The potential energy of a pair of hydrogen atoms separated by a large distance x is given by U(x)=-C(subscript)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 repulse?
I've taken the derivative of the equation and found it to be U(X)'= 6C_6/x^7
so the force should be inward/attractive. Is this all the info i need?
he potential energy of a pair of hydrogen atoms
the force F is is given by
F = -grad ( U )
Thus F = -C_6/x^7 and its attractive
To determine the force that one atom exerts on the other, you are correct that you need to take the derivative of the potential energy equation with respect to x.
The derivative of U(x) with respect to x, denoted as U'(x) or dU(x)/dx, gives you the force F(x) that one atom exerts on the other.
Taking the derivative of U(x) = -C_6/x^6, we have:
U'(x) = -d(C_6/x^6)/dx
= 6C_6/x^7
Therefore, the force F(x) that one atom exerts on the other is given by F(x) = 6C_6/x^7.
Since the force is proportional to 1/x^7, which is always positive for positive x, the force is attractive.
To determine the force exerted by one hydrogen atom on the other, you need to apply the negative derivative of the potential energy function with respect to distance.
You correctly calculated the derivative of U(x) = -C[subscript]6/x^6 as U'(x) = 6C[subscript]6/x^7. However, note that the negative sign in the potential energy function implies that the force will have an opposite sign.
Therefore, the force exerted by one atom on the other is given by F(x) = -U'(x) = -6C[subscript]6/x^7.
This force is attractive since it has a negative sign. This means the atoms are drawn towards each other.