Problem 110 - Physics for Scientists and Engineers with Modern Physics

Question

(III) The two atoms in a diatomic molecule exert an attractive force on each other at large distances and a repulsive force at short distances. The magnitude of the force between two atoms in a diatomic molecule can be approximated by the Lennard-Jones force, or \(F(r)=F_{0}\left[2(\sigma / r)^{13}-(\sigma / r)^{7}\right],\) where \(r\) is the separation between the two atoms, and \(\sigma\) and \(F_{0}\) are constant. For an oxygen molecule (which is diatomic) \(F_{0}=9.60 \times 10^{-11} \mathrm{~N}\) and \(\sigma=3.50 \times 10^{-11} \mathrm{~m} .\) (a) Integrate the equation for \(F(r)\) to determine the potential energy \(U(r)\) of the oxygen molecule. (b) Find the equilibrium distance \(r_{0}\) between the two atoms ( \(c\) ) Graph \(F(r)\) and \(U(r)\) between \(0.9 r_{0}\) and \(2.5 r_{0}\)

Step-by-Step Solution

Answer

(a) Integrate to find \( U(r) = -F_{0} \left[\frac{2\sigma^{13}}{12r^{12}} - \frac{\sigma^{7}}{6r^{6}}\right]\). (b) Equilibrium distance \(r_{0} \approx 3.92 \times 10^{-11} \mathrm{~m}\). (c) Graph \(F(r)\) and \(U(r)\) over appropriate range.

Step 1: Understanding the Lennard-Jones Force

The Lennard-Jones force is given by the formula: \ F(r)=F_{0}\left[2\left(\frac{\sigma}{r}\right)^{13}-\left(\frac{\sigma}{r}\right)^{7}\right], \ where \(F_{0}\) and \(\sigma\) are constants, and \(r\) is the separation between the two atoms in the diatomic molecule. We are given \(F_{0}=9.60 \times 10^{-11} \mathrm{~N}\) and \(\sigma=3.50 \times 10^{-11} \mathrm{~m}\) for an oxygen molecule.

Step 2: Integrate to find Potential Energy U(r)

The potential energy \(U(r)\) is found by integrating \(F(r)\) with respect to \(r\): \ U(r) = -\int F(r) \, dr. \ Substituting the expression for \(F(r)\), we integrate: \ \(U(r) = -F_{0} \int \left[2\left(\frac{\sigma}{r}\right)^{13}-\left(\frac{\sigma}{r}\right)^{7}\right] \, dr.\)This can be integrated to:\[ U(r) = -F_{0} \left[\frac{2\sigma^{13}}{12r^{12}} - \frac{\sigma^{7}}{6r^{6}}\right]. \]

Step 3: Find the Equilibrium Distance

The equilibrium distance \(r_{0}\) occurs when the force \(F(r)\) is zero. Solving \(F(r) = 0\) gives: \[ 2\left(\frac{\sigma}{r_{0}}\right)^{13} = \left(\frac{\sigma}{r_{0}}\right)^{7}. \]Simplifying:\[ 2 = \left(\frac{r_{0}}{\sigma}\right)^{6}. \]Taking the sixth root of both sides, we find:\[ r_{0} = \sigma \times 2^{1/6}. \]Plug in \(\sigma = 3.50 \times 10^{-11}\) m:\[ r_{0} \approx 3.92 \times 10^{-11} \mathrm{~m}. \]

Step 4: Graph F(r) and U(r)

To graph \(F(r)\) and \(U(r)\), calculate values between \(0.9r_{0}\) and \(2.5r_{0}\), where \(r_{0}\) is approximately \(3.92 \times 10^{-11} \mathrm{~m}\). Compute \(F(r)\) and substitute in the formula for \(U(r)\) over this range to visualize how both the force and potential energy change with \(r\). These graphs will show the force reaching zero at the equilibrium and becoming repulsive or attractive at different separation distances relative to \(r_{0}\).

Key Concepts

Diatomic Molecules

Diatomic molecules consist of two atoms that are bonded together. These molecules can be made up of two atoms of the same element, like an oxygen molecule (O₂), or of two different elements, like hydrogen chloride (HCl). In such molecules, the atoms are held together by chemical bonds, and their interactions can be modeled using the Lennard-Jones potential.
This potential accounts for the forces between the molecule’s atoms, which can be attractive at larger distances and repulsive when they are too close. Understanding these intermolecular forces is crucial because they dictate how the molecules behave in different conditions and environments.
The Lennard-Jones potential simplifies these interactions into manageable mathematical form, allowing us to predict bond lengths and the potential energy associated with the molecule based on the separation (or distance) between its constituent atoms.

Equilibrium Distance

The equilibrium distance refers to the specific separation between two atoms in a molecule, where they experience a balance of forces. In the case of diatomic molecules, this is the point at which the attractive and repulsive forces between the atoms exactly cancel out. This is the most stable configuration and it's where the potential energy of the molecule is at a minimum.
In mathematical terms, this distance is represented as \(r_{0}\). For a Lennard-Jones potential, you find \(r_{0}\) by setting the force \(F(r)\) equation to zero, which indicates no net force acting on the atoms. Solving the equation gives the distance at which the energy stabilization occurs.
In our example, the equilibrium distance for an oxygen molecule can be calculated using the given constants \( \sigma \) and \(F_{0}\). It's expressed as \(r_{0} = \sigma \times 2^{1/6}\).
This distance is fundamental as it helps us understand how the molecule retains its structure, and it influences the physical properties of the substance, such as boiling points and reactions with other molecules.

Potential Energy Integration

Potential energy integration is a mathematical process used to determine the potential energy \(U(r)\) of a system as a function of distance \(r\) by integrating the force \(F(r)\). In the context of the Lennard-Jones potential, this involves calculating how the potential energy changes as two atoms within a diatomic molecule are brought closer together or separated.
An integral \(-\int F(r) \, dr\) is used to derive \(U(r)\), which shows how different separations between atoms result in different energy levels. This energy configuration informs us about the stability and formation of chemical bonds within a molecule.
Understanding \(U(r)\) is key in predicting molecular behavior. It also assists in realizing how molecular interactions influence larger chemical and physical properties. By creating graphs of both \(F(r)\) and the resulting \(U(r)\), we get a visual representation of these interactions, highlighting equilibrium states and the forces at play at various distances.
This helps chemists and physicists understand and anticipate how molecules interact in complex systems.