Radius decays at a rate of that is proportional to its mass, and has a half-life of 1590 years. If 20 g of radium is present initially, how long will it take for 90% of this mass to decay?

so you want 10% to remain (the 20g is actually not relevant)

solve
.1 = (1/2)^(t/1590)
log .1 = log (.5^(t/1590))
t/1590(log .5) = log .1
t/1590 = log .1/log .5
etc.

let me know if didn't get t = 5282

To solve this problem, we need to use the concept of exponential decay. The rate of decay of the radium's mass is proportional to its mass itself, which means it follows an exponential decay model.

Let's start by finding the decay constant (k) using the half-life information. The half-life is defined as the time it takes for half of the initial amount of radium to decay. In this case, the half-life is given as 1590 years.

We can use the formula for exponential decay to relate the decay constant (k) with the half-life:

(1/2) = e^(-kt)

Where:
(1/2) is the fraction remaining after one half-life,
e is the base of natural logarithm (approximately 2.718),
k is the decay constant, and
t is the time in years.

Solving for the decay constant (k), we have:

e^(-kt) = 1/2
Taking natural logarithm on both sides:
ln(e^(-kt)) = ln(1/2)
-kt = ln(1/2)
k = ln(1/2) / -t

Now, we can substitute the given half-life (1590 years) into the equation:

k = ln(1/2) / -1590

Next, we can use the decay constant (k) to determine the time it takes for 90% of the initial mass to decay.

We have the initial mass (20 g) and want to find the time (t) when only 10% (90% decayed) remains.

Using the formula for exponential decay:

m(t) = m(0) * e^(-kt)

Where:
m(t) is the remaining mass at time t,
m(0) is the initial mass,
k is the decay constant, and
t is the time in years.

Substituting the values:

0.10 * m(0) = m(0) * e^(-k*t)

Dividing both sides by m(0):

0.10 = e^(-k*t)

Taking natural logarithm on both sides:

ln(0.10) = -k*t

Solving for t, we have:

t = ln(0.10) / -k

Now we can plug in the value of k we found earlier:

t = ln(0.10) / (ln(1/2) / -1590)

Evaluating this expression, we can find the time it takes for 90% of the initial mass to decay.