Thermodynamics and Propulsion

8.4 The Clausius-Clapeyron Equation (application of 1^st and 2^nd laws of thermodynamics)

Until now we have only considered ideal gases and we would like to show that the properties $u$ , $h$ , $s$ , etc. are true state variables and that the 1^st and 2^nd laws of thermodynamics hold when the working medium is not an ideal gas (i.e. a two-phase medium). An elegant way to do this is to consider a Carnot cycle for a two-phase medium. To state the fact that all Carnot engines operated between two given temperatures have the same efficiency is one way of stating the 2^nd law of thermodynamics. The working fluid need not be an ideal gas and may be a two-phase medium changing phases.

The idea is to run a Carnot engine between temperatures $T$ and $T-dT$ for a two-phase medium and to let it undergo a change in phase. We can then derive an important relation known as the Clausius-Clapeyron equation, which gives the slope of the vapor pressure curve. We could then measure the vapor pressure curve for various substances and compare the measured slope to the Clausius-Clapeyron equation. This can then be viewed as an experimental proof of the general validity of the 1^st and 2^nd laws of thermodynamics!

Figure 8.8: Carnot cycle devised to test the validity of the laws of thermodynamics

Consider the infinitesimal Carnot cycle $abcd$ shown in Figure 8.8. Heat is absorbed between states $a$ and $b$ . To vaporize an arbitrary amount of mass, $m$ , the amount of heat

$$Q = m h_{fg}$$ (8..1)

must be supplied to the system. From the 1^st and 2^nd laws of thermodynamics the thermal efficiency for a Carnot cycle can be written as

$$\eta = \frac{W}{Q} = \frac{Q-Q_R}{Q} = \frac{T-T_R}{T}.$$

Hence, for the infinitesimal cycle considered above,

$$\frac{dW}{Q} = \frac{T-(T-dT)}{T}=\frac{dT}{T}.$$ (8..2)

The work along $bc$ and $da$ nearly cancel such that the net work is the difference between the work along $ab$ and $cd$ , and $dW$ can be viewed as the area enclosed by the rectangle $abcd$ :

$$dW = p m(v_g-v_f) - (p-dp) m(v_g-v_f)$$

$$=m(v_g-v_f)dp.$$ (8..3)

Substituting Equations (8.1) and (8.3) into (8.2) one obtains

$$\frac{m(v_g-v_f)dp}{m h_{fg}} = \frac{dT}{T}.$$

Rearranging terms yields the Clausius-Clapeyron equation, which defines the slope of the vapor pressure curve:

$$\frac{dp}{dT} = \frac{h_{fg}}{T(v_g-v_f)}.$$ (8..4)

The beauty is that we have found a general relation between experimentally measurable quantities from first principles (1^st and 2^nd laws of thermodynamics).

In order to plot the Clausius-Clapeyron relation and to compare it against experimentally measured vapor pressure curves, we need to integrate Equation (8.4). To do so, the heat of vaporization and the specific volumes must be known functions of temperature. This is an important problem in physical chemistry but we shall not pursue it further here except to mention that if

• variations in heat of vaporization can be neglected,
• the vapor phase is assumed to be an ideal gas, and
• the specific volume of the liquid is small compared to that of the vapor phase,

$$h_{fg} \approx \textrm{const,} \qquad v_f \ll v_g \approx \frac{RT}{p},$$

the integration can be readily carried out^8.1. Making these approximations, the Clausius-Clapeyron equation becomes

$$\frac{dp}{dT} \approx \frac{h_{fg}p}{T^2R} \qquad \longrightarrow \qquad \frac{dp}{p} \approx \frac{h_{fg}}{R}\frac{dT}{T^2}.$$

Carrying out the integration, the resulting expression is

$$\ln p = - \frac{h_{fg}}{R} \frac{1}{T} + \ln C.$$

Note that the vapor pressure curves are straight lines if $\ln p$ is plotted versus $1/T$ and that the slope of the curves is $-h_{fg}/R$ , directly related to the heat of vaporization. Figures 8.9, 8.9, and 8.22 depict the vapor pressure curves for various substances. The fact that all known substances in the two-phase region fulfill the Clausius-Clapeyron equation provides the general validity of the 1^st and 2^nd laws of thermodynamics!

Figure 8.9: Clausius-Clapeyron Experimental Proof (1) [Gyarmathy, Thermodynamics I, Eidgen $\ddot{\textrm{o}}$ ssische Technische Hochschule Z $\ddot{\textrm{u}}$ rich, 1992]

Figure 8.10: Clausius-Clapeyron Experimental Proof (2) [Mahan, Elementary Chemical Thermodynamics, 1963]