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Chapter 5: Q 5.70 (page 199)

What happens when you add salt to the ice bath in an ice cream maker? How is it possible for the temperature to spontaneously drop below 0"C? Explain in as much detail as you can.

Short Answer

Expert verified

The ice cream bath is always below 0 degrees Celsius.

Step by step solution

01

Given information

Add salt to the ice bath in an ice cream maker.

02

Explanation

We know that water turns to the ice (solid) phase at 0°C, but the temperature of the ice bath must be less than 0°C, which can be accomplished by adding salt to the ice bath system. Adding salt to the ice bath system lowers the phase temperature, resulting in a lower transition temperature between the phases. The kinetic energy of the particles in the mixture determines the temperature of the mixture, and as the kinetic energy reduces, the temperature of the mixture decreases.

03

Explanation

The ice cream and salt mixture are insulated from the environment in the ice cream making system, which means that the energy required to break the chemical bonds of the salt crystal must be taken from the kinetic energy of the mixture particle, resulting in a decrease in kinetic energy and thus the temperature. As a result, the ice cream bath is always below 0 degrees Celsius.

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Most popular questions from this chapter

Consider an ideal mixture of just 100 molecules, varying in com- position from pure A to pure B. Use a computer to calculate the mixing entropy as a function of NA, and plot this function (in units of k). Suppose you start with all A and then convert one molecule to type B; by how much does the entropy increase? By how much does the entropy increase when you convert a second molecule, and then a third, from A to B? Discuss.

The enthalpy and Gibbs free energy, as defined in this section, give special treatment to mechanical (compression-expansion) work, -PdV. Analogous quantities can be defined for other kinds of work, for instance, magnetic work." Consider the situation shown in Figure 5.7, where a long solenoid ( Nturns, total length N) surrounds a magnetic specimen (perhaps a paramagnetic solid). If the magnetic field inside the specimen is B→and its total magnetic moment is M→, then we define an auxilliary field H→(often called simply the magnetic field) by the relation

H→≡1μ0B→-M→V,

where μ0is the "permeability of free space," 4π×10-7N/A2. Assuming cylindrical symmetry, all vectors must point either left or right, so we can drop the -→symbols and agree that rightward is positive, leftward negative. From Ampere's law, one can also show that when the current in the wire is I, the Hfield inside the solenoid is NI/L, whether or not the specimen is present.

(a) Imagine making an infinitesimal change in the current in the wire, resulting in infinitesimal changes in B, M, and H. Use Faraday's law to show that the work required (from the power supply) to accomplish this change is Wtotal=VHdB. (Neglect the resistance of the wire.)

(b) Rewrite the result of part (a) in terms of Hand M, then subtract off the work that would be required even if the specimen were not present. If we define W, the work done on the system, †to be what's left, show that W=μ0HdM.

(c) What is the thermodynamic identity for this system? (Include magnetic work but not mechanical work or particle flow.)

(d) How would you define analogues of the enthalpy and Gibbs free energy for a magnetic system? (The Helmholtz free energy is defined in the same way as for a mechanical system.) Derive the thermodynamic identities for each of these quantities, and discuss their interpretations.

The methods of this section can also be applied to reactions in which one set of solids converts to another. A geologically important example is the transformation of albite into jadeite + quartz:

NaAlSi3O8⟷NaAlSi2O6+SiO2

Use the data at the back of this book to determine the temperatures and pressures under which a combination of jadeite and quartz is more stable than albite. Sketch the phase diagram of this system. For simplicity, neglect the temperature and pressure dependence of both ∆S and ∆V.

Suppose that an unsaturated air mass is rising and cooling at the dry adiabatic lapse rate found in problem 1.40. If the temperature at ground level is 25 C and the relative humidity there is 50%, at what altitude will this air mass become saturated so that condensation begins and a cloud forms (see Figure 5.18)? (Refer to the vapor pressure graph drawn in Problem 5.42)

A mixture of one part nitrogen and three parts hydrogen is heated, in the presence of a suitable catalyst, to a temperature of 500° C. What fraction of the nitrogen (atom for atom) is converted to ammonia, if the final total pressure is 400 atm? Pretend for simplicity that the gases behave ideally despite the very high pressure. The equilibrium constant at 500° C is 6.9 x 10-5. (Hint: You'l have to solve a quadratic equation.)
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