(b) Explain why you expect there to be a value of r that minimizes V(r). Choose the correct answer below. O A. Find the first derivative of the given equation. Find r by equating the first derivative to 0. Now, find the second derivative and substitute the value of r. As the second derivative is negative, you expect there to be a value of r that maximize V(r). O B. Find the first derivative of the given equation. Find r by equating the first derivative to 0. Now, find the second derivative and substitute the value of r. As the second derivative is negative, you expect there to be a value of r that minimizes V(r). O C. Find the first derivative of the given equation. Find r by equating the first derivative to 0. Now, find the second derivative and substitute the value of r. As the second derivative is positive, you expect there to be a value of r that minimizes V(r). O D. Find the first derivative of the given equation. Find r by equating the first derivative to 0. Now, find the second derivative and substitute the value of r. As the second derivative is positive, you expect there to be a value of r that maximize V(r).
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
16-2:
![**Lennard-Jones 6-3 Potential Model**
One popular model for the interactions between two molecules is the Lennard-Jones 6-3 potential. According to this model, the energy of interaction between two molecules that are distance \( r \) apart is given by the following function. Molecules will attract or repel each other until they reach a distance that minimizes the function \( V(r) \). The coefficient \( A \) is a positive constant. Complete parts (a) through (c).
\[ V(r) = \frac{1}{r^6} - \frac{A}{r^3}, \; r > 0 \]
**Explanation:**
The function \( V(r) \) models how the potential energy changes with the distance \( r \) between the molecules. It has two main terms:
1. **Repulsive Term**: \( \frac{1}{r^6} \)
- This term becomes significant at short distances and leads to positive potential energy, indicating repulsion between the molecules as they get very close to each other.
2. **Attractive Term**: \( -\frac{A}{r^3} \)
- This term dominates at moderate distances and results in negative potential energy, reflecting an attractive force pulling the molecules together.
The balance between these two terms results in a minimum potential energy at a certain distance, which is the equilibrium distance where the molecules are most stable.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7dce3dae-6456-4f3a-a2bd-6bde155fd2f1%2F0b76f564-5197-4c2a-80b6-940b931f99fa%2Fmsl3jm_processed.png&w=3840&q=75)
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