Half life of P is 14.3 days.If you have 1.00gm of P today.then what would be the amount remaining in 10 days ?

Answer

To solve the problem of determining the remaining amount of substance P after 10 days, we will follow these steps:

### Step 1: Write down the given data
- Half-life of P (t_half) = 14.3 days
- Initial amount of P (n0) = 1.00 g
- Time elapsed (t) = 10 days

### Step 2: Calculate the decay constant (λ)
The decay constant (λ) can be calculated using the formula:

\[ \lambda = \frac{0.693}{t_{half}} \]

Substituting the given half-life:

\[ \lambda = \frac{0.693}{14.3} \approx 0.0487 \text{ days}^{-1} \]

### Step 3: Use the radioactive decay formula
The amount remaining after time t can be calculated using the formula:

\[ n = n_0 \cdot e^{-\lambda t} \]

Where:
- n = remaining amount after time t
- n0 = initial amount
- e = base of the natural logarithm (approximately 2.718)
- λ = decay constant
- t = time elapsed

### Step 4: Substitute the values into the formula
Now we substitute the values we have:

\[ n = 1.00 \cdot e^{-0.0487 \cdot 10} \]

Calculating the exponent:

\[ -0.0487 \cdot 10 = -0.487 \]

Now, calculating \( e^{-0.487} \):

\[ e^{-0.487} \approx 0.615 \]

So, substituting back into the equation:

\[ n \approx 1.00 \cdot 0.615 \approx 0.615 \text{ g} \]

### Step 5: Convert to milligrams (if necessary)
To express the remaining amount in milligrams:

\[ 0.615 \text{ g} = 615 \text{ mg} \]

### Final Answer
The amount of substance P remaining after 10 days is approximately **0.615 grams or 615 milligrams**.

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