The Radioactive Substance Cesium-137 Has A Half-life Of 30 Years. The Amount, A ( T A(t A ( T ] (in Grams), Of A Sample Is Given By The Following Exponential Function: A ( T ) = 725 ( 1 2 ) T 36 A(t) = 725 \left(\frac{1}{2}\right)^{\frac{t}{36}} A ( T ) = 725 ( 2 1 ) 36 T Find The Amount Of The
The Radioactive Substance Cesium-137: Understanding its Half-Life and Exponential Decay
Cesium-137 is a radioactive substance that has been widely used in various applications, including nuclear medicine, scientific research, and industrial processes. One of the key characteristics of cesium-137 is its half-life, which is a measure of the time it takes for the substance to decay to half of its original amount. In this article, we will explore the concept of half-life and how it relates to the exponential decay of cesium-137.
What is Half-Life?
The half-life of a radioactive substance is the time it takes for the substance to decay to half of its original amount. It is a fundamental concept in nuclear physics and is used to describe the rate of decay of radioactive materials. The half-life of a substance is typically denoted by the symbol "t1/2" and is usually expressed in units of time, such as years or seconds.
The Exponential Decay Function
The amount of a sample of cesium-137, denoted by the function A(t), is given by the following exponential function:
A(t) = 725 \left(\frac{1}{2}\right)^{\frac{t}{36}}
where t is the time in years and A(t) is the amount of the sample in grams.
Understanding the Exponential Decay Function
To understand the exponential decay function, let's break it down into its components. The function A(t) = 725 \left(\frac{1}{2}\right)^{\frac{t}{36}} can be rewritten as:
A(t) = 725 \cdot e^{\ln\left(\frac{1}{2}\right) \cdot \frac{t}{36}}
where e is the base of the natural logarithm and ln is the natural logarithm function.
Simplifying the Exponential Decay Function
Using the properties of logarithms, we can simplify the exponential decay function as follows:
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The Radioactive Substance Cesium-137: Understanding its Half-Life and Exponential Decay
Q&A: Understanding Cesium-137 and its Exponential Decay
Q: What is the half-life of cesium-137?
A: The half-life of cesium-137 is 30 years. This means that every 30 years, the amount of cesium-137 in a sample will decrease by half.
Q: What is the exponential decay function for cesium-137?
A: The exponential decay function for cesium-137 is given by the equation:
A(t) = 725 \left(\frac{1}{2}\right)^{\frac{t}{36}}
where t is the time in years and A(t) is the amount of the sample in grams.
Q: How does the exponential decay function relate to the half-life of cesium-137?
A: The exponential decay function is a mathematical representation of the rate at which cesium-137 decays. The half-life of 30 years is a key component of this function, as it determines the rate at which the amount of cesium-137 decreases over time.
Q: What is the significance of the base 1/2 in the exponential decay function?
A: The base 1/2 in the exponential decay function represents the rate at which cesium-137 decays. In this case, the rate of decay is 1/2, which means that every 30 years, the amount of cesium-137 will decrease by half.
Q: How can the exponential decay function be used to predict the amount of cesium-137 at a given time?
A: The exponential decay function can be used to predict the amount of cesium-137 at a given time by plugging in the value of t (time in years) into the equation. For example, if we want to know the amount of cesium-137 after 60 years, we would plug in t = 60 into the equation:
A(60) = 725 \left(\frac{1}{2}\right)^{\frac{60}{36}} = 725 \left(\frac{1}{2}\right)^{\frac{5}{3}} = 725 \cdot \frac{1}{32} = 22.578125
Q: What are some real-world applications of the exponential decay function for cesium-137?
A: The exponential decay function for cesium-137 has several real-world applications, including:
- Nuclear medicine: Cesium-137 is used in nuclear medicine to treat certain types of cancer. The exponential decay function is used to predict the amount of cesium-137 in the body over time.
- Scientific research: Cesium-137 is used in scientific research to study the properties of radioactive materials. The exponential decay function is used to predict the amount of cesium-137 in a sample over time.
- Industrial processes: Cesium-137 is used in industrial processes, such as in the production of nuclear fuels. The exponential decay function is used to predict the amount of cesium-137 in a sample over time.
Conclusion
In conclusion, the exponential decay function for cesium-137 is a mathematical representation of the rate at which this radioactive substance decays. The half-life of 30 years is a key component of this function, as it determines the rate at which the amount of cesium-137 decreases over time. The exponential decay function has several real-world applications, including nuclear medicine, scientific research, and industrial processes.