# 7: State Functions and The First Law

• 7.1: Changes in a State Function are Independent of Path
We can specify an equilibrium state of a system by giving the values of a sufficient number of the system’s measurable properties. We call any measurable property that can be used in this way a state function or a state variable. If a system undergoes a series of changes that return it to its original state, any state function must have the same value at the end as it had at the beginning. A system can return to its initial state only all state variables returns to their original values.
• 7.2: The Total Differential
The total differential of the function is the sum over all of the independent variables of the partial derivative of the function with respect to a variable times the total differential of that variable.
• 7.3: Line Integrals
The significance of the distinction between exact and inexact differential expressions comes into focus when we use the differential, df, to find how the quantity, f, changes when the system passes from the state defined by (x₁, y₁) to the state defined by (x₂, y₂).
• 7.4: Exact Differentials and State Functions
• 7.5: Determining Whether an Expression is an Exact Differential
Since exact differentials have these important characteristics, it is valuable to know whether a given differential expression is exact or not. That is, given a differential expression of the form df=M(x,y)dx+ N(x,y)dy, we would like to be able to determine whether df is exact or inexact. It turns out that there is a simple test for exactness: The differential df=M(x,y)dx+ N(x,y)dy is exact if and only if ∂M/∂y=∂N/∂x.
• 7.6: The Chain Rule and the Divide-through Rule
The divide-through rule is a convenient way to generate thermodynamic relationships.
• 7.7: Measuring Pressure-Volume Work
Pressure–volume work is done whenever a force in the surroundings applies pressure on the system while the volume of the system changes. Because chemical changes typically do involve volume changes, pressure–volume work often plays a significant role. Perhaps the most typical chemical experiment is one in which we carry out a chemical reaction at the constant pressure imposed by the earth’s atmosphere.
• 7.8: Measuring Work- Non-Pressure-Volume Work
For chemical systems, pressure–volume work is usually important. Many other kinds of work are possible. From our vector definition of work, any force that originates in the surroundings can do work on a system. The force drives a displacement in space of the system or some part of the system. Stretching a strip of rubber is a one-dimensional analog of pressure–volume work. Changing the surface area of a liquid is a two-dimensional analog of pressure–volume work.
• 7.9: Measuring Heat
When we want to measure the heat added to a system, measuring the temperature increase that occurs is often the most convenient method. If we know the temperature increase in the system, and we know the temperature increase that accompanies the addition of one unit of heat, we can calculate the heat input to the system. Evidently, it is useful to know how much the temperature increases when one unit of heat is added to various substances.
• 7.10: The First Law of Thermodynamics
While we can measure the heat and work that a system exchanges with its surroundings, neither the heat nor the work is necessarily zero when the system traverses a cycle. Heat and work are not state functions. Nevertheless, adding heat to a system increases its energy. Likewise, doing work on a system increases its energy. If the system surrenders heat to the surroundings or does work on the surroundings, the energy of the system is decreased.
• 7.11: Other Statements of the First Law
The first law has been stated in many ways. Some are intended to be humorous or evocative rather than precise statements; for example, “You can’t get something (useful work in some system) for nothing (no decrease in the energy of some other system).” Others are potentially ambiguous, because we construct them to be as terse as possible. To make them terse, we omit ideas that we deem to be implicit.
• 7.12: Notation for Changes in Thermodynamic Quantities - E vs. ∆E
From the outset of our study of energy, we recognize that we are always dealing with energy changes.
• 7.13: Heat Capacities for Gases- Cv, Cp
• 7.14: Heat Capacities of Solids- the Law of Dulong and Petit
• 7.15: Defining Enthalpy, H
Any mathematical expression that involves only state functions must itself be a state function. We can define several state functions that have the units of energy and that turn out to be particularly useful. One of them is named enthalpy and is customarily represented by the symbol H. We define enthalpy: H = E + PV .
• 7.16: Heat Transfer in Reversible Processes
• 7.17: Free Expansion of a Gas
To develop the theory of thermodynamics, we must be able to model the thermodynamic properties of gases as functions of pressure, temperature, and volume. To do so, we consider processes in which the volume of a gas changes. For the expansion (or compression) of a gas to be a reproducible process, the exchange of heat between the system and its surroundings must be controlled.
• 7.18: Reversible vs. Irreversible Pressure-Volume Work
Gas in a piston can be compressed only if the applied pressure exceeds the gas pressure. If the applied pressure equals the gas pressure, the piston remains stationary. If the applied pressure is greater than the gas pressure by any ever-so-small amount, the gas will be compressed. If the applied pressure is infinitesimally less than the gas pressure, the gas will expand. The work done under such conditions is reversible work.
• 7.19: Isothermal Expansions of An Ideal Gas
• 7.20: Adiabatic Expansions of An Ideal Gas
• 7.21: Problems