# What is the difference between these two works ?

## Main Question or Discussion Point

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 .

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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.

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 $$\sigma_{ij} = -P \delta_{ij}$$).

berkeman
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Thank you all.