please help me ...Experiments show that radium at rate proportional to the amount of radium present at the moment . if half life is 1590 years what percentage will disappear in one year ??
To find out what percentage of radium will disappear in one year, we first need to calculate the decay constant (k) using the half-life formula.
Given:
Half-life (T1/2) = 1590 years
We can use the formula:
k = ln(2) / T1/2
Calculating the decay constant:
k = ln(2) / 1590
Using a calculator, we find:
k ≈ 0.0004352
Now, we can use the exponential decay formula to determine the percentage of radium that will disappear in one year.
The exponential decay formula is given by:
A(t) = A(0) × e^(-kt)
Where:
A(t) is the amount of radium remaining at time t
A(0) is the initial amount of radium at t=0
t is the time in years
k is the decay constant
We are interested in finding the percentage of radium disappeared, so we can rearrange the formula as follows:
Percentage Disappeared = [1 - (A(t) / A(0))] × 100%
Substituting t = 1 year and simplifying, we get:
Percentage Disappeared = [1 - e^(-k)] × 100%
Calculating:
Percentage Disappeared = [1 - e^(-0.0004352)] × 100%
Percentage Disappeared ≈ 0.04349%
Therefore, approximately 0.04349% of the radium will disappear in one year.
To determine the percentage of radium that will disappear in one year, we can use the concept of radioactive decay and the formula for calculating half-life.
First, let's establish the equation for exponential decay:
N(t) = N₀ * (1/2)^(t / T)
Where:
N(t) = the amount of radium at time t
N₀ = the initial amount of radium
t = elapsed time
T = half-life
Given that the half-life of radium is 1590 years, we can substitute T = 1590 into the equation:
N(t) = N₀ * (1/2)^(t / 1590)
Since we want to calculate the percentage of radium that will disappear in one year, we need to find the ratio of the change in radium to the initial amount. Let's consider the change in radium over the course of one year:
N(t+1) = N₀ * (1/2)^((t + 1) / 1590)
Now, let's calculate the percentage change in radium:
Percentage Change = (N₀ - N(t+1)) / N₀ * 100%
Percentage Change = (N₀ - N₀ * (1/2)^((t + 1) / 1590)) / N₀ * 100%
Simplifying the expression:
Percentage Change = (1 - (1/2)^((t + 1) / 1590)) * 100%
Since we're considering one year, t = 1:
Percentage Change = (1 - (1/2)^(1 / 1590)) * 100%
To find the final answer, we can plug in the given half-life value and simplify the expression:
Percentage Change = (1 - (1/2)^(1 / 1590)) * 100% ≈ 0.00003947%
Therefore, approximately 0.00003947% of radium will disappear in one year.