Problem 32 - University Physics with Modern Physics

Question

The potential energy of a pair of hydrogen atoms separated by a large distance \(x\) is given by \(U(x)=-C_{6} / x^{6},\) where \(C_{6}\) is a positive constant. What is the force that one atom exerts on the other? Is this force attractive or repulsive?

Step-by-Step Solution

Answer

The force is \(F(x) = -\frac{6C_6}{x^7}\) and it is attractive.

Step 1: Relationship between Force and Potential Energy

The force between two objects can be determined from the potential energy by taking the negative gradient of the potential energy. In one-dimensional space, this relationship simplifies to:\[ F(x) = -\frac{dU}{dx} \]

Step 2: Differentiate the Potential Energy Function

Given the potential energy function \(U(x) = -\frac{C_6}{x^6}\), take its derivative with respect to \(x\):\[ \frac{dU}{dx} = \frac{d}{dx}\left(-\frac{C_6}{x^6}\right) = -C_6 \cdot \frac{d}{dx}(x^{-6}) = -C_6 \cdot (-6x^{-7}) = 6C_6 x^{-7} \]

Step 3: Calculate the Force

Using the relationship from Step 1, determine the force by substituting the derivative found in Step 2:\[ F(x) = -\left(6C_6 x^{-7}\right) = -\frac{6C_6}{x^7} \]

Step 4: Determine the Nature of the Force

The force expression \(F(x) = -\frac{6C_6}{x^7}\) is negative for any positive value of \(x\), indicating that the force is attractive. An attractive force implies the atoms are being pulled towards each other.

Key Concepts

Force Calculation

To understand the force between two hydrogen atoms, we start by considering the concept of potential energy. In this context, potential energy represents the energy stored due to the position of the atoms. When we talk about the force exerted by one atom on another, we are essentially describing how one atom influences the other atom's motion.
The relationship between force and potential energy is a fundamental concept in physics. In one-dimensional space, the force can be found by taking the negative derivative of the potential energy function with respect to distance. Mathematically, this is expressed as:

• \( F(x) = -\frac{dU}{dx} \)

This formula implies that if you know how the potential energy changes with distance, you can determine the force exerted. In our specific problem involving hydrogen atoms, the equation simplifies force calculation, allowing us to determine how the atoms interact over distance.

Gradient of Potential Energy

The gradient of potential energy plays a crucial role in calculating forces. It tells us how fast the potential energy changes as we move through space. In simpler terms, it's like checking how steep a hill is – the steeper the hill, the more it affects your movement.
For the problem at hand, the potential energy function given by \( U(x) = -\frac{C_6}{x^6} \) becomes our focal point. To find the gradient, or in this case, the derivative, we differentiate this function with respect to \(x\).
The derivative calculation proceeds as follows:

• \( \frac{dU}{dx} = \frac{d}{dx}(-\frac{C_6}{x^6}) = -C_6 \cdot (-6x^{-7}) = 6C_6 x^{-7} \)

This mathematical operation shows how potential energy decreases as the distance \( x \) increases, thus enabling us to understand the nature of the force generated.

Attractive Force

The nature of the force between the hydrogen atoms is determined by observing the sign of the force expression. With our derived force function:

• \( F(x) = -\frac{6C_6}{x^7} \)

We notice that it is negative for positive values of \( x \), which indicates an attractive force. An attractive force means the atoms tend to pull towards each other rather than push away.
This characteristic of the force is key in understanding molecular interactions. Attractive forces are essential in the formation of molecules and play a significant role in chemical bonding. In the realm of physics, attraction is commonly encountered whenever there is potential energy dependency, like in gravitational or electrostatic forces.
The negative sign in the force calculation not only quantifies the strength of the attraction but also signifies its direction, pulling one atom towards the other, thus reinforcing the concept of potential energy as a tool for predicting the behavior of interacting particles.