Work Energy Power

JEE Main level, numerical answer, medium

106 of 183

Question 106 of 183

The potential energy (U)\left(U\right) of a diatomic molecule is a function dependent on rr (interatomic distance) as U=αr10−βr5−3U=\dfrac{\alpha }{{r}^{10}}-\dfrac{\beta }{{r}^{5}}-3 where, α\alpha and β\beta are positive constants. The equilibrium distance between two atoms will be (2αβ)ab,{\left(\dfrac{2\alpha }{\beta }\right)}^{\dfrac{a}{b}}, where a=a=______ .

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Answer: 1

For equilibrium
dUdr=0\dfrac{dU}{dr}=0
−10αr11+5βr6=0\dfrac{-10\alpha }{{r}^{11}}+\dfrac{5\beta }{{r}^{6}}=0
5βr6=10αr11\dfrac{5\beta }{{r}^{6}}=\dfrac{10\alpha }{{r}^{11}}
r5=2αβ{r}^{5}=\dfrac{2\alpha }{\beta }
r5=2αβ{r}^{5}=\dfrac{2\alpha }{\beta }
r=(2αβ)15a=1r={\left(\dfrac{2\alpha }{\beta }\right)}^{\dfrac{1}{5}} a=1

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