Question #226371

Gold has a density of 19 700 kg m-3. 193 g of gold

contains 6.02 × 1023 atoms. Use this information

to estimate the volume of a gold atom, and hence

its radius. State any assumptions you make.


1
Expert's answer
2021-08-16T08:36:35-0400

6.02×10236.02 \times 10^{23} atoms of gold =193×103  kg= 193 \times 10^{-3} \;kg

mass of one atom of gold =193×1036.02×1023= \frac{193 \times 10^{-3}}{6.02 \times 10^{23}}

=32.06×1026  kg= 32.06 \times 10^{-26} \;kg

Volume of one atom of gold :

V=Mass  of  one  atom  of  goldDensity  of  goldV=32.06×102619700=1.63×1029  m3V= \frac{Mass \;of\; one \;atom \;of \;gold}{Density\; of\; gold} \\ V= \frac{32.06 \times 10^{-26}}{19700} = 1.63 \times 10^{-29} \;m^3

Assumption: gold atom is spherical.

Volume of one atom of gold =43πR3= \frac{4}{3} \pi R^3

R= radius of atom of gold

V=43πR3R=(3V4π)13R=(3×1.63×10294×3.14)13R=(0.3889×1029)13R=1.57×1010  mV= \frac{4}{3} \pi R^3 \\ R =( \frac{3V}{4 \pi})^{\frac{1}{3}} \\ R =( \frac{3 \times 1.63 \times 10^{-29}}{4 \times 3.14})^{\frac{1}{3}} \\ R= (0.3889 \times 10^{-29})^{\frac{1}{3}} \\ R= 1.57 \times 10^{-10} \;m


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