The atomic mass of carbon (diamond) is 12.01U and density is 2.22 × 10³kg m^–3, the radius of the carbon atom is

The question asks us to calculate the radius of a carbon atom in diamond, given the following data:

• Atomic mass of carbon: 12.01 U
• Density of carbon (diamond): 2.22 × 10³ kg/m³

Step-by-Step Solution

To determine the radius of a carbon atom in diamond, we will use the following relationship:

The volume V of a single carbon atom can be expressed as the volume of a sphere:

V = (4/3)πr³    

Where r is the radius of the carbon atom.

1. Calculate the Mass of a Single Carbon Atom

First, we need to calculate the mass of a single carbon atom using the atomic mass of carbon.

The atomic mass of carbon is 12.01 U (unified atomic mass units), where 1 atomic mass unit (1 U) = 1.67 × 10^-27 kg.

Thus, the mass of one carbon atom is:

Mass of one carbon atom = 12.01 × 1.67 × 10^-27 kg                               

= 2.00 × 10^-26 kg    

2. Use the Density to Find Volume

The density formula is:

Density = Mass / Volume    

Rearranging the formula to solve for volume:

Volume = Mass / Density    

Substitute the known values:

Volume of one carbon atom = (2.00 × 10^-26 kg) / (2.22 × 10³ kg/m³)                                  

= 9.00 × 10^-30 m³    

3. Calculate the Radius

Now that we know the volume, we can use the formula for the volume of a sphere:

V = (4/3)πr³    

Substitute the value for the volume:

9.00 × 10^-30 m³ = (4/3)πr³    

Solving for r:

r³ = (3 × 9.00 × 10^-30) / (4π)        

r³ = 2.14 × 10^-30 m³    

Taking the cube root:

r ≈ 1.29 × 10^-10 m    

4. Convert to Angstroms

Finally, to express the radius in more familiar units, we convert from meters to angstroms (1 angstrom = 1 × 10^-10 m):

r ≈ 1.29 Å    

Conclusion

The radius of a carbon atom in diamond, based on the atomic mass of carbon and the given density, is approximately 1.29 Å (angstroms).

In summary, we used the atomic mass of carbon and its density to calculate the volume of a single atom, and then derived the radius of the atom in the crystalline structure of diamond.