The potential energy ( U ) of a diatomic molecule is a function dependent on r (interatomic distance) as $\mathrm{U}=\frac{\alpha}{\mathrm{r}^{10}}-\frac{\beta}{\mathrm{r}^{5}}-3$ Where, $\alpha$ and $\beta$ are positive constants. The equilibrium between two atoms will be $\left(\frac{2 \alpha}{\beta}\right)^{\frac{\mathrm{a}}{\mathrm{b}}}$, where a = \_\_\_\_

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Work, Power & Energy

The potential energy ( U ) of a diatomic molecule is a function dependent on r (interatomic distance) as

U = \frac{α}{r ^{10}} - \frac{β}{r ^{5}} - 3

Where,

α

and

β

are positive constants. The equilibrium between two atoms will be

(\frac{2 α}{β})^{\frac{a}{b}}

, where a = ____

Answer: 1

Solution

The two atoms rest in equilibrium where the force between them vanishes, that is where the potential energy is stationary:

\frac{d U}{d r} = - \frac{10 α}{r ^{11}} + \frac{5 β}{r ^{6}} = 0

The constant term

- 3

drops out on differentiating. Rearranging,

\frac{5 β}{r ^{6}} = \frac{10 α}{r ^{11}} \Rightarrow r^{5} = \frac{2 α}{β}

The equilibrium separation is thus

(\frac{2 α}{β})^{1/5}

, so matching it against

(\frac{2 α}{β})^{a / b}

gives

a = 1

JEE Main · 2021 · 25 Feb 2021 · Shift 1 · Numerical answer

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