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Element X is a radioactive isotope such that every 13 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 90 grams, how long would it be until the mass of the sample reached 62 grams, to the nearest tenth of a year?

User Ccyrille
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Answer:

The mass of Element X will reach 62 grams in 7.0 years.

Explanation:

We can find the time of decay of Element X by using the exponential decay equation:


(dm)/(dt) = -\lambda t

The solution of the above equation is:


m_((t)) = m_(0)e^(-\lambda t) (1)

Where:

t: is the time =?

λ: is the decay constant


m_((t)): is the mass at time t = 62 grams


m_(0): is the initial mass = 90 grams

First, we need to calculate λ


\lambda = (ln(2))/(t_(1/2)) (2)

Where
t_(1/2) is the half-life = 13 years

By entering equation (2) into (1) and solving for "t" we have:


(m_((t)))/(m_(0)) = e^{-(ln(2))/(t_(1/2))*t}


ln((m_((t)))/(m_(0))) = -(ln(2))/(t_(1/2))*t


t = -ln((m_((t)))/(m_(0)))((t_(1/2))/(ln(2))) = -ln((62)/(90))((13)/(ln(2))) = 7.0 y

Therefore, the mass of Element X will reach 62 grams in 7.0 years.

I hope it helps you!

User Altunyurt
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