Question

The potential energy between two atoms in a molecule is given by $U ( x ) = \frac { a } { x ^ { 12 } } - \frac { b } { x ^ { 6 } }$; where a and b are positive constants and x is the distance between the atoms. The atom is in stable equilibrium when

• A. $x = \sqrt [ 6 ] { \frac { 11 a } { 5 b } }$
• B. $x = \sqrt [ 6 ] { \frac { a } { 2 b } }$
• C. $x = 0$
• D. $x = \sqrt [ 6 ] { \frac { 2 a } { b } }$

Answer 1

Answer: (D)

$x = \sqrt [ 6 ] { \frac { 2 a } { b } }$

Answer 2

Condition for stable equilibrium$F = - \frac { d U } { d x } = 0$

⇒ $- \frac { d } { d x } \left[ \frac { a } { x ^ { 12 } } - \frac { b } { x ^ { 6 } } \right] = 0$ ⇒ $- 12 a x ^ { - 13 } + 6 b x ^ { - 7 } = 0$

⇒ $\frac { 12 a } { x ^ { 13 } } = \frac { 6 b } { x ^ { 7 } }$⇒ $\frac { 2 a } { b } = x ^ { 6 }$ ⇒ $x = \sqrt [ 6 ] { \frac { 2 a } { b } }$