Ideal gas at constant volume

FIGURE 7.10 More energy levels become accessible in a lx>x of fixed width as the temperature is raised. The change from part (a) to part (b) is a model of the effect of heating an ideal gas at constant volume. The thermally accessible levels are shown by the tinted band. The average energy of the molecules also increases as the temperature is raised that is, both internal energy and entropy increase with temperature. 

Consider a system which does not do work, e.g., 1 mole of an ideal gas at constant volume. When energy is supplied to it in the form of heat, its temperature rises. There will be an increase in internal energy, or the energy contained within, corresponding to the heat put in. 

When an ideal gas is heated in a rigid container in which no change in volume occurs, there can be no PV work (AV = 0). Under these conditions all the energy that flows into the gas is used to increase the translational energies of the gas molecules. Thus Cv, the molar heat capacity of an ideal gas at constant volume, is fR, the result anticipated in the above discussion ... 

Now, consider changing the temperature of an ideal gas at constant volume from the point of view of thermodynamics. Because the volume is constant (the gas is confined in a vessel with rigid, diathermal walls), the pressure-volume work, w, must be zero therefore,... 

Entropy Capacities of Ideal Gases The entropy capacity Cy of an ideal gas at constant volume can be calculated using the law of equipartition. For the sake of simplicity, we will consider a monatomic gas because its particles cannot oscillate and have no rotational energy. (Because the mass lies upon the rotational axis, the... 

The physical explanation for this difference between Cp and Cy is that heating an ideal gas at constant volume does not work on the surroundings. In heating at constant pressure some of the heat is turned into work against the external pressure as the gas expands. A larger amount of heat is therefore required for a given change in the temperature than for a constant-volume process. In Chapter 4 we will be able to show that Cp cannot be smaller than Cy for any system. 

The thermodynamic temperature T can be experimentally determined with a gas thermometer. This thermometer utilizes that the pressure p in an ideal gas at constant volume V is proportional to the thermodynamic temperature T... 

We have seen how to use the individual laws to make predictions when only one variable is changed, such as heating a fixed amount of gas at constant volume. The ideal gas law enables us to make predictions when two or more variables are changed. 

The molar heat capacity of an ideal gas at constant pressure is greater than that at constant volume the two quantities are related by Eq. 13. 

Suppose that we were to increase the total pressure inside a reaction vessel by pumping in argon or some other inert gas at constant volume. The reacting gases continue to occupy the same volume, and so their individual molar concentrations and partial pressures remain unchanged despite the presence of an inert gas. In this case, therefore, provided that the gases can be regarded as ideal, the equilibrium composition is unaffected despite the fact that the total pressure has increased. 

The ideal gas law has many uses in chemistry, some of which we shall meet later in this text. To begin to see how useful the law can be, recall that we have seen how to use the individual laws to make predictions when only one variable is changed, such as heating a fixed amount of gas at constant volume. The ideal gas law enables us to make predictions when more than one variable is changed. For example, when we pump up an actual bicycle tire, the temperature of the gas in the pump increases as we press in the piston, so the compression is not strictly isothermal as we assumed in Example 4.3. 

The internal energy of an ideal gas at constant temperature is independent of the volume of the gas. 

Avogadro s law The volume occupied by an ideal gas at constant temperature and pressure is proportional to the number of moles (n) of gas present. 

HI. The heat required to raise the temperature from 300.0 K to 400.0 K for one mole of a gas at constant volume is 2079 J. The internal energy required to heat the same gas at constant pressure from 550.0 K to 600.0 K is 1305 J. The gas does 150. J of work during this expansion at constant pressure. Is this gas behaving ideally Is the gas a monatomic gas Explain. 

Recall from Section 12.1 that a true reversible process is an idealization it is a process in which the system proceeds with infinitesimal speed through a series of equilibrium states. The external pressure therefore, can never differ by more than an infinitesimal amount from the pressure, P, of the gas itself. The heat, work, energy, and enthalpy changes for ideal gases at constant volume (called isochoric processes) and at constant pressure (isobaric processes) have already been considered. This section examines isothermal (constant temperature) and adiabatic (q = 0) processes. 

It follows, therefore, that when gases approximate to ideal behavior, i.e., at very low pressures, the differences in their thermometric properties disappear. This fact presents the possibility of devising a temperature scale which shall be independent of the thermometric substance, the latter being a hypothetical ideal gas. Such a scale is the so-called absolute ideal gas scale, in which the (absolute) temperature is taken as direcUy proportional to the volume of a definite mass of an ideal gas at constant pressure or to the pressure at constant volume. For convenience, the magnitude of the degree on the absolute scale is usually taken to be the same as on the centigrade scale ( 2b), so that the absolute temperature T on the ideal gas scale is given by... 

There is an important consequence of the fact that the energy content of an ideal gas, at constant temperature, is independent of the volume. When an idesil gas expands against an appreciable external pressure, W has a finite value, but AE is zero it follows, therefore, from equation (7.2) that for the isothermal expansion of an ideal gas,... 

It is of interest to note that (d P/dT )v is zero for a van der Waals gas, as well as for an ideal gas hence, Cv should also be independent of the volume (or pressure) in the former case. In this event, the effect of pressure on Cp is equal to the variation of Cp — Cv with pressure. Comparison of equations (21.4) and (21.13), both of which are based on the van der Waals equation, shows this to be true. For a gas obeying the Berthelot equation or the Beattie-Bridgeman equation (d P/dT )v would not be zero, and hence some variation (f Cv with pressure is to be expected. It is probable, however, that this variation is small, and so for most purposes the heat capacity of any gas at constant volume may be regarded as being independent of the volume or pressure. The maximum in the ratio y of the heat capacities at constant pressure and volume, respectively, i.e., Cp/Cv, referred to earlier ( lOe), should thus occur at about the same pressure as that for Cp, at any temperature. 

Use the lattice model discussed in Appendix A9.1 to show that the state of maximum entropy for an ideal gas at constant temperature (and therefore energy) and contained in a volume V is the state of uniform density. 

As a first exarhple, we will consider the reversible compression of an ideal gas at constant pressure. Suppose that we have 1 mol of an ideal gas confined in a cylinder with a piston, at a pressure of Pu a volume of and a temperature (in kelvins) of... 

Charles s law states that the volume of a given amount of an ideal gas at constant pressure varies directly with its temperature (in kelvins). V = bT... 

Charle s Law is the empirical relationship, which states tliat which states that for an ideal gas at constant pressure, its volume is proportional to its absolute temperature. This relationship applies to ideal gases only. Real gases deviate considerable form this ideal relationship. 

This is the derivation of the ideal gas law, using purely thermodynamic arguments. We summarize once more what we have done. We have assumed that the energy of the ideal gas at constant temperature does on depend on the volume, as well as the enthalpy of the ideal gas does not depend on the pressure. 

The hyperbola has been rotated by 45°, and the asymptotes have become the coordinate axes themselves. These rectangular hyperbolas are shown in Fig. 5.7. More generally, hyperbolas of the formxy = const represent relations in which y is inversely proportional to x. An important example is Boyle s law relating the pressure and volume of an ideal gas at constant tenqierature, which can be expressed as pV= const. 

As the pressure of an ideal gas increases, its volume decreases proportionately. Charles law describes the relationship between the volume and temperature of an ideal gas at constant pressure ... 

FIGURE JIO. Specific heat capacity of chlorine gas at constant volume (ideal gas state). (R.M. Kapoor and J.J. Martin, Thermodynamic Properties of Chlorine, Engineering Research Institute, University of Michigan, Ann Arbor, MI (1957).)... 

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