How much time is required for a 5.75-mg sample of ^51 Cr to decay to 1.50 mg if it has a half-l

step 1 In this question, we have half lifetime of BI 210 is equal to 5 days. So K is equals to 0 .693 d

How much time is required for a 5.75-mg sample of ^51 Cr to...

How much time is required for a $5.75-\mathrm{mg}$ sample of ${ }^{51} \mathrm{Cr}$ to decay to $1.50 \mathrm{mg}$ if it has a half-life of $27.8$ days?
(a) $5.39$ days
(b) $2.69$ days
(c) $53.9$ days
(d) $5.49$ days

Key Concepts

Radioactive Decay

Radioactive decay is a stochastic process by which an unstable atomic nucleus loses energy by emitting radiation. This phenomenon is characterized by the spontaneous transformation of nuclei, which results in a decrease in the amount of the original substance over time. The process follows a predictable exponential decay pattern, making it useful in fields such as nuclear physics, archaeology, and medicine.

Half-life

The half-life of a radioactive substance is the time required for half of the original amount of the substance to decay. It is a constant that is independent of the initial quantity of the substance. This key concept enables scientists to calculate the time elapsed for a radioactive sample to decay to a specified fraction of its initial mass.

Exponential Decay Equation

The exponential decay equation models the decrease in the quantity of a substance over time. The equation generally takes the form N(t) = N? · (1/2)^(t/T), where N(t) is the remaining quantity at time t, N? is the initial quantity, and T is the half-life. This equation provides a mathematical framework for calculating the time required for a substance to decay to a given amount.

Logarithms

Logarithms are the inverse operations of exponentiation and are crucial for solving equations involving exponential decay. When the desired time is in the exponent, logarithms are used to isolate the variable representing time. This mathematical tool allows for the conversion of multiplicative relationships into additive ones, thereby facilitating the solution of decay problems.

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