Heat capacity monatomic

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 molar heat capacities of gases composed of molecules (as distinct from atoms) are Higher than those of monatomic gases because the molecules can store energy as rotational kinetic energy as well as translational kinetic energy. We saw in Section 6.7 that the rotational motion of linear molecules contributes another RT to the molar internal energy ... 

Explain why the heat capacities of methane and ethane differ from the values expected for an ideal monatomic gas and from each other. The values are 35.309 J-K " mol 1 for CH4 and 52.63 J-K -mol 1 for C2He. 

C06-0138. According to Table 6H, molar heat capacities of monatomic gases (He, Ar) are significantly smaller than those of diatomic gases (N2, O2, H2). Explain in molecular terms why more heat must be supplied to raise the temperature of I mol of diatomic gas by I K than to raise the temperature of I mol of monatomic gas by 1 K. 

For the reversible adiabatic expansion, a definite expression can be derived to relate the initial and final temperatures to the respective volumes or pressures if we assume that the heat capacity is independent of temperature. This assumption is exact at all temperatures for monatomic gases and above room temperature for diatomic gases. Again we start with Equation (5.39). Recognizing the restriction of reversibility, we obtain... 

Let us consider a system of M ideal monatomic gas molecules in a cubic box kept at a constant temperature T. For a very dilute gas, where the molecules do not interact with one another, the quantum mechanical solution is a number of electronic wave functions with three quantum numbers nx, riy, and for the translational energies in three dimensions. The energy of a molecule for a set of quanmm numbers, the observed average energy, and the heat capacity at constant volume are given by... 

Another observable property of gases is the heat capacity. The molar heat capacity of monatomic gases was measured and found to be equal to (3/2)R, the value predicted for a perfect (point particle) gas. But, actual atoms had a well defined physical size. Since finite spheres would be expected to rotate, where was the heat capacity due to rotation Maxwell worried about this failure of the kinetic theory. Another type of eyes was required to see this result in its proper context. 

The expressions in (3.72) and (3.73) are valid only for monatomic ideal gases such as He or Ar, and must be replaced by somewhat different expressions for diatomic or polyatomic molecules (Sidebar 3.8). However, the classical expressions for polyatomic heat capacity exhibit serious errors (except at high temperatures) due to the important effects of quantum mechanics. (The failure of classical mechanics to describe the heat capacities of polyatomic species motivated Einstein s pioneering application of Planck s quantum theory to molecular vibrational phenomena.) For present purposes, we may envision taking more accurate heat capacity data from experiment [e.g., in equations such as (3.84a)] if polyatomic species are to be considered. The term perfect gas is sometimes employed to distinguish the monatomic case [for which (3.72), (3.73) are satisfactory] from more general polyatomic ideal gases with Cv> nR. 

TABLE 11.2 Measured Thermodynamic Properties (in SI Units) of Some Common Fluids at 20° C, 1 atm Molar Heat Capacity CP, Isothermal Compressibility jS7, Coefficient of Thermal Expansion otp, and Molar Volume V, with Monatomic Ideal Gas Values (cf. Sidebar 11.3) Shown for Comparison... 

It is commonly assumed that (he independent cations end anions will behave as ideal monatomic gases with heat capacities (at constant volume) of R. m Born, M. Verhtirdl. Dent. Physlk. Ces. 1919. 21, 13 Haber, F. Ibid. 1919, 21. 750. 

Assuming the heat capacity of an ideal gas is constant with temperature, calculate the entropy change associated with lowering the temperature of 1.47 mol of monatomic ideal gas reversibly from 99.32°C to — 78.54°C at (a) constant pressure and (b) constant volume. 

For a monatomic gas, where the heat capacity involves only translational energy, V is independent of sound oscillation frequency (except at ultra-high frequencies, where a classical visco-thermal dispersion sets in). For a relaxing polyatomic gas this is no longer so. At sound frequencies, where the period of the oscillation becomes comparable with the relaxation time for one of the forms of internal energy, the internal temperature lags behind the translational temperature throughout the compression-rarefaction cycle, and the effective values of CT and V in equation (3) become frequency dependent. This phenomenon occurs at medium ultrasonic frequencies, and is known as ultrasonic dispersion. It is accompanied by... 

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. 

However, it should not be necessary to make any heat capacity measurements at all, or any assumptions as to the thermal properties of the solid and liquid states in order to calculate the correct value for the entropy of hydrogen. Since hydrogen gas at low temperatures consists entirely of molecules in the zero rotational state, its entropy will be that of a monatomic gas of atomic weight 2.016. The entropy at 298°K. will be obtained by adding the integral f CPd In T over the proper temperature range. The heat capacity may be separated into a constant term 5/2R and the rotational term Cr. 

These trends are explained by the larger heat capacities of more complex molecules and heavier species (except for monatomic gases) and the large latent heat contributions introduced upon melting and vaporization. 

Why is the heat capacity of monatomic gases, such as He and Ne, practically independent of temperature, whereas molecular heat capacities increase with temperature ... 

Thermodynamics can state nothing regarding the variation of specific heat with temperature. It actually happens, however, that the heat capacities of all gases approach the values that we have found theoretically for monatomic gases in Eq. (3.18), Chap. IV, namely,... 

A similar argument is used to deal with s this is based on the empirical observation that in an adiabatic process involving a noble gas at low pressures, the product PV7 is virtually constant. Here 7 is a fixed quantity (which will later turn out to be the ratio of molar heat capacities at constant pressure and volume) whose exact significance is irrelevant at this stage near room temperature and for monatomic gases 7 has a value close to 5/3. We therefore use the product PV7 as a measure of the empirical entropy through the simple relation... 

Determine an expression for Cv for electromagnetic radiation and determine its value for unit volume at 1 K and 300 K. Compare this with the heat capacity of a mole of an ideal monatomic gas at 300 K. At what temperature do these quantities become equal ... 

Clearly, a monatomic gas has no rotational or vibrational energy but does have a translational energy of RT per mole. The constant-volume heat capacity of a monatomic perfect gas is thus... 

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