#### Answer

The diameter of the sphere is 2.7 cm

#### Work Step by Step

The mass of 1 mole of gold is $196.967~g$. We can find the volume of this mass of gold as:
$V = \frac{m}{\rho}$
$V = \frac{0.196967~kg}{19.3\times 10^3~kg/m^3}$
$V = 1.02\times 10^{-5}~m^3$
We can find the radius of the sphere as:
$\frac{4}{3}\pi~R^3 = V$
$R^3 = \frac{3V}{4\pi}$
$R = (\frac{3V}{4\pi})^{(1/3)}$
$R = [\frac{(3)(1.02\times 10^{-5}~m^3)}{4\pi}]^{(1/3)}$
$R = 0.0135~m = 1.35~cm$
Since the diameter is $2R$, the diameter of the sphere is 2.7 cm.