11.1: Heat Capacity as a Function of Temperature The heat capacity of the solid substance decreases to zero as the absolute temperature decreases to zero; the curve meets the abscissa at the zero of temperature and does so asymptotically. That this is true for all substances seems like an odd sort of coincidence. Why should all solid substances exhibit essentially the same heat capacity (zero) at one temperature (absolute zero)? 11.2: Enthalpy as a function of Temperature The fact that Cₚ goes to zero asymptotically as the temperature goes to zero has no practical ramifications for the measurement or use of enthalpy. We can only measure changes in energy and enthalpy; no particular state of any system is a uniquely useful reference state for the enthalpy function. Experimental convenience is the only consideration that makes one reference state a better choice than another. 11.3: The Third Law The idea that the entropy change for a pure substance goes to zero as the temperature goes to zero finds expression as the third law of thermodynamics: If the entropy of each element in some crystalline state be taken as zero at the absolute zero of temperature, every substance has a positive finite entropy; but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances. 11.4: Genesis of the Third Law - the Nernst Heat Theorem The third law arises in a natural way in the development of statistical thermodynamics. It is probably fair to say that the classical thermodynamic treatment of the third law was shaped to a significant degree by the statistical thermodynamic treatment that developed about the same time. Nevertheless, we can view the third law as an inference from thermochemical observations. 11.5: Absolute Entropy At any given temperature, the entropy value that is obtained in this way is called the substance’s absolute entropy or its third-law entropy. When the entropy value is calculated for one mole of the substance in its standard state, the resulting absolute entropy is called the standard entropy. The standard entropy is usually given the symbol So . It is usually included in compilations of thermodynamic data for chemical substances. 11.6: The Standard State for Third-law Entropies The standard state for entropies is essentially the same as the standard state for enthalpies. For liquids and solids, the standard state for entropies is identical to that for enthalpies: At any given temperature, the standard state is the most stable form of the substance at that temperature and a pressure of 1 bar. 11.7: The Fugacity of a Gas For an ideal gas, the Gibbs free energy is a simple function of its pressure. It turns out to be useful to view the integral as a contribution to a “corrected pressure.” The “correction” is an adjustment to the pressure that, in our calculations, makes the real gas behave as an ideal gas. The idea is that we can express the Gibbs free energy as a function of this corrected pressure, which we call the fugacity. Fugacity is therefore a function of pressure. 11.8: A General Strategy for Expressing the Thermodynamic Properties of a Substance Our goal is to create a scheme in which the enthalpy, the entropy, or the Gibbs free energy of any substance in any arbitrary state is equal to the change in that thermodynamic property when the substance is produced, in that state, from its pure, separate, constituent elements, in their standard states at the same temperature. 11.9: The Standard Entropy and the Gibbs Free Energy of Formation 11.10: The Nature of Hypothetical States The hypothetical ideal gas standard state is a wholly theoretical construct. We create this “substance” only because it is convenient to have a name for the “unreal” state of substance A. 11.11: The Fugacity and Gibbs Free Energy of A Substance in Any System When we define the chemical activity of a substance in a particular system, we also introduce a new standard state. The primary criterion for our choice of this activity standard state is that we be able to measure how much the Gibbs energy of the substance differs between the activity standard state and other states of the system. A principal object of the next chapters is to introduce ideas for measuring the difference between the Gibbs energy of a substance in two states of a given system. 11.12: Evaluating Entropy Changes Using Thermochemical Cycles As for the standard enthalpy of reaction, we can obtain the standard entropy of reaction at a new temperature by evaluating entropy changes around a suitable thermochemical cycle. To do so, we need the standard entropy change at one temperature. We also need heat capacity data for all of the reactants and products. 11.13: Absolute Zero is Unattainable The third law postulates that the entropy of a substance is always finite and that it approaches a constant as the temperature approaches zero. The value of this constant is independent of the values of any other state functions that characterize the substance. For any given substance, we are free to assign an arbitrarily selected value to the zero-temperature limiting value. However, we cannot assign arbitrary zero-temperature entropies to all substances. 11.14: Problems