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Chapter 2: Problem 99

Potential Energy A commonly used potential-energy function for the interaction of two molecules is the Lennard-Jones \(6-12\) potential, given by $$ u(r)=u_{0}\left[\left(\frac{\sigma}{r}\right)^{12}-\left(\frac{\sigma}{r}\right)^{6}\right] $$ where \(u_{0}\) and \(\sigma\) are constants. The force corresponding to this potential is \(F(r)=-u^{\prime}(r)\). Find \(F(r)\).

Short Answer

Expert verified
The force function for the given potential-energy function can be found by differentiating the potential function with respect to \(r\) and taking the opposite of the derivative, which results in: $$ F(r) = \frac{u_0\sigma}{r^2}\left[12\left(\frac{\sigma}{r}\right)^{11} - 6\left(\frac{\sigma}{r}\right)^{5}\right] $$

Step by step solution

01

Differentiate the potential function, \(u(r)\), with respect to \(r\)

Given the potential function: $$ u(r) = u_{0} \left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right] $$ Recall that the derivative of a function with respect to \(r\), is the rate at which the function changes as \(r\) changes. So, we need to apply the power rule of differentiation and the chain rule: $$ \frac{du(r)}{dr} = \frac{d(u_0\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right])}{dr} $$ First, note that the \(u_0\) is a constant, so we can take it outside of the derivative: $$ \frac{du(r)}{dr} = u_0\frac{d(\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6})}{dr} $$ Applying the power rule and chain rule to each term separately: $$ \frac{du(r)}{dr} = u_0\left[-12\left(\frac{\sigma}{r}\right)^{11}\frac{d\left(\frac{\sigma}{r}\right)}{dr} + 6\left(\frac{\sigma}{r}\right)^{5}\frac{d\left(\frac{\sigma}{r}\right)}{dr}\right] $$ Now differentiate \(\frac{\sigma}{r}\) with respect to \(r\): $$ \frac{d\left(\frac{\sigma}{r}\right)}{dr} = \sigma\frac{d\left(\frac{1}{r}\right)}{dr} = -\frac{\sigma}{r^2} $$ Substitute this derivative back into the expression for \(\frac{du(r)}{dr}\): $$ \frac{du(r)}{dr} = u_0\left[-12\left(\frac{\sigma}{r}\right)^{11}\left(-\frac{\sigma}{r^2}\right) + 6\left(\frac{\sigma}{r}\right)^{5}\left(-\frac{\sigma}{r^2}\right)\right] $$
02

Compute the opposite (negative) of the derivative to obtain the force function, \(F(r)\)

The force \(F(r)\) is defined as the negative of the derivative of the potential function with respect to \(r\): $$ F(r) = -\frac{du(r)}{dr} = -u_0\left[-12\left(\frac{\sigma}{r}\right)^{11}\left(-\frac{\sigma}{r^2}\right) + 6\left(\frac{\sigma}{r}\right)^{5}\left(-\frac{\sigma}{r^2}\right)\right] $$ Simplify the expression by cancelling the negative signs: $$ F(r) = u_0\left[12\left(\frac{\sigma}{r}\right)^{11}\left(\frac{\sigma}{r^2}\right) - 6\left(\frac{\sigma}{r}\right)^{5}\left(\frac{\sigma}{r^2}\right)\right] $$ Finally, factor out the common term \(\frac{\sigma}{r^2}\): $$ F(r) = \frac{u_0\sigma}{r^2}\left[12\left(\frac{\sigma}{r}\right)^{11} - 6\left(\frac{\sigma}{r}\right)^{5}\right] $$ This is the desired force function for the given potential-energy function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Potential Energy
Potential energy is a fundamental concept in physics that describes the stored energy in a system due to its position or arrangement. In the context of molecular interactions, the Lennard-Jones potential is a mathematical model used to explain the potential energy between two non-bonded molecules or atoms.
The Lennard-Jones potential specifically considers short-range repulsive forces and long-range attractive forces. These forces exhibit a balance that stabilizes the molecules. The potential energy, as expressed in the Lennard-Jones model, is given by the formula: \( u(r) = u_{0}\left[\left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^{6}\right] \)
  • \(u_{0}\) is the depth of the potential well, indicating the strength of the interaction.
  • \(\sigma\) is the finite distance where the potential is zero, akin to the collision diameter.
Understanding potential energy in terms of the Lennard-Jones potential helps in analyzing molecular dynamics and predicting molecular behavior.
Force Calculation
To determine the force derived from the potential energy in molecular systems, we need to calculate the force function, which is the negative derivative of the potential energy function. This stems from the principle that force is the negative gradient of potential energy.
In our exercise, the force corresponding to the Lennard-Jones potential is given by:\( F(r) = -u'(r) \) This implies that to determine the force, we must first find the derivative of the potential function \(u(r)\) with respect to the distance \(r\). Physically, the result informs us how strongly two molecules will repel or attract each other at a given distance. The derivative will help us capture these changes in force as potentially stabilizing or destabilizing interactions.
Differentiation
Differentiation is a mathematical process used to determine how a function changes as its input changes. In the exercise, we need to differentiate the potential energy function with respect to the variable \(r\), which represents the distance between molecules.
To compute the derivative, we apply familiar calculus rules, particularly focusing on each term in the Lennard-Jones potential:
  • For \(\left(\frac{\sigma}{r}\right)^{12}\), the goal is to find how this part changes with \(r\).
  • Similarly, for \(\left(\frac{\sigma}{r}\right)^{6}\), we determine its rate of change.
These changes are analyzed using the power rule in differentiation, capturing how potential energy shifts with molecular separation, leading us to understand molecular forces better.
Power Rule
The power rule is a fundamental principle in differentiation, simplifying the process when calculating derivatives for terms involving powers of variables. It states that if you have a function \( x^n \), its derivative is \( nx^{n-1} \).
In the case of the Lennard-Jones potential, this rule enables us to find the derivatives of each term. For example:
  • For \(\left(\frac{\sigma}{r}\right)^{12}\), using the power rule gives \(-12\left(\frac{\sigma}{r}\right)^{11} \times \frac{d(\frac{\sigma}{r})}{dr}\).
  • Similarly, \(\left(\frac{\sigma}{r}\right)^{6}\) becomes \(-6\left(\frac{\sigma}{r}\right)^{5} \times \frac{d(\frac{\sigma}{r})}{dr}\).
These results are integral in dissecting the full derivative, ultimately guiding us to calculate the resultant force. Application of the power rule makes differentiation more manageable when tackling polynomial expressions in physics.
Chain Rule
The chain rule is a crucial concept in differentiation, especially when differentiating composite functions. It allows us to differentiate a function inside another function. In this context, it involves expressions like \(\left(\frac{\sigma}{r}\right)^n\).
To apply the chain rule, consider:
  • Firstly, differentiate the outer function as per the power rule.
  • Then, differentiate the inner function \(\frac{\sigma}{r}\), treating it as \(\sigma \times r^{-1} \).
  • Finally, multiply these differential results to obtain the derivative.
In the scenario of the Lennard-Jones potential differentiation, the chain rule helps in obtaining the derivative \(-\frac{\sigma}{r^2}\), contributing to an accurate force calculation. This approach decomposes complex functions into manageable parts, streamlining the differentiation process.

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Most popular questions from this chapter

Bimolecular Reaction In a bimolecular reaction \(A+B \rightarrow M, a\) moles per liter of \(A\) and \(b\) moles per liter of \(B\) are combined. The number of moles per liter that have reacted after time \(t\) is given by $$ x=\frac{a b\left[1-e^{(b-a) k t}\right]}{a-b e^{(b-a) k t}} \quad a \neq b $$ where the positive number \(k\) is called the velocity constant. Find an expression for \(x\) if \(a=b\), and find \(\lim _{t \rightarrow \infty} x\). Interpret your results.

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