The interatomic distance for a metal is 3 x 10⁻¹⁰ m . If the interatomic force constant is 3.6 x 10⁻⁹ N / Å, then the Young’s modulus in N / m² will be

To find the Young’s modulus of the metal, we can use the relationship between the Young's modulus, the interatomic force constant, and the interatomic distance.

1. Identify the given values:
Interatomic distance,
$r_0=3\times {10}^{-10}\text{m}$
Interatomic force constant,
$k=3.6\times {10}^{-9}\text{N/Å}$

2. Convert the force constant to SI units (N/m):
Since $1\text{Å}={10}^{-10}\text{m}$, we can substitute this value into the expression for $k$:
$k=\frac{3.6\times {10}^{-9}\text{N}}{{10}^{-10}\text{m}}$
$k=3.6\times {10}^{-9-(-10)}\text{N/m}$
$k=3.6\times {10}^1\text{N/m}=36\text{N/m}$

3. Calculate Young’s modulus (Y):
The relation between Young's modulus ($Y$), interatomic force constant ($k$), and interatomic distance ($r_0$) is given by:
$Y=\frac{k}{r_0}$
Substituting the values of $k$ and $r_0$ into the formula:
$Y=\frac{36\text{N/m}}{3\times {10}^{-10}\text{m}}$
$Y=12\times {10}^{10}{\text{N/m}}^2$
$Y=1.2\times {10}^{11}{\text{N/m}}^2$

Therefore, the Young's modulus of the metal is $1.2\times {10}^{11}{\text{N/m}}^2$.