# What is the difference between these two works ?

1. Mar 18, 2008

### variation

Some examples in text book make me confused when these two works are discussed at the same time.

One of the works is the (mechanical) work in work-energy theorem:

$$\Delta K = \sum_iW_i,$$

where $$K$$ is the kinetic energy and $$W_i$$ was the work done by the $$i$$-th force.

The other is the (thermodynamical) work in the first law of thermodynamics:

$$\Delta U = Q + W,$$

where $$U$$ is the internal energy of the system, $$Q$$ is the heat transfered, and $$W$$ is the work done on the system by surroundings.

Are the two works the same when we want to use work-energy theorem and the first law of thermodynamics at the same time?

Can any one give some criterion to distinguish these two works ?

Thank you .

2. Mar 18, 2008

### wongdaisiu

Hi Variation:

You might want to refer to the overall energy balance equation:

dPE + dKE + dU = Q - W

Now the first equation have certain assumptions, that the change in potential energy is zero (such as it is on a flat surface) and the change in internal energy is zero (no state changes, or temperature and pressure changes). This is your classic pulling a block on a flat surface problem.

The second equation, you have thermodynamic work. You have state changes, and this assumes that no mechanical energy, potential energy or kinetic energy. So basically on a flat surface and not moving. A classic problem is the cylinder with the piston. You put it near the flame and it would expand. Flame provides heat (Q) and expansion is work (W)

I hope this helps.

3. Mar 19, 2008

### SeniorTotor

This two works are barely the same. Let me give you the simplest example. Internal energy is
$$U = U(S, V, N)$$
with
$$dU = TdS - PdV + \sum \mu dN$$
$$-PdV = dW$$
being the elementary mechanical work.
You get precisely the same elementary work with mechanics of continuous media (consider the simplest case of a diagonal stress tensor $$\sigma_{ij} = -P \delta_{ij}$$).

4. Mar 19, 2008

### Staff: Mentor

5. Mar 25, 2008

### variation

Thank you all.