EALC The potential energy of a pair of hydrogen atoms...

EALC 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?

Key Concepts

Interpretation of Force Sign (Attractive vs. Repulsive)

The sign of the calculated force indicates whether the interaction is attractive or repulsive. A negative force value implies that the force acts to decrease the separation between particles (i.e., it is attractive), while a positive force would imply a repulsive interaction. This interpretation is crucial for analyzing the behavior of systems under various potential energy curves.

Relationship Between Potential Energy and Force

In physics, force is related to potential energy by taking the negative gradient (or derivative in one dimension) of the potential energy function. This relationship, given by F(x) = -dU/dx, ensures that the force acts in the direction that decreases the potential energy, which is essential for understanding how particles move in conservative force fields.

Differentiation of Power-Law Functions

When the potential energy functions follow a power-law form, such as U(x) ? 1/x?, calculus, specifically the power rule, is used to differentiate the function. This mathematical tool allows us to determine how rapidly the potential energy changes with distance, which in turn gives us the force acting between the particles.

Transcript

00:01 Problem 732, the potential energy of a pair of hydrogen atoms separate by large editions x is given by u of x equal to negative c divided by x to 6th, or c is a positive constant.

00:13 What is the force that one atom exerts on the other, and is it attractive or repulsive? okay, so we have this equation here, and what i'm going to do is in order to find the force, and apply what we have for equation 716.

00:28 The equation 716 is going to be telling us that we have a force that is somehow related to the potential energy and the way that that is related is going to be the negative of the derivative of the potential energy in that one dimension.

00:53 Okay, so since we know what u of x is, we can just plug that in.

00:57 So we have f of x is going to be equal to negative d or.

01:06 Over d x, we have negative c, and then i'll write down to x to the negative six.

01:20 So we have it over x of six, that is equivalent to x to the negative six.

01:28 Okay, so we have a constant negative c that can come out of the equation, and the negatives will cancel.

01:34 So we have c, d over d x, x to the negative six, in order to take a derivative of something to some negative power, you just need to write down c times the power itself that it's currently at.

01:56 And then you get x and you need to subtract 1.

01:59 So you need to get negative 7 here.

02:02 So then in total, what we get is negative 6c x to the negative 7 as our force for the interaction between these two hydrogen atoms.

02:21 Okay, so then the question is, is this force attractive or repulsive? so there's one or two ways that you could look at this...

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