Answer to Question #120759 in Molecular Physics | Thermodynamics for Abhishek

Question

Derive Clausius — Clapeyron equation. On the basis of this equation, explain why mountaineers find it difficult to cook their food at high altitudes.

Answer 1

Derivation of the Clapeyron-Clausius equation by the cycle method

An infinitesimal Carnot cycle is considered, which consists of a two-phase system consisting of a liquid and its saturated vapor. The mass of the system is equal to one kilogram.

Actual process properties

• The pressure of a saturated gas is uniquely determined by its temperature.

• The isotherm T=const is the same time isobar p=const.

• Isothermal temperatures in cyclic mode dt.

It is required to obtain an expression for the derivative along the evaporation curve.

The Carnot cycle performed by the system is shown in the figure.

As can be seen from the figure, the work completed in a cycle is equal to the area bounded by a parallelogram 1-2-3-4 in the diagram.

$$A=(v_1-v_2)dp$$

The amount of heat entering the system from the heater per cycle is equal to the specific heat of vaporization q.

We write the efficiency for the Carnot cycle through the temperatures of the refrigerator and heater and through the ratio of the perfect system of work to the amount of heat entering the system and equate these expressions

$$\frac {dT}{T}=\frac{dp(v_1-v_2)}{q}$$

Then

$$\frac{dp}{dT}=\frac{q}{T(v_1-v_2)}$$

At high altitude, the pressure is lower. Therefore, boiling water does not begin at the usual 100 degrees, but at a lower temperature. For example, at an altitude of about 2500 meters, water boils at a temperature of 90 degrees, and at a peak of 4000 meters already at 85. Molecules and water atoms during boiling are ejected beyond the liquid boundary, because the process of atom movement accelerates due to heating, and they overcome external pressure. Accordingly, the higher you go up into the mountains, the lower the temperature the water will begin to boil. And for cooking meat, like other products, you need the usual 100 degrees. Otherwise, all the components of the broth will simply remain raw.