Question

The potential energy function for either one of the two atoms in a diatomic molecule is often approximated by U(x) = −a/x12 − b/x6 where x is the distance between the atoms.

a) At what distance of separation does the potential energy have a local minimum (not at x = [infinity])

Answer 1

x = ( 2a / b ) ^( 1 / 6 )

Step-by-step explanation:

Solution:-

- The The potential energy function for either one of the two atoms in a diatomic molecule is often approximated by:

`U ( x ) = - (a)/(x^1^2) - (b)/(x^6)`

Where, x : The separation between two atoms.

- We are to find the separation where the potential energy between two atoms is minimum. For that we have to resort to the methods of calculus. The given function U ( x ) is a single variable function of separation "x" which differential over all the real numbers interval.

- We will determine the first derivative of the potential energy function U ( x ) and set it to zero to calculate the critical values of separation x.

`(d U ( x ))/(dx ) = 12*(a)/(x^1^3) + 6*(b)/(x^7) \\\\(d U ( x ))/(dx ) = 12*(a*x^7)/(x^1^3*x^7) + 6*(b*x^1^3)/(x^1^3*x^7) \\\\(d U ( x ))/(dx ) = (12ax^7 + 6bx^1^3)/(x^2^0) = 0\\\\\\frac{12ax^7 + 6bx^1^3}{x^2^0} = 0\\\\12ax^7 + 6bx^1^3 = 0\\\\x^7* ( 2a + bx^6 ) = 0\\\\x^7 \\eq 0 , 2a + bx^6 = 0 \\\\\\x = ( (2a)/(b))^(1)/(6)`

- The potential energy function U ( x ) has a local minima at x = ( 2a / b ) ^( 1 / 6 )

Answer 2

Answer:
`x = \sqrt[6]{2a/b}`

Step-by-step explanation:

If we want to find a local minimum, we can serach for the points where

U'(x) = 0

here we have that U(x) = -a/x^12 - b/x^6

then U'(x) = (-12)*-a/x^11 -(-6)*b/x^5

this must be zero.

12a/x^11 + 6b/x^5 = 0

12a/x^11 = 6b/x^5

12a = (6b/x^5)*x^11 = 6b*x^(11-5) = 6bx^6

x^6 = 12a/6b = 2a/b

`x = \sqrt[6]{2a/b}`

This is the distance at wich the potential energy has a local minimum.