We will be left with 5 g of Cesium-137 after 90 years, given its half-life of 30 years and the initial quantity we have as 40 g.
What do we mean by the half-life of a substance?
Many elements tend to decompose over some time. The period over which an element reduces to half of itself in quantity is called the half-life of that substance.
The formula for the quantity of the element at a certain time t can be given as:
[tex]N_{t} = N_{0}\left ( \frac{1}{2} \right )^{\frac{t}{t_{1/2}}}[/tex]
Where,
[tex]N_{t}[/tex] is the quantity of the element at time t,
[tex]N_{0}[/tex] is the initial quantity of the element,
[tex]{t_{1/2}[/tex] is the half-life of the element.
How do we solve the given question?
In the question, we are given that an element, Cesium-137 has a half-life of approximately 30 years.
We are asked to find the quantity of this element left after 90 years when we had an initial sample of 40 g.
To solve for the quantity left, we can use the formula for the quantity of the element at a certain time t can be given as:
[tex]N_{t} = N_{0}\left ( \frac{1}{2} \right )^{\frac{t}{t_{1/2}}}[/tex]
Where,
[tex]N_{t}[/tex] is the quantity of the element at time t,
[tex]N_{0}[/tex] is the initial quantity of the element,
[tex]{t_{1/2}[/tex] is the half-life of the element,
by taking t = 90 years, [tex]N_{0}[/tex] = 40 g, [tex]{t_{1/2}[/tex] = 30 years, and [tex]N_{t}[/tex] = ?.
∴ [tex]N_{t} = N_{0}\left ( \frac{1}{2} \right )^{\frac{t}{t_{1/2}}}[/tex]
or, [tex]N_{t} = 40\left ( \frac{1}{2} \right )^{\frac{90}{30}}} g[/tex]
or, [tex]N_{t} = 40\left ( \frac{1}{2} \right )^{3}} g[/tex]
or, [tex]N_{t} = 40\left ( \frac{1}{8} \right ) g[/tex]
or, [tex]N_{t} = 5 g[/tex] .
∴ We will be left with 5 g of Cesium-137 after 90 years, given its half-life of 30 years and the initial quantity we have as 40 g.
Learn more about the half-life of a substance at
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