concept in the study of exponential and logarithmic equations, which describe processes that exhibit exponential growth or decay.

5 Must Know Facts For Your Next Test

1. The decay rate is typically represented by the Greek letter $\lambda$ (lambda) and is the inverse of the half-life of a quantity.
2. Exponential decay follows the equation $y = y_0 e^{-\lambda t}$, where $y_0$ is the initial value, $t$ is time, and $e$ is the base of the natural logarithm.
3. The half-life of a quantity is related to the decay rate by the equation $t_{1/2} = \frac{\ln 2}{\lambda}$, where $t_{1/2}$ is the half-life.
4. Logarithmic equations can be used to solve for the decay rate or half-life of a quantity given other variables, such as the initial value and the value at a specific time.
5. Decay rate is an important concept in fields such as radioactive dating, population dynamics, and signal processing, where exponential and logarithmic relationships are observed.

Review Questions

• Explain how the decay rate is related to the half-life of a quantity.
  • The decay rate, represented by the Greek letter $\lambda$, is inversely related to the half-life of a quantity. The half-life is the time it takes for a quantity to decrease to half of its initial value. The relationship between the decay rate and half-life is given by the equation $t_{1/2} = \frac{\ln 2}{\lambda}$, where $t_{1/2}$ is the half-life and $\lambda$ is the decay rate. This means that a higher decay rate corresponds to a shorter half-life, and vice versa.
• Describe how the decay rate is used in the equation for exponential decay.
  • The decay rate, $\lambda$, is a key parameter in the equation for exponential decay, which is given by $y = y_0 e^{-\lambda t}$. In this equation, $y_0$ is the initial value of the quantity, $t$ is the time, and $e$ is the base of the natural logarithm. The decay rate, $\lambda$, determines the rate at which the quantity decreases over time. A higher decay rate results in a faster decrease in the quantity, while a lower decay rate leads to a slower decrease.
• Explain how logarithmic equations can be used to solve for the decay rate or half-life of a quantity.
  • Logarithmic equations can be used to solve for the decay rate or half-life of a quantity that exhibits exponential decay. For example, if the initial value, $y_0$, and the value at a specific time, $y$, are known, the decay rate can be calculated using the equation $\lambda = -\frac{\ln(y/y_0)}{t}$. Similarly, if the decay rate is known, the half-life can be calculated using the equation $t_{1/2} = \frac{\ln 2}{\lambda}$. These logarithmic relationships allow for the determination of key parameters related to the decay process.

Related terms

Half-Life: The time it takes for a quantity to decrease to half of its initial value, which is inversely related to the decay rate.

Exponential Decay: A process where a quantity decreases by a constant proportion over equal intervals of time, resulting in an exponential relationship between the quantity and time.

Logarithmic Equations: Equations that describe relationships between quantities that exhibit exponential growth or decay, and can be used to solve for unknown variables related to the decay rate.