the mass of cobalt-60 in a sample is found to have decreased from 0.800g to 0.200g in a period of 10.5 years. from this info calculate the half-life of cobalt-60
To calculate the half-life of cobalt-60, we can use the formula:
t₁/₂ = (ln2) / k
Where:
- t₁/₂ is the half-life
- ln2 is the natural logarithm of 2 (approximately 0.693)
- k is the decay constant
First, let's calculate the decay constant (k) using the given information:
ln(N₀ / N) = kt
Where:
- N₀ is the initial mass of the sample (0.800g)
- N is the final mass of the sample (0.200g)
- t is the time period (10.5 years)
ln(0.800g / 0.200g) = k * 10.5 years
ln(4) = 10.5k
Now, let's solve for k:
k = ln(4) / 10.5 years
k ≈ 0.237 years⁻¹
Finally, substitute the value of k into the formula to find the half-life:
t₁/₂ = 0.693 / 0.237 years⁻¹
t₁/₂ ≈ 2.92 years
Therefore, the half-life of cobalt-60 is approximately 2.92 years.
To calculate the half-life of cobalt-60, we need to use the formula for exponential decay:
N(t) = N₀ * (1/2)^(t / t₁/₂)
Where:
N(t) = the final amount of cobalt-60
N₀ = the initial amount of cobalt-60
t = the time elapsed
t₁/₂ = the half-life of cobalt-60
Here, we are given:
N₀ = 0.800g (initial mass)
N(t) = 0.200g (final mass)
t = 10.5 years
Now, let's substitute these values into the formula:
0.200g = 0.800g * (1/2)^(10.5 / t₁/₂)
Next, let's simplify the equation:
0.200 / 0.800 = (1/2)^(10.5 / t₁/₂)
Divide both sides by 0.800:
0.25 = (1/2)^(10.5 / t₁/₂)
To remove the exponent, we need to take the logarithm of both sides of the equation (base 1/2):
log(0.25) = log[(1/2)^(10.5 / t₁/₂)]
Using the property of logarithms, we can bring the exponent down:
log(0.25) = (10.5 / t₁/₂) * log(1/2)
Now, let's solve for t₁/₂:
10.5 / t₁/₂ = log(0.25) / log(1/2)
Calculate the logarithms:
10.5 / t₁/₂ ≈ -0.60206 / -0.30103
Simplify the right side:
10.5 / t₁/₂ ≈ 1.995
Cross-multiply and solve for t₁/₂:
t₁/₂ ≈ 10.5 / 1.995
t₁/₂ ≈ 5.26 years
Therefore, the half-life of cobalt-60 is approximately 5.26 years.
ln(No/N) = kt
No = what you started with initially.
N = what you ended up with.
k = solve for this.
t = time in years.
Then plug k into the below equation.
k = 0.693/t1/2 and solve for t1/2