Solve each of the following:
Conversion: 1 amu = 1.661 x 10-27 kg
a. The atomic mass of carbon is 12.011 amu. What is the mass in grams of 6.02 x 1023 carbon atoms?
b. The atomic mass of gold is 196.96657 amu (per atom). How many gold atoms are there in 197 grams of gold?
The correct answer and explanation is :
Given Information:
- 1 amu = 1.661 x 10^-27 kg
- Avogadro’s number = 6.02 x 10^23 atoms/mol
Let’s break down each part of the problem:
a) Mass in Grams of 6.02 x 10^23 Carbon Atoms
We are asked to find the mass in grams of 6.02 x 10^23 carbon atoms, given that the atomic mass of carbon is 12.011 amu.
Step 1: Convert Atomic Mass to Kilograms
The atomic mass of carbon is given as 12.011 amu. First, we need to convert this to kilograms:
[
12.011 \, \text{amu} = 12.011 \times 1.661 \times 10^{-27} \, \text{kg}
]
[
= 1.993 \times 10^{-26} \, \text{kg}
]
This means that one carbon atom has a mass of 1.993 x 10^-26 kg.
Step 2: Convert Kilograms to Grams
Since we need the mass in grams, we convert kilograms to grams (1 kg = 1000 g):
[
1.993 \times 10^{-26} \, \text{kg} = 1.993 \times 10^{-23} \, \text{g}
]
Step 3: Find the Mass of 6.02 x 10^23 Atoms
Now, we can find the total mass of 6.02 x 10^23 carbon atoms:
[
\text{Total mass} = (6.02 \times 10^{23}) \times (1.993 \times 10^{-23} \, \text{g})
]
[
= 12.011 \, \text{g}
]
So, the mass of 6.02 x 10^23 carbon atoms is 12.011 grams.
b) Number of Gold Atoms in 197 Grams of Gold
Next, we are asked to determine how many gold atoms are there in 197 grams of gold, given that the atomic mass of gold is 196.96657 amu.
Step 1: Convert Atomic Mass of Gold to Kilograms
The atomic mass of gold is 196.96657 amu. We first convert it to kilograms:
[
196.96657 \, \text{amu} = 196.96657 \times 1.661 \times 10^{-27} \, \text{kg}
]
[
= 3.270 \times 10^{-25} \, \text{kg}
]
This means that one gold atom has a mass of 3.270 x 10^-25 kg.
Step 2: Convert the Mass of Gold from Grams to Kilograms
Now, convert the 197 grams of gold to kilograms:
[
197 \, \text{g} = 197 \times 10^{-3} \, \text{kg} = 0.197 \, \text{kg}
]
Step 3: Find the Number of Gold Atoms
To find the number of atoms, divide the total mass of gold by the mass of one gold atom:
[
\text{Number of atoms} = \frac{\text{Total mass of gold}}{\text{Mass of one gold atom}}
]
[
= \frac{0.197 \, \text{kg}}{3.270 \times 10^{-25} \, \text{kg/atom}}
]
[
= 6.02 \times 10^{23} \, \text{atoms}
]
So, there are approximately 6.02 x 10^23 gold atoms in 197 grams of gold.
Explanation
Part a:
The process for determining the mass of 6.02 x 10^23 carbon atoms involves converting atomic mass units (amu) to kilograms and then to grams. Using the fact that the atomic mass of carbon is approximately 12.011 amu, we calculate the mass of a single carbon atom in grams and multiply by the number of atoms (Avogadro’s number). The result is 12.011 grams, which aligns with the fact that 1 mole of any substance contains Avogadro’s number of atoms and weighs its atomic mass in grams. Since 6.02 x 10^23 is exactly one mole, the total mass is exactly 12.011 grams.
Part b:
The second part of the problem focuses on how many atoms are present in a given mass of gold (197 grams). The key is recognizing that 196.96657 amu is approximately the molar mass of gold (in grams per mole). By dividing the given mass by the molar mass and using Avogadro’s number, we determine that 197 grams of gold contains approximately 6.02 x 10^23 atoms. This is consistent with the concept of a mole in chemistry, where 1 mole of any element corresponds to Avogadro’s number of atoms and has a mass equal to its atomic mass in grams.