I am confused about whether work is path independent or not. If one were to drop a pen from a height of 5 m or to throw a pen parabolically from 5 m, the work done would be the same-- thus indicating that work is path independent. However, when one is talking about engines and refrigerators, work is deemed to be path-dependent based on U=Q+W. Would someone please clarify and reconcile this confusion. Thanks.

Work is path-independent if and only if the force acting on the particle is conservative. A conservative force is one which can be written as minus the gradient of a scalar potential function: [math]\mathbf{F}=-\nabla \phi[/math]. The work is path independent for conservative forces because the work done along an arbitrary curve C with endpoints A and B is given by:

 

[math]W=\int_C \mathbf{F}\cdot \mathrm{d}\mathbf{r}[/math]

 

If [math]\mathbf{F}[/math] is conservative, then:

 

[math]\int_C \mathbf{F}\cdot \mathrm{d}\mathbf{r}=-\int_C \nabla \phi \cdot \mathrm{d}\mathbf{r}=\phi(A)-\phi(B)[/math]

 

Gravity is conservative because it can be written as the gradient of the potential function [math]\phi=-\frac{Gm_1m_2}{r}[/math]. Magnetic forces, frictional forces, etc. are not conservative, so the work done in moving a particle through a magnetic field or when working against friction will depend on which path is taken.

That clarified a lot. Thank you for the awesome explanation. So then the forces involved in heat engines are not conservative and thus work is path-dependent. However, why is heat (q) in U=Q+W, path dependent-- also b/c it is not caused by conservative forces?

That clarified a lot. Thank you for the awesome explanation. So then the forces involved in heat engines are not conservative and thus work is path-dependent. However, why is heat (q) in U=Q+W, path dependent-- also b/c it is not caused by conservative forces?

 

[math]Q[/math] is the heat added to the system. This means that there's some heat transfer process, for example conduction, which is a complicated process that involves molecules bouncing around and smashing into each other. It depends on a number of factors, like temperature, the material, cross-sectional area, the thickness of the material, etc.