The Young’s modulus of a metal is 1.2x10¹¹ N / m² and the inter-atomic force constant is 3.6 x 10⁻⁹ N / Å. The mean distance between the atoms of the metal is

The correct option is 3 Å.

To find the mean inter-atomic distance between the atoms of the metal, we use the relationship between Young's modulus ($Y$), the inter-atomic force constant ($k$), and the mean inter-atomic distance ($r_0$):

$Y=\frac{k}{r_0}$

Rearranging this formula to solve for the mean inter-atomic distance ($r_0$):

$r_0=\frac{k}{Y}$

First, write down the given values and convert them into standard SI units:

Young's modulus, $Y=1.2\times {10}^{11}{\text{N/m}}^2$
Inter-atomic force constant, $k=3.6\times {10}^{-9}\text{N/Å}$

Since $1\text{Å}={10}^{-10}\text{m}$, convert the unit of $k$ to Newtons per meter (N/m):

$k=\frac{3.6\times {10}^{-9}\text{N}}{{10}^{-10}\text{m}}=3.6\times {10}^1\text{N/m}=36\text{N/m}$

Now, substitute the values of $k$ and $Y$ into the equation for $r_0$:

$r_0=\frac{36}{1.2\times {10}^{11}}\text{m}$

$r_0=30\times {10}^{-11}\text{m}=3\times {10}^{-10}\text{m}$

Convert the final distance back to Angstroms ($\text{Å}$):

$r_0=3\text{Å}$