To solve this problem, we need to use the formula for radioactive decay which is:

N = N0 * (1/2)^(t/T)

where:
N is the final quantity of the substance
N0 is the initial quantity of the substance
t is the time elapsed
T is the half-life of the substance

We are given:
N0 = 2.80 mg
N = 1.20 mg
T = 2.23 minutes

We need to solve for t.

Substituting the given values into the formula, we get:

1.20 = 2.80 * (1/2)^(t/2.23)

To isolate the term with t, we divide both sides by 2.80:

1.20/2.80 = (1/2)^(t/2.23)

0.42857 = (1/2)^(t/2.23)

To get rid of the exponent, we take the natural logarithm (ln) of both sides:

ln(0.42857) = ln((1/2)^(t/2.23))

This simplifies to:

-0.8473 = (t/2.23) * ln(1/2)

To solve for t, we divide both sides by ln(1/2) and multiply by 2.23:

t = -0.8473 / ln(1/2) * 2.23

t = 2.23 minutes

So, it would take approximately 2.23 minutes for the sample to decay from 2.80mg to 1.20mg.

Question

To solve this problem, we need to use the formula for radioactive decay which is:

N = N0 * (1/2)^(t/T)

where:
N is the final quantity of the substance
N0 is the initial quantity of the substance
t is the time elapsed
T is the half-life of the substance

We are given:
N0 = 2.80 mg
N = 1.20 mg
T = 2.23 minutes

We need to solve for t.

Substituting the given values into the formula, we get:

1.20 = 2.80 * (1/2)^(t/2.23)

To isolate the term with t, we divide both sides by 2.80:

1.20/2.80 = (1/2)^(t/2.23)

0.42857 = (1/2)^(t/2.23)

To get rid of the exponent, we take the natural logarithm (ln) of both sides:

ln(0.42857) = ln((1/2)^(t/2.23))

This simplifies to:

-0.8473 = (t/2.23) * ln(1/2)

To solve for t, we divide both sides by ln(1/2) and multiply by 2.23:

t = -0.8473 / ln(1/2) * 2.23

t = 2.23 minutes

So, it would take approximately 2.23 minutes for the sample to decay from 2.80mg to 1.20mg.

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