A piece of charcoal of mass 35g is found in the ruins of an ancient city. The sample shows a carbon-14 activity or R(t)=4.167 decays/sec.

Convert the decay constant of ^14C from (1)(a)in terms of seconds instead of yrs.
Find the number of atoms N(t)using constant you found.

To convert the decay constant of ^14C from years to seconds, we need to know that 1 year is equal to 3.154 × 10^7 seconds. The decay constant (λ) is the probability of decay per unit time and is given by the formula:

λ = ln(2) / T

Where ln is the natural logarithm and T is the half-life of the radioactive substance. The half-life of carbon-14 is approximately 5730 years.

To convert the decay constant to seconds, we can use the following steps:

1. Calculate the decay constant in years:
λ(years) = ln(2) / T
= ln(2) / 5730

2. Convert the decay constant to seconds:
λ(seconds) = λ(years) / (3.154 × 10^7)

Now, let's calculate the decay constant in seconds:

1. λ(years) = ln(2) / 5730
= 0.693 / 5730
≈ 1.21 × 10^(-4) (rounded to 4 decimal places)

2. λ(seconds) = 1.21 × 10^(-4) / (3.154 × 10^7)
≈ 3.83 × 10^(-12) (rounded to 4 decimal places)

So, the decay constant of carbon-14 (^14C) is approximately 3.83 × 10^(-12) decays per second.

To find the number of atoms N(t), we can use the formula:

N(t) = N(0) * e^(-λt)

Where N(0) is the initial number of radioactive atoms, λ is the decay constant, and t is the time in seconds.

However, we need to know the initial number of carbon-14 atoms (N(0)) to calculate N(t). Please provide the initial number of carbon-14 atoms present in the charcoal sample.

To convert the decay constant of 14C from years to seconds, we can use the conversion factor of 1 year = 3.1536 x 10^7 seconds.

The decay constant (lambda) is given by the equation:

lambda = (0.693 / T½)

where T½ is the half-life of the radioactive isotope. The half-life of 14C is approximately 5730 years.

lambda = 0.693 / 5730 years

To convert this to seconds, we multiply by the conversion factor:

lambda = (0.693 / 5730 years) x (3.1536 x 10^7 seconds / 1 year)

Simplifying,

lambda = 1.209 x 10^-4 seconds^-1

Now, to find the number of atoms N(t) using the decay constant:

N(t) = N0 * e^(-lambda * t)

Here, N0 is the initial number of atoms, t is the time in seconds, and e is the base of the natural logarithm.

Since the activity R(t) is given as 4.167 decays/sec, we can use this to find the initial number of atoms N0.

R(t) = lambda * N0

4.167 decays/sec = 1.209 x 10^-4 seconds^-1 * N0

Simplifying,

N0 = 4.167 decays/sec / (1.209 x 10^-4 seconds^-1)

N0 = 34,409 atoms

Therefore, the number of atoms N(t) can be found using N0 = 34,409 atoms and the decay constant lambda = 1.209 x 10^-4 seconds^-1.