Heat capacity constant-volume molar

Relation between the constant-pressure and constant-volume molar heat capacities of an ideal gas ... 

The heat capacities that have been discussed previously refer to closed, single-phase systems. In such cases the variables that define the state of the system are either the temperature and pressure or the temperature and volume, and we are concerned with the heat capacities at constant pressure or constant volume. In this section and Section 9.3 we are concerned with a more general concept of heat capacity, particularly the molar heat capacity of a phase that is in equilibrium with other phases and the heat capacity of a thermodynamic system as a whole. Equation (2.5), C = dQ/dT, is the basic equation for the definition of the heat capacity which, when combined with Equation (9.1) or (9.2), gives the relations by which the more general heat capacities can be calculated. Actually dQ/dT is a ratio of differentials and has no value until a path is defined. The general problem becomes the determination of the variables to be used in each case and of the restrictions that must be placed on these variables so that only the temperature is independent. 

The total energy of a system is a difficult quantity to measure directly. It is much easier to measure energy changes dE/dT—for example, the number of joules necessary to raise the temperature of one mole of gas by one degree Kelvin. If the gas is kept in a constant volume container, this is called the constant-volume molar heat capacity cv, and equals 3R/2 (independent of temperature) for a monatomic gas. Each possible direction of motion (x,y, or z) contributes RT / 2 to the total energy per mole, or R/2 to the heat capacity. 

If the volume is held constant, the integral vanishes. The added energy required to heat one mole of gas from temperature T to temperature T2 (q( T2) — q T )), and the constant-volume molar heat capacity cv, are given by ... 

The heat capacity per unit mass, cr, is called the specific heat at constant volume. The heat capacity per mole is called the molar heat capacity at constant volume, CVm. For homogeneous systems, the system heat capacity can be calculated as... 

C, Constant-volume molar heat capacity J/(kmol K)... 

Ideal gas constant-volume molar heat capacity (J/mol K) Enthalpy (J)... 

If we are speaking of molar heat capacities, the volume in the derivative is the molar volume from the equation of state, V= RT/p. Differentiating with respect to temperature, keeping the pressure constant, yields dVIdT) = R/p. Putting this value in Eq. (7.41) reduces it to the simple result... 

Heat capacity at constant pressure, molar (J moP ) Heat capacity at constant volume, molar (J moP )... 

Therefore, for an ideal gas, the constant-pressure molar heat capacity al ys exceeds the constant-volume molar heat capacity by exactly R. The values of Cp for many substances are given in Appendix 2. 

Name (synonyms) Isobar molar heat capacity (300K) Isochor molar heat capacity (300K) Isentropic exponent (300K) Thermal conduct ivity (298K) Attraction constant Co-volume Critical pressure Critical tempera- ture Critical density Critical compress. factor Solubility water (0°C) Refractive index (589nm)... 

When a gas is heated at constant volume, AV is zero i.e., no work is done. All the heat absorbed by the system is used to increase the internal energy of the system. Consider 1 mole of an ideal gas whose temperature is raised by 1° at constant volume. The increase in internal energy itself gives the molar heat capacity at constant volume,... 

Suppose that the molar internal energy of a substance over a Umited temperature range could be expressed as a polynomial in T as Um(T) = a + bT+ cP. Find an expression for the constant-volume molar heat capacity at a temperature T. 

The constant volume molar heat capacity of molten salts, Cy, is not measured directly but is derived from Cp and requires volumetric data ... 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Cv specific molar heat capacity or heat capacity at constant volume, J/mol K... 

Thus, for the ideal gas the molar heat capacity at constant pressure is greater than the molar heat capacity at constant volume by the gas constant R. In Chapter 3 we will derive a more general relationship between Cp m and CV m that applies to all gases, liquids, and solids. 

As an example, the molar constant-volume heat capacity of argon is 12.8 J-K 1-mol 1, and so the corresponding constant-pressure value is 12.8 + 8.3... 

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. 

We can see how the values of heat capacities depend on molecular properties by using the relations in Section 6.7. We start with a simple system, a monatomic ideal gas such as argon. We saw in Section 6.7 that the molar internal energy of a monatomic ideal gas at a temperature T is RT and that the change in molar internal energy when the temperature is changed by AT is A(Jm = jRAT. It follows from Eq. 12a that the molar heat capacity at constant volume is... 

The high-temperature contribution of vibrational modes to the molar heat capacity of a solid at constant volume is R for each mode of vibrational motion. Hence, for an atomic solid, the molar heat capacity at constant volume is approximately 3/. (a) The specific heat capacity of a certain atomic solid is 0.392 J-K 1 -g. The chloride of this element (XC12) is 52.7% chlorine by mass. Identify the element, (b) This element crystallizes in a face-centered cubic unit cell and its atomic radius is 128 pm. What is the density of this atomic solid ... 

Estimate the molar heat capacity (at constant volume) of sulfur dioxide gas. In addition to translational and rotational motion, there is vibrational motion. Each vibrational degree of freedom contributes R to the molar heat capacity. The temperature needed for the vibrational modes to be accessible can be approximated by 6 = />vvih/, where k is Boltzmann s constant. The vibrational modes have frequencies 3.5 X... 

Hz, 4.1 X 1013 Hz, and 1.6 X 1013 Elz. (a) What is the high-temperature limit of the molar heat capacity at constant volume (b) What is the molar heat capacity at constant volume at 1000. K (c) What is the molar heat capacity at constant volume at room temperature ... 

Use the estimates of molar constant-volume heat capacities given in the text (as multiples of R) to estimate the change in reaction enthalpy of N2(g) + 3 H,(g) —> 2 NH.(g) when the temperature is increased from 300. K to 500. K. Ignore the vibrational contributions to heat capacity. Is the reaction more or less exothermic at the higher temperature ... 

A sample of nitrogen gas of volume 20.0 L at 5.00 kPa is heated from 20.°C to 400.°C at constant volume. What is the change in the entropy of the nitrogen The molar heat capacity of nitrogen at constant volume, CVm, is 20.81 J-K -mol . Assume ideal behavior. 

In these equations x and y denote independent spatial coordinates T, the temperature Tib, the mass fraction of the species p, the pressure u and v the tangential and the transverse components of the velocity, respectively p, the mass density Wk, the molecular weight of the species W, the mean molecular weight of the mixture R, the universal gas constant A, the thermal conductivity of the mixture Cp, the constant pressure heat capacity of the mixture Cp, the constant pressure heat capacity of the species Wk, the molar rate of production of the k species per unit volume hk, the speciflc enthalpy of the species p the viscosity of the mixture and the diffusion velocity of the A species in the y direction. The free stream tangential and transverse velocities at the edge of the boundaiy layer are given by = ax and Vg = —ay, respectively, where a is the strain rate. The strain rate is a measure of the stretch in the flame due to the imposed flow. The form of the chemical production rates and the diffusion velocities can be found in (7-8). 

It is thus seen that heat capacity at constant volume is the rate of change of internal energy with temperature, while heat capacity at constant pressure is the rate of change of enthalpy with temperature. Like internal energy, enthalpy and heat capacity are also extensive properties. The heat capacity values of substances are usually expressed per unit mass or mole. For instance, the specific heat which is the heat capacity per gram of the substance or the molar heat, which is the heat capacity per mole of the substance, are generally considered. The heat capacity of a substance increases with increase in temperature. This variation is usually represented by an empirical relationship such as... 

Figure 1.2 Molar heat capacity at constant pressure and at constant volume, isobaric expansivity and isothermal compressibility of AI2O3 as a function of temperature.

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

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