The Young’s modulus for steel is Y = 2 x 10¹¹ N /m² . If the inter-atomic distance is 3.2Å, the inter atomic force constant in N/ Å will be

The correct option is 6.4 x 10⁻⁹.

To find the inter-atomic force constant, we can use the relationship between Young's modulus, the inter-atomic distance, and the inter-atomic force constant.

The formula relating these quantities is given by:
$k=Y\times r_0$
where:
• $k$ is the inter-atomic force constant,
• $Y$ is the Young's modulus of the material, and
• $r_0$ is the equilibrium inter-atomic distance.

From the problem statement, we are given:
• Young's modulus, $Y=2\times {10}^{11}{\text{N/m}}^2$
• Inter-atomic distance, $r_0=3.2\text{Å}=3.2��{10}^{-10}\text{m}$

Substituting these values into the formula to find the force constant in $\text{N/m}$:
$k=(2\times {10}^{11}{\text{N/m}}^2)\times (3.2\times {10}^{-10}\text{m})$
$k=6.4\times {10}^1\text{N/m}=64\text{N/m}$

The question asks for the force constant in units of $\text{N/Å}$. We know that:
$1\text{m}={10}^{10}\text{Å}$
Therefore, the conversion factor for the unit is:
$1\text{N/m}=\frac{1\text{N}}{{10}^{10}\text{Å}}={10}^{-10}\text{N/Å}$

Now, we convert our calculated value of $k$ to $\text{N/Å}$:
$k=64\times {10}^{-10}\text{N/Å}$
$k=6.4\times {10}^{-9}\text{N/Å}$