5. An oxygen atom has a diameter of 1.2 x 10-10 m. What is the volume, in liters, of 6.46 x 1024 oxygen atoms? The volume of a sphere is (4⁄3)πr3.
ANSWER: 0.0058 L or 5.8 x 10-3 L
SOLUTION:
In this problem, we are asked the volume, in L, of 6.46 x 1024 oxygen atoms. We are given the diameter of one oxygen atom.
In this problem, we are asked the volume, in L, of 6.46 x 1024 oxygen atoms. We are given the diameter of one oxygen atom.
Given: diameter of an oxygen atom
Desired: Volume of 6.46 x 1024 oxygen atoms.
We can determine the volume of one oxygen atom by using the diameter. Once we find the volume of one atom, we can determine the volume of 6.46 x 1024 oxygen atoms. To find the volume of one oxygen atom, we can use the formula for the volume of a sphere. Notice, we need the radius of the oxygen atom. We can determine that from the diameter because the radius is 0.5 diameter.
r = 0.5 x (1.2 x 10-10m) = 6.0 x 10-11 m.
Now we can find the volume of one oxygen atom.
\(\displaystyle V\;=\;\frac{4}{3}\times\pi\times{(6.0\times10^{-11}m)^3}\;=\;\mathbf{9.0\times10^{-31}\;m^3}\)
The volume of one oxygen atom is 9.0 x 10-31 m.
We can now determine the volume of 6.46 x 1024 oxygen atoms.
Our equivalence is 9.0 x 10-31 m = 1 oxygen atom.
The conversion factors are
\(\displaystyle\frac{1\;atom}{9.0\times{10}^{-31}\;m^3}\;or\;\frac{9.0\times{10}^{-31}\;m^3}{1\;atom}\)
Now we can determine the volume of 6.46 x 1024 oxygen atoms.
\(\displaystyle\require{cancel}6.46\times{10}^{24}\;\cancel{atoms}\times\frac{9.0\times{10}^{-31}\;m^3}{\cancel{atom}}\;=\;\mathbf{5.8\times{10}^{-6}\;m^3}\)
Next, we convert m3 to L. There are 1000 L in a cubic meter. The equivalence is 1000 L = 1 m3. The conversion factors are:
\(\displaystyle\frac{1000\;L}{1\;m^3}\;or\;\frac{1\;m^3}{1000\;L} \)Now, we can solve our problem.
\(\displaystyle 5.8\times{10}^{-6}\;m^3\times\frac{1000\;L}{m^3}=\mathbf{0.0058\;L}\)
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