Clausius Clapeyron Equation

Dhristi JEE 2022-24

Clausius Clapeyron Equation Thermodynamics

The Clausius Clapeyron equation is a way of describing a discontinuous phase transformation between two phases of matter of a single constituent, named after Rudolf Clausius and Benoît Paul Émile Clapeyron. A straight line does not exist between a liquid's temperature and its vapour pressure.


The vapour pressure in the water, for example, rises at a much faster rate than the system's temperature. The Clausius equation will describe this action. The temperature of a system determines the balance between a liquid and its vapour; an increase in temperature induces a subsequent rise in the vapour pressure of the liquid.

 

The Clausius Clapeyron equation Thermodynamics is as follows,

\[ln \frac{P_{2}}{P_{1}} = \frac{\Delta H_{vap}}{R} (\frac{1}{T1} - \frac{1}{T2})\]

To determine the ranges of hydrate stability, the Clausius Clapeyron equation can be applied to a hydrating system and used to estimate the equilibrium water behaviour for a hydrate pair occurring in equilibrium at various temperatures.


For hydrate systems, the Clausius Clapeyron equation was obtained by assuming that the higher and lower hydrates occur in equilibrium as three pure phases of water vapour and that the average volume transition involved in this equilibrium process is approximated by the volume of the emitted water vapour which behaves ideally.


A Thermogravimetric analyzer with an attached water vapour distribution system was used to dynamically assess the equilibrium water vapour pressure for the nedocromil sodium monohydrate and trihydrate systems at various temperatures.


The enthalpy of dehydration obtained by applying the Clausius Clapeyron equation to experimentally defined equilibrium water vapour pressures agrees very well with the enthalpy of dehydration resulting from differential scanning calorimetry (13.7 +/- 0.6 kcal/mol of water loss, n = 5), meaning that the Clausius Clapeyron equation can be applied to organic hydrate structures.


The Clausius Clapeyron Equation Derivation

The Clausius Clapeyron equation predicts the rate at which vapour pressure increases per unit increase in temperature for a substance's vapour pressure (P) and temperature (T).

\[\frac{d  ln  p}{dT} = \frac{\Delta H_{vap}}{RT_{2}}\]

The molar enthalpy of vaporisation of the liquid, the ideal gas constant, and the temperature of the system determines the rate at which the natural logarithm of the vapour pressure of a liquid varies with temperature, according to this equation.

If Hvap is assumed to be independent of the system's temperature, the Clausius Clapeyron equation can be written in the integrated form below, where C is a constant.

Where Hvap is the liquid's enthalpy of evaporation, R is the gas constant, and A is a constant whose value is determined by the substance's chemical identity.

In this equation, the temperature (T) must be in kelvin. Since the relationship between vapour pressure and temperature is not linear, the equation is often rewritten in the logarithmic form to provide the following linear equation:

\[ln(P) = \frac{-\Delta H_{vap}}{RT} + ln A\]

If the enthalpy of evaporation and vapour pressure at a given temperature is defined for every liquid, the Clausius Clapeyron equation can be used to calculate the vapour pressure at a different temperature.

The linear equation can be formulated in a two-point format to accomplish this. If the vapour pressure at temperature T1 is P1 and the vapour pressure at temperature T2

 is P2, the corresponding linear equations are:

\[ln (P_{1}) = -\frac{\Delta H_{vap}}{RT_{1}} + ln A\]

And

\[ln(P_{2}) = -\frac{\Delta H_{vap}}{RT_{2}} + ln A\]

Since the constant, A, is the same in both equations, they can be rearranged to separate ln A and then made equal: 

\[ln (P_{1}) = -\frac{\Delta H_{vap}}{RT_{1}} + ln A  =  ln(P_{2}) = -\frac{\Delta H_{vap}}{RT_{2}} + ln A\]

\[ln (P_{1}) = -\frac{\Delta H_{vap}}{RT_{1}} = ln(P_{2}) = -\frac{\Delta H_{vap}}{RT_{2}}\]

which can be combined into:

\[ln \frac{P_{2}}{P_{1}} = \frac{\Delta H_{vap}}{R} (\frac{1}{T1} - \frac{1}{T2})\]

Hence it is the Clausius clapeyron equation.


Potential Uses of Clausius Clapeyron Equation 

The Clausius clapeyron equation has many Potential uses. Some examples are:

  • We can use thermodynamic data to calculate the slope of a metamorphic reaction to see whether it could be used as a geothermometer or geobarometer. A geobarometer could be a reaction with a shallow dP/dT slope because it is more sensitive to pressure changes. A reaction with a steep (nearly vertical) slope is vulnerable to temperature and maybe a geothermometer.

  • We can quantify the slope and extrapolate to other conditions instead of conducting more time-consuming tests if we have experimental effects on a reaction at one temperature (or pressure).

  • The Clausius equation is used to calculate the thermodynamic values of reactions and phases. The slope of an experimentally defined reaction can be used to measure the S of the reaction and the entropy of formation (Sf) of a given process when combined with volume results. The amounts of phases are often well-known, but the entropy data can be subject to significant uncertainty.

  • We can use the Clausius equation to accurately place reactions around an invariant point if we do a Schreinemakers analysis.


Clausius Clapeyron Equation Derivation in Thermodynamics

The temperature of the system affects the balance between water and water vapour.

The saturation pressure of water vapour rises as the temperature rises. The Clausius Clapeyron equation calculates the rate of increase in vapour pressure per unit increase in temperature. Let T be the temperature and p be the saturation vapour pressure. The Clausius Clapeyron equation for liquid-vapour equilibrium is then used.

\[\frac{dp}{dT} = \frac{L}{(T(V_{v} - V_{l}))}\]

where L is the latent heat of evaporation and Vv and Vl are the vapour and liquid phases' real volumes at temperature T, respectively.

The Clausius Clapeyron equation, in a larger perspective, describes the relationship between pressure and temperature in two-phase equilibrium.

For sublimation, the two phases could be vapour and solid, or solid and liquid for melting.

Finally, the Clausius Clapeyron equation is derived using thermodynamic principles.

Note that \[V_{v}\]  is very much greater than Vl so that is a good approximation.

\[\frac{dp}{dT} = \frac{L}{TV_{v}}\]

Furthermore, the ideal gas equation is applied to the vapour, i.e,

\[V_{v} = \frac{RT}{p}\]

where R is the universal gas constant.

Thus,

\[(\frac{1}{p}) (\frac{dp}{dT}) = \frac{L}{RT^{2}}\]

In differential form this is,

\[d(ln(p)) = \frac{L}{R} d\frac{-1}{T}\]

If L is independent of temperature then the solution of the above differential equation is,

\[P = c_{1} exp (-\frac{L}{RT})\]

where C1 is a constant.

The Clapeyron Clausius equation can be written for different phase equilibria - solid-vapour, solid-liquid and liquid-vapour.

For the integration of the Clapeyron-Clausius equation, it is essential to know the specific volume and the explicit relationship between pressure and temperature, and for the enthalpy of phase transition. 

Despite the respective functions being complicated and unknown, the integration can still be carried out in some cases. 

At moderate pressures, the vapour specific volume is several times greater than the liquid or solid specific volume. It is therefore possible to neglect the values of vapour solid and vapour equations. 

Even then, the vapour's specific volume can, with reasonable accuracy, be derived from the perfect gas equation of state.


Applications of the Clausius Clapeyron equation

The equation can be used for the following:

  1. To determine the slope of a metamorphic reaction from thermodynamic data. This is used to find out if it could be a potential geothermometer or geobarometer. 

  2. The equation can help us determine thermodynamic values for reactions or phases. 

  3. When we perform a Schreinemakers analysis of an invariant point, we can use the Clausius Clapeyron equation to position reactions correctly.

  4. By using the experimental results on a certain reaction at one temperature or pressure, we can calculate the slope, and look at other conditions, instead of conducting other experiments that could make it time-consuming.

FAQs on Clausius Clapeyron Equation

1. Who Discovered the Clausius Clapeyron Equation?

The Clausius-Clapeyron equation is named after Rudolf Clausius, a 19th-century German physicist, and Émile Clapeyron, a 19th-century French engineer.

This relationship suggests that an improvement in the ability of air to retain water vapour is a result of the rising temperature of the amount of air, assuming there is a liquid water surface in equilibrium with the atmosphere.

2. What is the Unit of Clausius Clapeyron?

  • The natural log term on the left-hand side (LHS) is unitless. It makes no difference what pressure units you use; the only requirement is that P₁ and P₂ must be represented in the same pressure unit.

  • The unit on the temperature term will be K⁻¹. The unit on the R is J mol⁻¹ K⁻¹.

  • The K⁻¹ in the temperature will cancel with the K⁻¹ associated with R.

  • Hence it means that the unit on ΔH must be J/mol.

3. Who derived the Clausius Clapeyron equation?

  • The Clausius Caleyron equation is named after 19th-century German physicist Rudolf Clausius and 19th-century French engineer Émile Clapeyron, who derived it.

  • Rudolf Clausius had deduced the Clausius–Clapeyron relation, which is a method of characterising the phase transition between two states of matter (solid-liquid, liquid-vapour, and solid-vapour), from thermodynamics. 

  • Originally though, this relation had been developed by Émile Clapeyron in 1834. 

  • Clapeyron had also developed the idea of a reversible process, which had already been suggested by Nicholas Carnot earlier. Clapeyron made a definitive statement of Carnot's principle, now known as the second law of thermodynamics. 

  • These foundations were what enabled Clapeyron to make substantive extensions of Clausius' work, including the formula, now known as the Clausius–Clapeyron relation. 

4. How is the Clausius–Clapeyron used in cloud study?

  • The Clausius–Clapeyron equation is used to do a comparative study of the properties of the clouds that form in planetary atmospheres. 

  • For this process, the simple static atmospheric models for various planets, satellite Titan, and the extrasolar planet HD209458b are used together. The saturation vapour pressure curves of different kinds of molecules are also used to find the following: the density, pressure, and the scale height of the clouds. 

  • This application helps us further our knowledge and understanding of water clouds on other planets.

5. What are the assumptions in the determination of the vapour pressure of water, using the Clausius-Clapeyron equation?

The determination of the vapour pressure of water, using the Clausius-Clapeyron equation, requires the following assumptions:

  • The saturated vapour is an ideal gas.

  • The heat of vapourization is constant over the temperature range of interest.

  • The saturated vapour’s molar volume is much greater than that of the saturated liquid.

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