Your friend is working on a design as part of a science project. She plans to minimize the excessive heating of a solid (object A) by automatically bringing it into thermal contact with another solid at lower temperature (object B) to transfer any excessive heat. She has been provided with graphs of the objects' entropy (S) as a function of internal energy (U).As part of the project, she needs to justify why her project will work. She knows heat will flow spontaneously from object A to B due to their differences in temperature, but she wants an additional explanation that does not rely purely on the temperature difference.How could she use the provided graphs to justify why her project will work? Assume that there are no thermal expansions/compressions and the system is designed such that the heat loss to the environment is minimized.

Note: UA.initial and UB.initial for both A and B represent the internal energies associated with each object separately before they are brought into thermal contact. The two graphs provided have the same scale.

b) Semiconductors are the substances whose conductivity lies between that of conductors and insulators. Conductivity of semiconductors can be enhanced through the process of doping in which an impurity (e.g., Gallium) is added to a crystal (e.g., Germanium).The addition of each Gallium atom to a Germanium crystal results in the formation of a vacancy (V) which can accommodate an electron from a neighbouring atom as shown inFigure-1. Thermal fluctuations can cause electrons in the crystal to jump' and occupy the vacancy, resulting in the formation of a new vacancy in the electron's initial position as shown in Figure-2. The number of lattice sites in the crystal is N (where N > 3).

Determine the change in the entropy of the Germanium-Gallium structure by calculating the number of microstates available for this vacancy upon addition of

(i)one atom and

(ii) two atoms of Gallium.

Express your answer in terms of N and fundamental constants. Is the process of doping entropically favorable? Explain your reasoning with appropriate principles.

c) Make a qualitatively accurate sketch of G vs. pressure P for the three phases of water at exactly 0°C. Label which portions of the graph correspond to the liquid, gas, and solid phases and explain (with words or equations) any other important features of your graph.Include in your explanation discussion of all physically relevant features associated with the slope of your line at all regions/points.



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