The potential energy of the two atoms in a diatomic (two-atom) molecule can be approximated as (Lennard-Jones potential) U(r) = -(a/r⁶) + (b/r¹²), where r is the distance between the two atoms and a and b are positive constants. At what values of r is U(r) a minimum? A maximum?

검증된 단계별 안내

Start by understanding the problem: The Lennard-Jones potential is given as U(r) = -(a/r⁶) + (b/r¹²), where r is the distance between two atoms, and a and b are constants. To find the values of r where U(r) is a minimum or maximum, we need to analyze the derivative of U(r) with respect to r.

Take the first derivative of U(r) with respect to r. Using the power rule for differentiation, the derivative is: dU/dr = 6a/r⁷ - 12b/r¹³.

Set the first derivative equal to zero to find the critical points: 6a/r⁷ - 12b/r¹³ = 0. Simplify this equation to isolate r: 6a/r⁷ = 12b/r¹³. Divide through by 6 and simplify further: a/r⁷ = 2b/r¹³.

Solve for r by multiplying through by r¹³: a * r⁶ = 2b. Then isolate r by dividing both sides by 2b and taking the sixth root: r = (2b/a)^(1/6). This gives the value of r where U(r) is a minimum.

To determine if there is a maximum, take the second derivative of U(r) with respect to r and analyze its sign at the critical points. If the second derivative is positive, the critical point is a minimum; if negative, it is a maximum. For this potential, U(r) has a minimum at r = (2b/a)^(1/6) and no maximum within the physical range of r.

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주요 개념

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Lennard-Jones Potential

The Lennard-Jones potential is a mathematical model that describes the interaction between a pair of neutral atoms or molecules. It accounts for both attractive forces, which dominate at longer distances, and repulsive forces, which become significant at very short distances. The potential is expressed as U(r) = -(a/r⁶) + (b/r¹²), where 'a' and 'b' are constants that characterize the strength of these interactions.

Potential Energy Minimum and Maximum

In the context of potential energy, a minimum occurs at a point where the potential energy is lower than at surrounding points, indicating a stable configuration. Conversely, a maximum occurs where the potential energy is higher than at neighboring points, indicating an unstable configuration. To find these points, one typically takes the derivative of the potential energy function and sets it to zero to solve for critical points.

Critical Points and Stability

Critical points in a function occur where the first derivative is zero or undefined, indicating potential minima or maxima. To determine the nature of these critical points, the second derivative test is used: if the second derivative is positive, the point is a minimum (stable), and if negative, it is a maximum (unstable). This analysis is crucial for understanding the stability of molecular configurations in the Lennard-Jones potential.