Suppose that a device is using a 5.00-mg sample of silicon that is doped with 1 × 10-5% (by mass) phosphorus. How many phosphorus atoms are in the sample?
For who needs it in the future:
5.00mg Si / 1000 mg Si = 0.005g Si
(0.005)((1e-5)/100g Si) = 5.00e-10
((5.00e-10)(6.02e23))/(31 g P) = 9.71e12 atoms P
P = 5.00 mg * 1E-5 % = 5.00E-10 g
P atoms = (5.00E-10 / 31.0) * 6.02E23
Well, let me do some quick calculations here. Okay, carry the one... ah, there we go! It looks like we've got approximately "very small" number of phosphorus atoms in that sample. But don't worry, you won't have to count them all one by one. Trust me, it's a lot!
To find the number of phosphorus atoms in the sample, we need to calculate the mass of phosphorus and then convert it to the number of atoms.
First, let's calculate the mass of phosphorus in the sample:
Mass of phosphorus = (1 × 10^-5%) x mass of silicon
= (1 × 10^-5/100) x 5.00 mg
= 5.00 × 10^-5 mg
Next, we need to convert the mass of phosphorus to the number of atoms. We can use Avogadro's number, which states that 1 mole of any substance contains 6.022 × 10^23 atoms:
Number of phosphorus atoms = (mass of phosphorus / molar mass of phosphorus) x Avogadro's number
The molar mass of phosphorus is 30.97 g/mol. Since we have mass in mg, we need to convert it to grams:
Mass of phosphorus (g) = (5.00 × 10^-5 mg) x (1 g / 1000 mg)
= 5.00 × 10^-8 g
Now we can calculate the number of phosphorus atoms:
Number of phosphorus atoms = (5.00 × 10^-8 g / 30.97 g/mol) x (6.022 × 10^23 atoms/mol)
= 9.79 × 10^14 atoms
Therefore, there are approximately 9.79 × 10^14 phosphorus atoms in the sample.
To find the number of phosphorus atoms in the sample, we need to follow these steps:
1. Calculate the mass of phosphorus in the sample:
- Mass of silicon sample = 5.00 mg
- Mass fraction of phosphorus = 1 × 10^(-5)% = 1 × 10^(-7)
- Mass of phosphorus in the sample = mass of silicon sample × mass fraction of phosphorus
- Mass of phosphorus = 5.00 mg × (1 × 10^(-7)) = 5.00 × 10^(-9) mg
2. Convert the mass of phosphorus to the number of phosphorus atoms:
- We know the molar mass of phosphorus (P) is 30.97 g/mol or 30.97 mg/mmol.
- Calculate the number of moles of phosphorus: Moles of phosphorus = mass of phosphorus / molar mass of phosphorus
- Moles of phosphorus = (5.00 × 10^(-9) mg) / (30.97 mg/mmol)
- Moles of phosphorus = 1.615 × 10^(-10) mmol
3. Use Avogadro's number (6.022 × 10^23 atoms/mol) to convert moles to atoms:
- Number of phosphorus atoms = Moles of phosphorus × Avogadro's number
- Number of phosphorus atoms = (1.615 × 10^(-10) mmol) × (6.022 × 10^23 atoms/mol)
- Number of phosphorus atoms = 9.73 × 10^13 atoms
Therefore, the sample contains approximately 9.73 × 10^13 phosphorus atoms.