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Chapter 5: Q 5.43 (page 177)

Assume that the air you exhale is at 35掳C, with a relative humidity of 90%. This air immediately mixes with environmental air at 5掳C and unknown relative humidity; during the mixing, a variety of intermediate temperatures and water vapour percentages temporarily occur. If you are able to "see your breath" due to the formation of cloud droplets during this mixing, what can you conclude about the relative humidity of your environment? (Refer to the vapour pressure graph drawn in Problem 5.42.)

Short Answer

Expert verified

The relative humidity required to produce cloud droplets is around 25%.

Step by step solution

01

Given information

Consider the initial water vapour pressure of air mixed at temperatures of 10掳C and 35掳C. When the air particles have the same mass, the air mixture should have a half-temperature of between 10 and 35 degrees Celsius.

The reason for this is that air's heat capacity is independent of its temperature. In the same way, the water partial pressure is half way between the air particle starting pressures. The air mixture temperature and water partial pressure lie on the straight line in the figure for any value of the initial mass ratio.

02

Explanation

Use the vapour pressure graph from problem 5.42 to graph the compositions of exhaled air at 35掳 and 90% relative humidity, as well as the composition of outside air, as illustrated in the diagram below.

P is the partial pressure of water in bar, and T is the temperature in degrees Celsius. In the diagram above, the lower dot represents the external air temperature (10掳C) and the top dot represents the exhaled air temperature (35掳C).

03

Conclusion

Cloud droplets are formed when the state of the mixture is above the equilibrium curve, as indicated in the diagram. Because both initial states are below the curve, this is conceivable.

When the outdoor air dot goes vertically downward, the vapour pressure curve may be crossed. The minimal partial pressure in this situation is 0.003 bar, while the relative humidity is around 25%, according to the graph.

As a result, the relative humidity required to produce cloud droplets is around 25%.

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Most popular questions from this chapter

Sketch a qualitatively accurate graph of G vs. T for a pure substance as it changes from solid to liquid to gas at fixed pressure. Think carefully about the slope of the graph. Mark the points of the phase transformations and discuss the features of the graph briefly.

Functions encountered in physics are generally well enough behaved that their mixed partial derivatives do not depend on which derivative is taken first. Therefore, for instance,

鈭�鈭�V鈭�U鈭�S=鈭�鈭�S鈭�U鈭�V

where each 鈭�/鈭�Vis taken with Sfixed, each 鈭�/鈭�Sis taken with Vfixed, and Nis always held fixed. From the thermodynamic identity (forU) you can evaluate the partial derivatives in parentheses to obtain

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a nontrivial identity called a Maxwell relation. Go through the derivation of this relation step by step. Then derive an analogous Maxwell relation from each of the other three thermodynamic identities discussed in the text (for H,F,andG ). Hold N fixed in all the partial derivatives; other Maxwell relations can be derived by considering partial derivatives with respect to N, but after you've done four of them the novelty begins to wear off. For applications of these Maxwell relations, see the next four problems.

Graphite is more compressible than diamond.

(a) Taking compressibilities into account, would you expect the transition from graphite to diamond to occur at higher or lower pressure than that predicted in the text?

(b) The isothermal compressibility of graphite is about 3 x 10-6 bar-1, while that of diamond is more than ten times less and hence negligible in comparison. (Isothermal compressibility is the fractional reduction in volume per unit increase in pressure, as defined in Problem 1.46.) Use this information to make a revised estimate of the pressure at which diamond becomes more stable than graphite (at room temperature).

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(d) Make a rough estimate of the pressure under the blade of an ice skate, and calculate the melting temperature of ice at this pressure. Some authors have claimed that skaters glide with very little friction because the increased pressure under the blade melts the ice to create a thin layer of water. What do you think of this explanation?

By subtracting 渭Nfrom localid="1648229964064" U,H,F,orG,one can obtain four new thermodynamic potentials. Of the four, the most useful is the grand free energy (or grand potential),

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(d) As a simple application, let the system be a single proton, which can be "occupied" either by a single electron (making a hydrogen atom, with energy -13.6eV) or by none (with energy zero). Neglect the excited states of the atom and the two spin states of the electron, so that both the occupied and unoccupied states of the proton have zero entropy. Suppose that this proton is in the atmosphere of the sun, a reservoir with a temperature of 5800Kand an electron concentration of about 2脳1019per cubic meter. Calculate 桅for both the occupied and unoccupied states, to determine which is more stable under these conditions. To compute the chemical potential of the electrons, treat them as an ideal gas. At about what temperature would the occupied and unoccupied states be equally stable, for this value of the electron concentration? (As in Problem 5.20, the prediction for such a small system is only a probabilistic one.)
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