Perfect gas at constant volume

The specific heat of a perfect gas at constant volume should, therefore, be independent of the temperature and equal to %R. 

In other words, the specific heat of a perfect gas at constant volume is independent of the absolute magnitude of the volume. That is, suppose we consider one gram of a perfect gas occupying a volume v, and we raise the temperature i°, keeping the volume at v, we require to add a certain quantity of heat C If we consider the same mass of gas at the same temperature as before, and at quite a different volume vl (the pressure being, of course, different now), and we raise the temperature 1°, keeping vl constant, it will be found that the amount of heat le-quired is again C ... 

These relations are true in general and hence hold for the perfect gas. We can demonstrate this more explicitly for the perfect gas as follows. For the perfect gas at constant volume... 

Referring back to Fig. 16, we assume that the fraction of space between the globs just above the melting point is AF/F, and this is the volume available for perfect gas atoms. However, since the thickness of the inter-globular shell is assumed to maintain a constant value r, then the space is a two-dimensional gas. Since the bulk coefficient of expansion of a perfect gas at constant pressure is IjT, that of a two-dimensional gas is fT. So the bulk coefficient of thermal expansion of a liquid just above the melting point should be obtained by prorating the expansions due to the solid fraction 1 — AF/F and the gaseous fraction AF/F ... 

For a perfect gas at constant pressure, P = const., Eq 2.1 gives the relationship between the heat capacity at constant pressure and constant volume, viz. Cp - = R (R is the gas constant). Similarly,... 

Absolute zero The temperature at which a perfect gas kept at constant volume exerts no pressure it is equal to -273.16 °C (0 K). 

The first term inside the brackets evidently is the energy of a solute molecule J in the perfect gas (cf. Eq. 10) hence we have for the energy of formation of the clathrate from and the gaseous solute at constant volume per molecule of Q... 

The early application of volumetric data for hydrocarbons made use of the perfect gas laws. They were not sufficiently descriptive of the actual behavior to permit their widespread use at pressures in excess of several hundred pounds per square inch. The need for accurate metering aroused interest in the volumetric behavior of petroleum and its products at elevated pressures. Table II reviews references relating to the volumetric behavior of a number of components of petroleum and their mixtures. For many purposes the ratio of the actual volume to the volume of a perfect gas at the same pressure and temperature has been considered to be a single-valued function of the reduced pressure and temperature or of the pseudo-reduced (38) pressure and temperature. The proposals of Dodge (15), Lewis (12), and Brown (8) with their coworkers serve as examples of the nature of these correlations. The Beattie-Bridgeman (2) and Benedict (4) equations of state describe the volumetric behavior of many pure substances and their mixtures with an accuracy adequate (31) for most purposes. However, at pressures above 3000 pounds per square inch the accuracy of representation with existing constants leaves something to be desired. 

The Absolute Temperature Scale. The idea of the absolute zero of temperature was developed as a result of the discovery of the law of Charles and Gay-Lussac the absolute zero would be the temperature at which an ideal gas would have zero volume at any finite pressure. For some years (until 1848) the absolute temperature scale was defined in terms of a gas thermometer the absolute temperature was taken as proportional to the volume of a sample of gas at constant pressure. Since, however, no real gas approaches a perfect gas closely enough at practically useful pressures to permit an accurate gas thermometer... 

The value of the gas constant R is found experimentally by determining the volume occupied by i mole of a perfect gas at standard conditions. One mole ol oxygen weiglis exactly 32 g, and the density of oxygen gas at standard conditions is found by experiment to be 1.429 g. The quotient 32/1.429 = 22.4 I is accordingly the volume occupied by 1 mole of gas (Avogadro s number of gas molecules) at standard conditions. 

The same gas might be heated between the same limits of temperature but keeping it at a volume V different from V. From what we have seen in the preceding article, if the gas is a perfect one, the values of Uq and U will undergo no change and similarly for c. Thus the mean value between two given temperor tures of the specific heat at constant volume of a perfect gas does not depend upon the value of this volume al which the gas is kept. 

When the compound is a perfect gas, the excess of the heat of for-motion at constant pressure over the heat of formation at constant volume is equivalent to the external work done by the formation of a gramme of the compound under the constant pressure considered. 

The quantity of heat set free by a system which undergoes a transformation does not depend solely upon the initial and final states, page 86.-32. Example from the study of perfect gases, 37.—33. Case in which the quantity of heat set free by a system depends solely upon the initial and final states, 88.—34. Utility, in chemical calorimetry, of the preceding law, 89.-35. Exothermic and endothermic reactions, 41.—36. Heats of formation under constant pressure and at constant volume, 44.—37. Case in which the two heats of formation are equal to each other, 45.— 38. General relation between the two heats of formation, 45.—39. Case in which the compound is a perfect gas, 46.—40. The distinction between the two heats of formation has small importance in practice, 46.—41. Infiuence of temperature on the heats of formation, 47.-42. Heat of formation referred to a temperature at which the reaction considered is impossible, 48.—... 

Characteristic function of a system, 91.—76. Characteristic function of a perfect gas, 92.-77. The characteristic function considered as available energy, 98.-78. Definite form of the equilibrium condition of a system kept at a given constant temperature, 94. —79. Internal thermodynamic potential, 94.—80. Total thermodynamic potential under constant pressure or at constant volume,... 

Thus a van der Waals gas has the same specific heat at constant volume as the perfect gas at the same temperature. This result is common to all equations of state in which the pressure is linearly related to the temperature. This can be shown quite generally since (c/. 4.2)... 

El. 3(b) The relation between pressure and temperature at constant volume can be derived from the perfect gas law... 

D2.3 The difference results from the definition H = U + PV hence AH = AU + A(PV). As A(PV) is not usually zero, except for isothermal processes in a perfect gas, the difference between AH and AU is a non-zero quantity. As shown in Sections 2.4 and 2.5 of the text, AH can be interpreted as the heat associated with a process at constant pressure, and AU as the heat at constant volume. 

The molar Helmholtz energy A = Af /n of a pure perfect gas may be obtained by integration of eq 3.15 subject to the equation of state, p = —(5 P /5Fm) = nKT/ V, and an expression for the perfect-gas molar heat capacity at constant volume, C y T) = T dS /dT)y. Starting from a reference state defined by temperature T and amount-of-substance density pjf", the result is ... 

The rate constants for second-order reactions can be expressed in dm mol sec, or, if the rate is measured from the change of gas pressure at constant volume, as described in eq. (2.12), in (pressure) V(time)". The conversion between units of pressure and those of concentration for gas phase reactions which are not first order, can be obtained from the perfect gas equation, from which, for the general case of a reaction of order n, the variation in the total gas pressure can be related to the change in the total gas concentration hy p = cRT... 

FIGURE 7.4 The c hange in entropy as a sample of perfect gas expands at constant temperature. Here we have plotted IS/nR. The entropy increases logarithmically with volume. 

In terms of the original fuel volume, VF, at and p x, by the perfect gas relationship at constant pressure and approximately constant molecular weights,... 

At a constant temperature, the volume of a fixed amount of a perfect gas varies inversely with its pressure. 

---