Molar heat capacity at constant volume

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. 

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 ... 

Table 5.15 Mean molar heat capacities at constant volume J mol 1 K 1...

In calculating the heat of explosion we assumed that the temperature of explosion Tt was 4000 K. If this assumption was correct then the heat liberated by the explosion at 4000 K should equal 1149 kJ mol-1. The calculated value for the heat liberated, where the initial temperature is taken as 300 K, is presented in Equation 6.12. The values for the mean molar heat capacities at constant volume can be found in Table 5.15. 

The molar heat capacity at constant volume is therefore... 

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... 

For monatomic gases such as helium, argon, or xenon, etc., the molar heat capacity at constant volume is approximately 3(cal)/(g mole)(K). In the same system of units R is about 2 so the heat capacity at constant pressure for an ideal, monatomic gas is around 5(cal)/(g mole)(K). 

If the expression for the translational partition function is inserted into equation (16.8), it is readily found, since tt, m, fc, h and V are all constant, that the translational contribution Et to the energy, in excess of the zero-point value, is equal to %RT per mole, which is precisely the classical value. The corresponding molar heat capacity at constant volume is thus f P. As stated earlier, therefore, translational energy may be treated as essentially classical in behavior, since the quantum theory leads to the s ame results as does the classical treatment. Nevertheless, the partition function derived above [[equation (16.16) [] is of the greatest importance in connection with other thermodynamic properties, as w ill be seen in Chapter IX. 

For the special case of 1 mole of an ideal gas, dE may be replaced by CvdT Qcf. equation (9.22)], where Cv is the molar heat capacity at constant volume, and P may be replaced by BT/V equation (19.18) thus becomes... 

MW = molecular weight, g/gmol Q = molar heat capacity at constant volume, cal/(gmol °C) fx = absolute viscosity, g/(cm s)... 

The molar heat capacities at constant volume and constant pressure are defined by the relations... 

The molar heat capacity at constant volume (Cv) is the increase in the internal energy U of one mole of a material per degree Kelvin increase in its temperature at constant volume. It is... 

Fig. 9,7, The molar heat capacity at constant volume of diatomic gases hydrogen, x nitrogen, O iodine.

Equation 3.17 is not very convenient in this form as it is far easier to measure the molar heat capacity at constant volume, C . 

The molar heat capacity at constant volume and at constant pressure with assumption of ideal gas behaviour, are expressed as follows ... 

The volume is rather a basic concept of a region, as it appears in the derivation of all kinds of equations of continuity. For this reason, we introduce quantities that refer to the unit of volume. We will use the superscript X in order to indicate that this is X per volume. Remember that we are using C for the molar heat capacity (at constant volume), Cv for the specific heat capacity, C for a reduced heat capacity, i.e., divided by a reference value, and now C for the heat capacity density. The quantities X are basically X densities, for example, the mass density or the molar density ... 

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