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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.

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.

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.