Clausius-Clapeyron equation and absorbed heat

[MaxLinus]

Homework Statement

Work out the Clausius-Clapeyron equation using the efficiency of a Carnot cycle.

Relevant Equations

Clausius-Clapeyron equation; Carnot efficiency

In working out the Clausius-Clapeyron equation in an elementary method, we usually consider the work done in a Carnot cycle, built with two isothermobarics at $p$ , $T$ and $p- dp$,$T-dT$ pressures and temperatures and two adiabatics, as $\mathcal{L} = dp(V_g - V_\ell)$, where $V_g$ is the volume of $n$ vaporized moles and $V_\ell$ is their volume in the liquid phase. The heat absorbed in the isothermobaric vaporization is usually set to $Q_a = n \lambda$, where $\lambda$ is the molar latent heat of vaporization. So the efficiency is $\eta =\frac{\mathcal{L}}{Q_a} = \frac{dp(V_g - V_\ell)} {n \lambda}$. Then the solution is easy: the efficiency must be equal to $dT/T$ (Carnot cycle), and you get
$$ \frac{dp}{dT} = \frac{n \lambda}{T(V_g - V_\ell)} $$But why the work $\mathcal{L}_1 = p (V_g - V_\ell)$ done in the isothermobaric expansion is not considered in the balance? The heat absorbed should account for vaporization and for expansion, whose effect appears not so easily neglibile: latent heat of water is about $2 \, MJ/kg$, while an expansion of $V_g - V_\ell = 1 \, m^3$ (not an exaggerated value) at atmospheric pressure ($\sim 100 \, kPa$) should contribute with approximately $100 \, kJ \sim 0.1 \, MJ$.

 

[TSny]

The heat absorbed should account for vaporization and for expansion

If I'm understanding it correctly, the definition of heat of vaporization does include the expansion work $p(V_g-V_l)$. For example, see the definitions here and here.

 

[MaxLinus]

Ok. The first link you posted states it very clearly.
Many thanks.

 

[MaxLinus]

Back again, just a bit.
In first problems about calorimetry, we usually do not deal with volumes, but only with moles and masses. Yet, vaporization takes place at a definite pressure. So, if I understand correctly your hint, we can still define a work in the expansion even if it occurs (for example) in an open vessel, but in an environment with pressure. Something related to partial volumes ...