The half life of carbon14 is 5600 years If a piece of charco

The half life of carbon-14 is 5600 years. If a piece of charcoal made from the wood of a tree shows only 63% of the carbon-14 expected in living matter, when did the tree die? The tree died about years ago.

Solution

Half life of Carbon - 14 is 5600 years.

Using formula   t_1/2 = -0.693 / k.          // here t_1/2 is the half life   and k is the rate constant

5600 = -0.693 / k

k = -0.693 / 5600

k = -1.23 x 10^-4

Using

y = y_o * e^{k * t}                      // here y_o is the initial amount , y is amount left of the substance   and t is time taken.

63/100 * y_o = y_o * e^{k * t}

0.63 y_o = y_o * e ^{k * t}

0.63 = e ^{-1.23 x 10^(-4) * t}

taking natural log

ln 0.63 = ln e ^{-1.23 x 10^(-4) * t}

-0.462 = -1.23 x 10 ^(-4) * t * ln e

0.462 = 1.23 x 10 ^-4 * t                                    // ln e = 1

t = 0.462 / 1.23 x 10 ^-4

t = 3756 years

the tree was 3756 years