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In physics, the force between two objects is a crucial concept, especially when dealing with atomic interactions like that between hydrogen atoms. Imagine two hydrogen atoms separated by a distance, interacting with each other due to a potential energy that changes with distance. The potential energy for hydrogen atoms is given by the formula \( U(x) = -\frac{C_6}{x^6} \), where \( C_6 \) is a positive constant and \( x \) is the separation distance. To understand the force one atom exerts on the other, we recall that force is derived from the potential energy's derivative. If you think about potential energy like a hill, the steepness of the hill at any point represents how strongly it pulls or pushes objects.

• Force is the gradient (or slope) of the potential energy surface.
• It tells us about how atoms attract or repel each other.
• Derived using calculus, specifically the concept of differentiation.

In essence, the force between hydrogen atoms arises because of changes in this potential energy as atoms move closer or farther apart.

To determine the force from potential energy, we use calculus, focusing specifically on derivatives. Derivatives help measure how one quantity changes in response to changes in another. Here, it involves how potential energy changes as two hydrogen atoms move relative to each other. Looking at the potential energy function: \[ U(x) = -\frac{C_6}{x^6} \]We want to find how this energy changes as \(x\) changes. This requires taking the derivative of \( U(x) \) with respect to \( x \). Using the power rule, a common method in calculus, the derivative is: \[ \frac{dU}{dx} = 6C_6 x^{-7} \] The result represents how quickly potential energy changes over distance. In physical terms, the steeper this change, the greater the force. A negative sign accompanies the force expression, aligning with how force is the potential energy's negative rate of change. This derivative is crucial as it directly leads to the force value.

Understanding whether a force is attractive or repulsive is important in atomic interactions. Here, we derive the nature of the force between hydrogen atoms by examining the sign of the calculated force. The formula we found for the force is: \[ F(x) = -6C_6 x^{-7} \] Because \( C_6 \) is a positive constant, the overall expression \( -6C_6 x^{-7} \) resolves to a negative value for any positive \( x \). The fact that the force is negative gives us crucial information:

• A negative force means that it pulls objects towards each other, which qualifies it as attractive.
• This is similar to how gravity works, pulling objects towards Earth.
• In our context, it means hydrogen atoms naturally draw nearer when left alone, due to this attractive force.

Thus, the negative sign of this mathematical expression aligns with the physical reality that hydrogen atoms attract one another when influenced by this force.