Evaluating the location r_m of the minimum of potential

In summary, the conversation discusses the modeling of a gas of neutral atoms as an ideal gas and the use of the equation V(r) = 4(Epsilon)[(delta/r)^12 - (delta/r)^6] to calculate the potential energy in dimensionless form. The conversation also mentions finding the extrema of a function and determining the equilibrium position of the system. The solution suggests finding the location of r_m, the minimum potential, by setting the derivative of the potential to zero.
  • #1
tasleem moossun
8
0

Homework Statement


A gas of neutral atoms is often modeled as an ideal gas, where the gas atoms are considered to be "elastic billiard balls" that only interact by bouncing off each other in a manner that conservers the total kinetic energy.

Homework Equations


V(r) = 4(Epsilon)[(delta/r)^12 - (delta/r)^6]

in dimensionless form
U(xi) = 4 [(1/xi)^12 - (1/xi)^6]

xi= delta/r, U = V(r)/epsilon

The Attempt at a Solution


so i differentiated V(r) to get F(r),since F(r) = - dV(r)/dr.The force at the minimum of the potential is zero and the potential minimum defines the equilibrium position of the system.
 
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  • #2
Do you have a question?

Generally speaking, you can find the extrema (minimum, maximum, saddle point) of any function f(x) by calculating when its derivative is 0. You don't need to invoke the force here.
 
  • #3
my question was how do i evaluate the location of r_m of the minimum potential.
 
  • #4
Like this:
tasleem moossun said:
The force at the minimum of the potential is zero and the potential minimum defines the equilibrium position of the system.
 
  • #5
so basically the location of r_m would be -4(epsilon)[(-12(delta^12)/r^13 + 6delta^6/r^7]
 
  • #6
No, it is the r where this expression is zero.
 

What is the significance of evaluating the location r_m of the minimum of potential?

Evaluating the location r_m of the minimum of potential is important because it helps determine the most stable position of a system. This is crucial in understanding the behavior and properties of the system.

How is the location r_m of the minimum of potential calculated?

The location r_m of the minimum of potential is calculated by finding the point where the potential energy function reaches its lowest value. This can be done analytically or numerically depending on the complexity of the system.

What factors can affect the location r_m of the minimum of potential?

The location r_m of the minimum of potential can be affected by various factors such as the shape and size of the system, the distance between particles, and external forces acting on the system. These factors can alter the potential energy landscape and shift the location of the minimum.

How does the location r_m of the minimum of potential impact the stability of a system?

The location r_m of the minimum of potential is directly related to the stability of a system. A lower potential energy at the minimum indicates a more stable system, while a higher potential energy suggests a less stable system. It is important to evaluate and understand the location of the minimum in order to predict the behavior of the system.

Can the location r_m of the minimum of potential change over time?

Yes, the location r_m of the minimum of potential can change over time in dynamic systems. This can be due to changes in external conditions or internal interactions within the system. In some cases, the location of the minimum can also shift due to random fluctuations in the system.

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