Entropy of a gas

Adiabatic compression (termed adiabatic isentropic or constant entropy) of a gas in a centrifugal machine has the same characteristics as in any other compressor. That is, no heat is transferred to or from the gas during the compression operation. The characteristic equation... 

The mathematical machinery of thermod3mamics allows this qualitative statement to be expressed quantitatively. Experiments and theory show that the molar entropy of a gas or solute varies logarithmically with concentration... 

The integral in Equation 4.84 can be evaluated from an equation of state3. However, before this entropy departure function can be applied to calculate entropy, the reference state must be defined. Unlike enthalpy, the reference state cannot be defined at zero pressure, as the entropy of a gas is infinite at zero pressure. To avoid this difficulty, the standard state can be defined as a reference state at low pressure P0 (usually chosen to be 1 bar or 1 atm) and at the temperature under consideration. Thus,... 

Worked Example 4.1 Show mathematically how the entropy of a gas is higher than the entropy of its respective liquid. 

There is a small change in the number of moles principally, the amount of gas increases. As was seen above, the entropy of a gas is greater than its respective liquid, so we see a net increase in the entropy of the reaction, making AS positive. 

For a Gas. The procedure for the calculation of the entropy of a gas in its standard state is substantially the same as the that for a solid or liquid except for two factors. If the heat capacity data have been obtained at a pressure of 1 atm (101.325 kPa), the resultant value of Sjj, is appropriate for that pressure and must be corrected to the standard state pressure of 1 bar (0.1 MPa). This correction is given by... 

C) All reactions proceed spontaneously in the direction that increases the entropy (disorder) of the system plus surroundings. The entropy of a gas increases when its pressure decreases at constant temperature, while the entropy decreases when pressure increases. The more we expand a gas, the more space the gas molecules will have and so the less ordered they will be. 

Because the molar entropy of a gas is so much greater than that of solids and liquids, a change in the number of moles of gas normally dom-... 

The entropy change depends on the pressure change. This equation tells us that as the pressure is raised the entropy of a gas decreases i.e., molecules in the gas become more ordered. 

If the entropy is zero at T = 0 K, then Eq. (5.11) lends itself to the calculation of absolute entropies. With T — 0 as the lower limit of integration, the absolute entropy of a gas at temperature T based on calorimetric data follows from Eq. (5.11) integrated to give ... 

Explain how pressure affects the entropy of a gas and affects changes between the liquid and vapor states. 

Boiling points are pressure dependent because pressure has a large effect on the entropy of a gas. When a gas is expanded (pressure is decreased), its entropy increases because the degree of disorder of the molecules increases. At sea level, water boils at 100°C. In Denver, Colorado, where the elevation is 1.6 km, atmospheric pressure is about 0.84 times the pressure at sea level. At that elevation, water boils at about 95°C. On Pike s Peak, where the elevation is 4.3 km, water boils at about 85°C. People often use pressure cookers at that altitude to increase the boiling point of water. 

From Equation 13.9 we see that the entropy of a gas increases during an isothermal expansion (V2 > Vi) and decreases during a compression (V2 < Vt). Boltzmann s relation (see Eq. 13.1) provides the molecular interpretation of these results. The number of microstates available to the system, H, increases as the volume of the system increases and decreases as volume decreases, and the entropy of the system increases or decreases accordingly. 

The equation tells us that as the pressure is raised the entropy of a gas decreases. 

We shall now see the usefulness of statistical thermodynamics by calculating the thermodynamic properties of gaseous nitrogen at 298 K. The translational entropy of a gas at 1 atm can be calculated from the equations in Section 9.7. 

Figure 20.7 The large decrease in entropy of a gas when it dissolves in a liquid. The chaotic movement and high entropy of molecules of O2 are reduced greatly when the gas dissolves in water.

The formula for the conventional molar entropy of a gas of diatomic molecules can be written... 

All parts of this problem rest on two principles. First, the entropy of a solid is always less than the entropy of a liquid, and the entropy of a liquid is always much smaller than the entropy of a gas. Second, in comparing systems in the same phase, the one with the most complex particles has the higher entropy. 

As we will see in Sect. 9.3, the temperature coefficient a corresponds to the negative molar entropy Sm, i.e., a = Sm- Anticipating this can help us to remember the two rules above more easily First, in Chap. 3, we demonstrated that the molar entropy is always positive the negative sign of the temperature coefficient easily results from this (the rare exceptions mentioned above will be discussed in detail in Sect. 8.4). Second, the fact that the molar entropy of a liquid is greater than that of a solid, and the molar entropy of a gas is much greater than that of a Uquid (see Sect. 3.9), leads to the sequence above.)... 

H2C204(s) < C2H50H( ) < CH4(g). The entropy of a gas is generally much greater than that of a liquid, which in turn is generally greater than that of a solid. 

According to equation (12 64) the translational entropy of a gas is larger the greater is the molecular weight. Why is this ... 

Recall from your physical chemistry courses that the stability of an ideal gas is in part related to the volume that the gas occupies. The entropy of a gas is proportional to nR nV). This discussion is a simple statistical analysis, stating that the number of ways to arrange a set number of gas molecules (n) with a volume V increases with larger V. Here, the entire ensemble of gas molecules is considered to be more stable when V increases. It is important to note that the chemical structures of the individual gas molecules themselves have not become more stable just because they occupy a larger volume. 

Va is related to the universal gas constant R by R= Na/c with k (and R, respectively) as the fundamental constant relating temperature to energy (Section 3.1.4). Max Planck (1858-1947) first introduced k and gave an accurate value for it in his derivation of the law of black body radiation in 1900. The constant k, although Ludwig Boltzmann himself never introduced it, was named the Boltzmann constant, as he was the first (1877) who stated that the entropy of a gas is proportional to the logarithm of the number of microstates a gas can occupy with k as the factor of proportionality. 

To this energy we must add the entropy contribution, to arrive at the free energy. The entropy of a gas of fermions consisting of spin-up and spin-down particles with occupation numbers k is given by (see Appendix D)... 

In terms of these average occupation numbers, the entropy of a gas of spinless fermions is found to be (see Problem 1)... 

Why does the entropy of a gas increase when it expands into a vacuum ... 

When solving this type of problem, we should check the sign of AS. Here, we expect an increase in entropy (AS is positive) because, as discussed on page 587, the entropy of a gas is much higher than that of a liquid. 

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