Infinitesimal changes

Wlien a specimen is moved in or out of an electric field or when the field is increased or decreased, the total work done on the whole system (charged condenser + field + specimen) in an infinitesimal change is... 

A particular path from a given initial state to a given final state is the reversible process, one in which after each infinitesimal step the system is in equilibrium with its surroundings, and one in which an infinitesimal change in the conditions (constraints) would reverse the direction of the change. 

Equation (A2.1.15) involves only state fiinctions, so it applies to any infinitesimal change in state whether the actual process is reversible or not (although, as equation (A2.1.14) suggests, dS is not experimentally accessible unless some reversible path exists). 

Essentially this requirement means that, during die irreversible process, innnediately inside die boundary, i.e. on the system side, the pressure and/or the temperature are only infinitesimally different from that outside, although substantial pressure or temperature gradients may be found outside the vicinity of the boundary. Thus an infinitesimal change in p or T would instantly reverse the direction of the energy flow, i.e. the... 

There exists a state function S, called the entropy of a system, related to the heat Dq absorbedfrom the surroundings during an infinitesimal change by the relations... 

Alternatively, authors have repeatedly invoked the internal pressure of water as an explanation of the rate enhancements of Diels-Alder reactions in this solvent ". They were probably inspired by the well known large effects of the external pressure " on rates of cycloadditions. However, the internal pressure of water is very low and offers no valid explanation for its effect on the Diels-Alder reaction. The internal pressure is defined as the energy required to bring about an infinitesimal change in the volume of the solvents at constant temperature pi = (r)E / Due to the open and... 

The coordinates of thermodynamics do not include time, ie, thermodynamics does not predict rates at which processes take place. It is concerned with equihbrium states and with the effects of temperature, pressure, and composition changes on such states. For example, the equiUbrium yield of a chemical reaction can be calculated for given T and P, but not the time required to approach the equihbrium state. It is however tme that the rate at which a system approaches equihbrium depends directly on its displacement from equihbrium. One can therefore imagine a limiting kind of process that occurs at an infinitesimal rate by virtue of never being displaced more than differentially from its equihbrium state. Such a process may be reversed in direction at any time by an infinitesimal change in external conditions, and is therefore said to be reversible. A system undergoing a reversible process traverses equihbrium states characterized by the thermodynamic coordinates. 

Equation (1.4) states that if we add together all of the infinitesimal changes dZ over a closed path, the sum is equal to zero. This is a necessary condition for a state function. 

In equation (1.4), the infinitesimal change dZ is an exact differential. Later, we will describe the mathematical test and condition to determine if a differential is exact. 

Changing the pressure will have a similar effect. If we increase p by dp, the solid melts. This process can be reversed at any time by decreasing the pressure by dp. Note that at p = 1 atm (101.325 kPa), only at T = 273.15 K can the phase change be made to occur reversibly because this is the temperature where solid and liquid are in equilibrium at this pressure. If we tried to freeze liquid water aip— atm and a lower temperature such as 263.15 K, the process, once started, would proceed spontaneously and could not be reversed by an infinitesimal change in p or T. 

This is true for finite changes, AZ, or for infinitesimal changes, dZ, where we replace the summation in equation (A 1.11) with an integral sign and a circle to represent an integration over a closed path. That is... 

For a stationary material the change in the internal energy is equal to the difference between the net amount of heat added to the system and the net amount of work done by the system on its surroundings. For an infinitesimal change ... 

To calculate the work of reversible, isothermal expansion of a gas, we have to use calculus, starting at Eq. 3 written for an infinitesimal change in volume, dV ... 

The work done by any system on its surroundings during expansion against a constant pressure is calculated from Eq. 3 for a reversible, isothermal expansion of an ideal gas, the work is calculated from Eq. 4. A reversible process is a process that can be reversed by an infinitesimal change in a variable. 

Now suppose that the temperature of a sample is increased from T, to T2. overall change in entropy is the sum (integral) of all these infinitesimal changes ... 

To find the relation between the Gibbs free energy and the maximum nonexpansion work, we start with Eq. 15 for an infinitesimal change (denoted d) in G at constant temperature ... 

We then use Eq. 9 of Chapter 6 H = U + PV) to express the infinitesimal change in enthalpy at constant pressure in terms of the change in internal energy dH = dU + Pd V at constant pressure Then we substitute this expression into the first ... 

With the entropy term introduced, it is possible to make a combined statement of the first and the second laws. To do this, one may consider an infinitesimal change of state of a closed system. For this, the first law gives... 

In the case of an infrared spectrum, the intensity is related to the square of the infinitesimal change of the electric dipole moment p with respect to the normal coordinates,46 q,... 

An infinitesimal change in internal energy is an exact differential and is a unique function of temperature and pressure (for a given composition). Since the density of a given material is also uniquely determined by temperature and pressure (e.g., by an equation of state for the material), the internal energy may be expressed as a function of any two of the three terms T, P, or p (or v = 1/p). Hence, we may write ... 

For a constant-volume system, an infinitesimal change in temperature gives an infinitesimal change in internal energy and the constant of proportionality is the heat capacity at constant volume... 

Generally, a function f(x,y) for which an infinitesimal change may be expressed... 

The Gibbs energy is related to enthalpy and entropy through G = H - TS. For an infinitesimal change in the system... 

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