In the first law of thermodynamics, we learned that $W$ and $Q$ are path-dependent quantities, but how are $Q$ and $W$ defined?

I mean $W = \int_{\gamma} p(s) ds$ would be one possibility, where $\gamma$ is a path from volume 1 to volume 2, but how can I calculate $p$ as a function of $V$ in general? So given an arbitrary system, how can I calculate the pressure as a function of $V$?

Similarly for $Q$. If $Q = \int_{\gamma} T(s) ds $ where $T$ is a path from entropy $S_1$ to entropy $S_2$, I have no idea how I get for an arbitrary system $T$ as a function of $S$?


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The first law of thermodynamics, is as the name suggest a law. It states that if you consider some process a thermodynamic system undergoes then

$$\Delta U = W + Q$$

The point is that $W$ and $U$ are things that depend on what you are studying. Pick the infinitesimal version just for simplicity

$$dU = \delta W + \delta Q$$

Then for a gas it trully makes sense to put $\delta W = - P dV$, if you reason a little bit you will se this agrees with your intuition of mechanical work for a gas. For a rubber band under some tension $\tau$ it makes sense to put $\delta W = \tau dL$. And there is not just mechanical work, but also chemical work.

The law ends up being something like: if you know for a certain thermodynamic system how to define it's energy and define work properly then heat will be the change in the energy which is not in the form of work.

Why to think in this way? Thermodynamics is concerned with the description of equilibrium states of macroscopic systems. These equilibrium states are defined as those which can be described with just some extensive parameters, without counting all degrees of freedom the system really has.

Inside $W$ you count the change in the energy associated with changes on the measured degrees of freedom, the extensive parameters. But there is also change in energy due to changes on the unmeasured parameters. This change in energy due to tons of parameters together which you cannot control directly is then called heat and the first law states exactly that: the change in energy of a thermodynamic system consists of work and heat where work is change in energy due to change on the measured degrees of freedom and heat is the change in energy due to change on the unmeasured degrees of freedom.

Now, $W$ and $Q$ are not functions of state because one can see intuitively they are properties of a process and not of the states themselves.

Edit: From the point of view of calculations, usually the point is that $U = U(S,V,N_1,\dots, N_k)$ is something you know. In that case, you'll have

$$P = - \dfrac{\partial U}{\partial V}$$

This is often taken as definition of pressure. Then from $\delta W = - PdV$ you can obtain $W$ by integrating over the path. Finally, $Q$ is something that usually you get by it's "definition" $Q = \Delta U - W$, but you can also get by your method. In that case, one uses

$$T = \dfrac{\partial U}{\partial S}$$

For more information on this, I recommend seeing Thermodynamics and an Introduction to Thermostatistics by Herbert B. Callen.


As you wrote $W = \int_{\gamma} p(s) ds$ you are implying that the pressure, I assume you denoted pressure by $p$, is function of only the volume $V$ and the dummy integration variable is in fact the volume. That is not the case, the thermal equation of state even for the simplest system involves the (absolute or empirical) temperature, as well. That is $p=p(T,V)$. The same consideration goes for your other formula of heat exchange in the case of reversible process $Q_{rev} = \int_{\gamma'} T(s) ds$ where now $s$ is entropy along $\gamma'$. The integrand (caloric equation of state) should be written as $T=T(S,V)$, say, if you take volume as the "configuration coordinate". Note, too, that I have written $\gamma'$ for the path in the $S,V$ to calculate $Q_{rev}$ but $\gamma$ for the $T,V$ plane to calculate $W$. The two paths are different since they are paths on different coordinate planes.

As to the question where does one get the thermal or caloric equations of state $p=p(V,T)$ and $T=T(S,V)$ phenomenological (classical) thermodynamics says that in general you should measure them. For some specific homogeneous systems, such as for gases or crystalline solids, one can actually calculate them by using statistical mechanics.


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