How did they arrive at this equation?

[pivoxa15]

Homework Statement

n(x)F(x)=d[p(x)]/dx in relation to a collection of gases


n(x) is the number density
F(x) is the force along the x direction
p(x) is the pressure of the gas along the x direction

The Attempt at a Solution

Is F(x)=-d[U(x)]/dx related?

 

[malawi_glenn]

what is U(x) ?

 

[nrqed]

Homework Statement

n(x)F(x)=d[p(x)]/dx in relation to a collection of gases


n(x) is the number density
F(x) is the force along the x direction
p(x) is the pressure of the gas along the x direction

The Attempt at a Solution

Is F(x)=-d[U(x)]/dx related?

Here's what comes mind (without having thought about it too deeply):

Consider a small rectangular "slab" of gas of thickness dx and surface area A. Consider the net pressure on the slab from the inside toward the outside (say). The slab contains N particles. Then the force on the left side will be minus the pressure there times the surface area, whereas the pressure on the right side will be the pressure there times the surface area (I neglect the pressure on the other sides of the slab which is very small).

We get

Total force = P(x+dx) *A - P(x) * A \approx dV \frac{P(x+dx) - P(x)}{dx} \approx dV \frac{d P(x) }{dx}.

I think it should be easy now to complete this by taking into account a certain number of particles with a density n(x).


Hope this helps

Patrick