The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration.

a) How long will it take for 10% of the U-238 atoms in a sample of U- 238 to decay?

b)If a sample of U-238 initially contained 1.9×1018 atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms will it contain today?

a) Well, let's do some math and make this half-life fun! If we know that the half-life of U-238 is 4.5 billion years, we can use this information to figure out how long it will take for 10% of the U-238 atoms to decay.

So, if the half-life of U-238 is 4.5 billion years, that means after one half-life, 50% of the atoms will have decayed. Now, we want to find out when 10% of the atoms will have decayed. Since 10% is one-fifth of 50%, we need to multiply the half-life by five.

So, 4.5 billion years (one half-life) multiplied by five equals 22.5 billion years. That's how long it will take for 10% of the U-238 atoms to decay. But remember, I'm just a clown bot, so don't take my answer too seriously!

b) Alright, let's dust off our math skills and calculate how many U-238 atoms will be left today. If the universe was formed 13.8 billion years ago and we started with 1.9×10^18 U-238 atoms, we need to figure out how many half-lives have passed since then.

Since the half-life of U-238 is 4.5 billion years, we can divide the age of the universe (13.8 billion years) by the half-life to find out how many half-lives have passed. Let's do the math:

13.8 billion years divided by 4.5 billion years equals approximately 3.07 half-lives. We round down because we must consider only full half-lives.

Now that we know 3 half-lives have passed, we can calculate how many atoms are left. After one half-life, 50% of the atoms will remain, after two half-lives, 25% will remain, and after three half-lives, 12.5% will remain.

So, we multiply the initial number of atoms (1.9×10^18) by 12.5%, or 0.125, to find out how many atoms are left. Let me punch the numbers:

1.9×10^18 atoms multiplied by 0.125 equals 2.375×10^17. Therefore, today, there would be approximately 2.375×10^17 U-238 atoms left in the sample.

And that's it! But remember, I'm just a clown bot doing my best to make science a little more amusing. So don't quote these answers as the gospel truth!

a) To find out how long it will take for 10% of the U-238 atoms to decay, we can use the formula for exponential decay:

N(t) = N0 * (1/2)^(t / T)

Where N(t) is the number of atoms remaining at time t, N0 is the initial number of atoms, T is the half-life, and t is the time elapsed.

Since we want to find the time it takes for 10% of the atoms to decay, we can set N(t) to 0.1 * N0:

0.1 * N0 = N0 * (1/2)^(t / T)

Divide both sides by N0:

0.1 = (1/2)^(t / T)

Take the logarithm of both sides (any base can be used):

log(0.1) = log((1/2)^(t / T))

Using the property of logarithms, we can bring the exponent down:

log(0.1) = (t / T) * log(1/2)

Divide both sides by log(1/2):

(t / T) = log(0.1) / log(1/2)

Now we have an equation to solve for t, the time it takes for 10% of the atoms to decay. Plugging in the given half-life of U-238 (T = 4.5 billion years) and calculating:

(t / 4.5 billion years) = log(0.1) / log(1/2)

(t / 4.5) ≈ -1.301 / -0.301

(t / 4.5) ≈ 4.321

t ≈ 4.321 * 4.5 billion years

t ≈ 19.45 billion years

Therefore, it will take approximately 19.45 billion years for 10% of the U-238 atoms to decay.

b) To calculate the number of U-238 atoms in the sample today, we need to determine how many half-lives have passed since the universe was formed 13.8 billion years ago.

Since the half-life of U-238 is 4.5 billion years, the number of half-lives can be found by dividing the elapsed time (13.8 billion years) by the half-life:

Number of half-lives = 13.8 billion years / 4.5 billion years

Number of half-lives ≈ 3.067

Round down to the nearest whole number since it represents the number of completed half-lives, not partial ones:

Number of half-lives = 3

Now, we can calculate the number of U-238 atoms remaining using the formula for exponential decay:

N(t) = N0 * (1/2)^(t / T)

Where N(t) is the number of atoms remaining at time t, N0 is the initial number of atoms, T is the half-life, and t is the time elapsed.

N(t) = 1.9×10^18 * (1/2)^(3)

N(t) = 1.9×10^18 * (1/8)

N(t) = 0.2375×10^18

N(t) ≈ 2.375×10^17

Therefore, the sample of U-238 will contain approximately 2.375×10^17 atoms today.

a) To determine how long it will take for 10% of the U-238 atoms in a sample to decay, we can use the concept of half-life. The half-life of U-238 is given as 4.5 billion years, meaning that after 4.5 billion years, half of the U-238 atoms would have decayed.

To find the time it takes for 10% of the atoms to decay, we need to find how many half-lives it would take to get to that point. Since 10% is half of 20% (which is equivalent to one-fifth), it would take two half-lives for the initial concentration of U-238 to reduce to 10%.

One half-life is 4.5 billion years, so two half-lives would be 2 * 4.5 billion years = 9 billion years. Therefore, it will take approximately 9 billion years for 10% of the U-238 atoms in the sample to decay.

b) To determine the number of U-238 atoms that the sample will contain today, we need to find out how many half-lives have passed since the universe was formed 13.8 billion years ago.

Since the half-life of U-238 is 4.5 billion years, we can divide the total time elapsed (13.8 billion years) by the half-life to find the number of half-lives.

Number of half-lives = 13.8 billion years / 4.5 billion years

Number of half-lives ≈ 3.07

This means that approximately 3.07 half-lives have passed since the sample was initially formed.

To find the number of U-238 atoms remaining today, we can use the formula:

Remaining atoms = Initial atoms * (1/2)^(number of half-lives)

Where:
- Initial atoms = 1.9×10^18 atoms (as given in the question)
- Number of half-lives = 3.07

Remaining atoms ≈ 1.9×10^18 * (1/2)^3.07

Remaining atoms ≈ 3.66×10^17 atoms

Therefore, the sample of U-238 would contain approximately 3.66×10^17 atoms of U-238 today.

k = 0.693/t1/2

Then
ln(No/N) = kt
I would call No = 100
Then N = 90(after 10 have decayed--which is 10%)
k from above. Solve for t in years.

b.
Same procedure. No = atoms you started with. N = atoms remaining.