Heating at constant volume or pressure

Consider the process of changing the temperature of a phase at constant volume. The rate of change of internal energy with T under these conditions is the heat capacity at constant volume Cv = (9t//dT)v (Eq. 7.3.1). Accordingly, an infinitesimal change of U is given by 

Three comments, relevant to these and other equations in this chapter, are in order  

Cy may depend on both V and T, and we should integrate with V held constant and Cy treated as a function only of T. 

Suppose we want to evaluate AU for a process in which the volume is the same in the initial and final states (V2 = V ) but is different in some intermediate states, and the temperature is not uniform in some of the intermediate states. We know the change of a state function depends only on the initial and final states, so we can still use Eq. 7.4.2 to evaluate AU for this process. We integrate with V held constant, although V was not constant during the actual process. 

In general A finite change AX of a state function, evaluated under the condition that another state function Y is constant, is the same as AX under the less stringent condition F2 = Fi. 

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We wish to determine (a) the behavior of radiation as a gas, (b) the specific heat (at constant volume or pressure) of radiation, and (c) the relation governing the isen tropic process of radiation. 

Entropy is sometimes said to he a measure of disorder. According to this idea, the entropy increases whenever a closed system becomes more disordered on a microscopic scale. This description of entropy as a measure of disorder is highly misleading. It does not explain why entropy is increased by reversible heating at constant volume or pressure, or why it increases during the reversible isothermal expansion of an ideal gas. Nor does it seem to agree with the freezing of a supercooled hquid or the formation of crystalline solute in a supersaturated solution these processes can take place spontaneously in an isolated system, yet are accompanied by an apparent decrease of disorder. 

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES 7.4 Heating at Constant Volume or Pressure... 

The SI unit for heat capacity is J-K k Molar heat capacities (Cm) are expressed as the ratio of heat supplied per unit amount of substance resulting in a change in temperature and have SI units of J-K -moC (at either constant volume or pressure). Specific heat capacities (Cy or Cp) are expressed as the ratio of heat supplied per unit mass resulting in a change in temperature (at constant volume or pressure, respectively) and have SI units of J-K -kg . Debye s theory of specific heat capacities applies quantum theory in the evaluation of certain heat capacities. 

Heat capacity is always positive on the MCAT the temperature will always increase when energy is added to a substance at constant volume or pressure. In the real world, heat capacity also changes with temperature tire amount of energy that a substance can absorb per change in temperature varies with the temperature. However, unless otherwise indicated, for the MCAT, assume that heat capacity does not change with temperature. 

If the external work of expansion due to heating is zero, as it is when a material is heated at constant volume, or if it is negligible, as it is when solids are heated at atmospheric pressure, all the heat supplied goes into internal energy and we can approximate AHt by AEt. It is values of AH298 that you will find tabulated. The variation of AG with temperature is illustrated in Figure 3.5. 

Cp Molar heat capacities at constant volume or pressure [thermics + physical chemistry]... 

If gases are heated or cooled at constant volume, the pressure changes, also at the rate of 7-5 of its value at 0°C. Then the pressure of a gas... 

Corollary. dVJ is a perfect differential when the pressure is constant, and Qi is independent of the path. The independence of the heat effect on the path requires that the change shall occur either at constant volume or at constant pressure. If the volume is maintained constant (dv = 6) the pressure may be changed in any way if the pressure is maintained constant (dp = o) the volume may be altered in any manner so that the limiting conditions are satisfied but if both pressure and volume change... 

This is equivalent to the statement that the evolution of heat is dependent solely on the initial and final states, and independent of the intermediate states. In this form, however, the principle is indefinite, because the evolution of heat will depend on the conditions of the system whilst the reaction is occurring. As a matter of fact Hess s principle is strictly true only when by quantity of heat evolved we understand that evolved when the reaction progresses at constant volume, or when it occurs under a constant pressure. 

Any characteristic of a system is called a property. The essential feature of a property is that it has a unique value when a system is in a particular state. Properties are considered to be either intensive or extensive. Intensive properties are those that are independent of the size of a system, such as temperature T and pressure p. Extensive properties are those that are dependent on the size of a system, such as volume V, internal energy U, and entropy S. Extensive properties per unit mass are called specific properties such as specific volume v, specific internal energy u, and specific entropy. s. Properties can be either measurable such as temperature T, volume V, pressure p, specific heat at constant pressure process Cp, and specific heat at constant volume process c, or non-measurable such as internal energy U and entropy S. A relatively small number of independent properties suffice to fix all other properties and thus the state of the system. If the system is composed of a single phase, free from magnetic, electrical, chemical, and surface effects, the state is fixed when any two independent intensive properties are fixed. 

The actual amount of heat we measure experimentally for a given reaction depends somewhat on (a) the temperature of the experiment and (b) whether the experiment is run at constant volume or constant pressure. The basic reasons for this are that (a) each reactant and product has a characteristic specific heat that varies individualistically with temperature, and (b) at constant pressure, some of the heat of reaction may expand or compress gases if they are... 

We next offer a simple way to calculate the heat effect at constant pressure from that observed at constant volume, or vice versa. First, note that the product ofP and V always has the units of energy. A simple bit of evidence for this observation comes from the ideal gas law, using the value of 1.987 cal/mole K for R ... 

A system undergoes a two-step process. In step 1, it absorbs 50 J of heat at constant volume. In step 2, it gives off 5 J of heat at a constant pressure of 1.00 atm as it is returned to the same internal energy it had originally. Find the change in volume of the system during the second step and identify it as an expansion or compression. 

The state of a multivariant system is defined by assigning values to either the temperature, volume, and mole numbers of the components or the temperature, pressure, and mole numbers. Thus, we define heat capacities at constant volume or heat capacities at constant pressure for such closed systems. The equations and method of calculation are exactly the same as those outlined for univariant systems when the heat capacity at constant volume is desired. For the heat capacity at constant pressure, Equation (9.14) or (9.15) and the set of equations, one for each component, illustrated by Equation (9.18) are still applicable. The method of calculation is the same, with the exception that the volume of the system is a dependent variable... 

The heat required to heat a substance depends on just how that heating is conducted. The most common ways of heating are at constant volume or at constant pressure. In applying the first law to the process of heating a system at... 

The specific heat capacity is the heat that must be added per kg of a substance to raise the temperature by one Kelvin or one degree Celsius. The molar heat capacity is the specific heat multiplied by the molar mass (the molar mass of a structural unit in the case of polymers). Specific and molar heat capacity may be defined at constant volume or at constant pressure. The heat added causes a change in the internal energy (It) and in the enthalpy (heat content, H) of the substance. The following notations can be formulated ... 

The calculated values do not exactly agree with those obtained by experiment if the explosion takes place in a bomb, the true compositions of the explosion products are different and, moreover, vary with the loading density. In accurate calculations these factors must be taken into account. In difficult cases (strongly oxygen-deficient compounds and side reactions, such as the formation of CH4, NH3, HCN, or HCI), the only way is to analyze the explosion products. For standard values of heats of formation at constant volume or constant pressure -> Energy of Formation. 

The reaction-point.— Take, at a very low temperature, a system in the state of false equilibrium and gradually raise the temperature at a certain moment the system will cease to be in false equilibrium and a reaction will be produced. The temperature at which a given system, under a given pressure or maintained at a given volume, ceases to be in the state of false equilibrium and becomes the seat of a chemical modification, is called the reax tim-point of this system. Thus the reaction-point of a system which contains hydrogen and selenium, without trace of selenhydric acid, and which is heated at constant volume, is close to 250 at this temperature selenhydric acid begins to be formed. 

The quantities Cy and Cp are defined as shown in (1.13.15), and are known as heat capacities at constant volume or at constant pressure. These can be experimentally determined as a function of T over wide temperature ranges, normally by standard calorimetric methods (see also Section 1.16). Integration of the experimental heat capacities with respeet to temperature then yields 2 , //, or S as a function of T (see Section 1.17), with either K or 7 as parameters. Alternatively, by inserting the equation of state into (1.13.16) or (1.13.17), followed by integration, one can find the dependence of on T or /f on P, with T as a parameter. Eqs. (1.13.16) and (1.13.17) are known as caloric equations of state. 

Note If a physico-chemical change occurs in a system either at constant volume or constant pressure, the heat received during this change depends only upon the initial and final states so that... 

Section 12.2 defines specific heat capacity as the amount of heat required to increase the temperature of 1 g of material by 1 K. That definition is somewhat imprecise, because, in fact, the amount of heat required depends on whether the process is conducted at constant volume or at constant pressure. This section describes precise methods for measuring the amonnt of energy transferred as heat during a process and for relating this amonnt to the thermodynamic properties of the system under investigation. 

Suppose that 1.00 kj of heat is transferred to 2.00 mol argon (at 298 K, 1 atm). What will the final temperature Tf be if the heat is transferred (a) at constant volume, or (b) at constant pressure Calculate the energy change, AU, in each case. 

Cp = specific heat at constant pressure, Btu/lb°F C = specific heat at constant volume, Btu/lb°F D = impeller diameter or rotor, ft Dj = specific diameter, ft Ejd = adiabatic efficiency... 

Since the cryohydric point is a quadruple point in a two-component system, it represents an invariant system. The condition of the system is, therefore, completely defined the four phases, ice, salt, solution, vapour, can coexist only when the temperature, pressure, and concentration of the solution have constant and definite values. Addition or withdrawal of heat, therefore, can cause no alteration of the condition of the system, except a variation of the relative amounts of the phases. Addition of heat at constant volume will ultimately lead to the system salt—solution— vapour or the system ice—solution— vapour, according as ice or salt disappears first. This is readily apparent from the diagram (Fig, 72), for the systems ice—salt—solution and ice—salt— vapour can exist only at temperatures below the cryohydric point (provided the curve for ice—salt—solution slopes towards the pressure axis). 

The term capacity originates from the Latin words capax as able to hold much, or from capere as to take or to catch. In ordinary life we are talking, for example, of a vat that has the capacity of 101, which means that we can fill in a maximum of 101. The best-known capacity in thermodynamics is the heat capacity, either at constant volume or at constant pressure. Other terms concerning heat capacity appear in... 

Changes in U and H determine the heat taken in or evolved when a change occurs in a calorimeter, at constant volume or at constant pressure respectively, under such conditions that no work is done (other than that of simple expansion). 

Notice that although this process is carried out at constant pressure, AE is still given by wCyAT. This result seems contradictory at first glance, but actually it makes good sense. Because E for an ideal gas depends only on T (it does not depend on pressure or volume, for example), AE = wCyAT when an ideal gas is heated whether the process occurs at constant volume or constant pressure. 

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