Ideal Gas Calculations

This section shows how to do calculations such as those necessary to answer Ted s and Amelia s questions about gas density and volume, and in addition, it considers some of the gas-related issues that Lilia s sister Rebecca and her co-workers need to resolve. For one thing, the design team needs to know the amount of gas that they can safely add to their reaction vessel, and then Rebecca needs to determine the maximum temperature at which the reaction can be run without causing the pressure of that amount of gas to reach dangerous levels. 

All these calculations, and others like them, can be done with the aid of two useful equations that we will now derive from the relationships described in Section 13.1. 

We discovered in Section 13.1 that pressure of an ideal gas is direcdy proportional to the number of gas particles (expressed in moles), direcdy proportional to temperature, and inversely proportional to the volume of the container. 

These three relationships can be summarized in a single equation nT 

The constant in this equation is the same for all ideal gases. It is called the universal gas constant and is expressed with the symbol R. The value of R depends on the units of measure one wishes to use in a given calculation. Two choices are given below, showing R for different pressure units (atmospheres, atm, and kilopascals, kPa). 

While the energy balance (or first law) is required to evaluate the energy changes of the system, the ideal gas law describes the relationship between the gas properties. For a closed system, the energy balance was written as 

Let s suppose we are interested in evaluating the work associated with an isothermal compression. Since an ideal gas has no molecular interactions, the internal energy is a function of temperature only, and thus the energy balance reduces to 

we substitute our known definition for the work associated with the change in volume  

For 1 mole of an ideal gas, PV=RT, and substitution of this relationship into the definition of work provides a relationship that can be integrated. We find 

Substituting the ideal gas relationship into this last expression, we can also obtain the work requirement in terms of the pressure change  

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JThis calculation is one of the most satisfying in science. The values of the thermodynamic properties of the ideal gas calculated from molecular parameters are usually more accurate than the same thermodynamic results obtained from experimental measurements. 

A mole of steam is condensed reversibly to liquid water at 100°C and 101.325-kPa (constant) pressure. The heat of vaporization of water is 2256.8 Jg. Assuming that steam behaves as an ideal gas, calculate W, Q, AUra, AH, ASra, AGm. and AAm for the condensation process. 

Without doing calculations, the signs of Dp, DV, DT, DE, DH and DS for one mole of an ideal gas calculated taken through each of the following four steps of a Carnot cycle. Cp and Cyas constants assumed ... 

Ideal gas particles have a volume that is insignificant compared to the volume the gas occupies as a whole. The relatively small volume of a 20-ounce soda bottle, for example, completely dwarfs the individual gas particles inside the bottle, making their sizes irrelevant to any ideal gas calculation. 

Example 7.4 Consider again the nozzle of Example 7.3, assuming now that stea behaves as an ideal gas. Calculate ... 

In Sec. 10.5 we treated dew- and bubble-point calculations for multicomponent systems that obey Raoult s law [Eq. (10.16)], an equation valid for low-pressure VLE when an ideal-liquid solution is in equilibrium with an ideal gas. Calculations for the general case are carried out in exactly the same way as for Raoult s law,... 

The results of an ideal gas calculation are usually a sufficient approximation for most substances and mixtures. The calculations for real gases will not be explored in this text. 

Nitrogen gas initially at 8.5 bar expands isenhopically to 1 bar and 423.15 K (150°C). Assuming nitrogen to be an ideal gas, calculate the initial temperature and the work produced per mole of luhogen. 

The use of the procedure of Rau et al. (2) leads to a set of thermal functions and associated enthalpies of formation which reproduce the observed sulfur vapor pressure data. That Is, the sum of the calculated partial pressures of all eight sulfur vapor species, S (g) to Sg(g), does closely reproduce the observed vapor pressure. [A difference between the calculated boiling point (at 1 atm) and the secondary reference temperature boiling point is due to the difference between the Ideal gas calculation and the real observed value.]... 

The extent of thermal dissociation of phosgene at 0.5, 1.0 and 10 bar pressure (0.05, 0.1 and 1 MPa, respectively) has been calculated [1764] based on the accepted ideal gas thermodynamic values [359aa], and is illustrated in Fig. 5.3, whilst the enthalpy of formation for this reaction has been based on actual measurements of the equilibrium constant in the temperature range of 645-725 K by heating together carbon monoxide and dichlorine [218], see Section 6.1. The equilibrium reaction depicted in Equation (5.1) has been measured experimentally both by dissociation of phosgene and by association of carbon monoxide and dichlorine [216]. At 603, 553 and 503 C, the dissociation was found to be 91, 80 and 67%, respectively, in reasonable agreement with the values based on the ideal gas calculations illustrated in Fig. 5.3. At temperatures above 800 C, the dissociation is essentially complete [216]. 

Cy can be measured calorimetrically or, for the ideal gas, calculated from spectroscopic data. For a real gas or liquid, it can be obtained by combining the ideal-gas value with equation-of-state calculations to be discussed in Section 4.2.7, where the isothermal variation of U with V will also be discussed. 

Consider first the conditions in the absorber. From the flow conditions of the entering gas and the given tower diameter, using the ideal gas, calculate the molar gas velocity entering at the bottom of the tower. (In all calculations in this example, we will use a subscript a to indicate flows and concentrations in the absorber, and a subscript s to refer to conditions in the stripper.)... 

Given each of the following sets of values for an ideal gas, calculate the unknown quantity. 

Comments The ideal-gas calculation over estimates the temperature by about 50 K (about 3%) and the work by... 

Comments The temperature and amount of steam in the tank are fairly close to the value obtained using the steam tables. However, the ideal-gas calculation predicts no temperature drop during throttling and thus overestimates the exit temperature by about 16 °C. 

Problem 9.1 A stream that contains a mixture of methane (25% by mol) and carbon monoxide is compressed from 1 bar, 35 to 12 bar. The compressor efficiency is 90%. Treating the mixture as an ideal gas, calculate the required work. 

Assume that an element exists that has two isotopes, and that each isotope has an abundance of 50.00%. Using the formula for the entropy of mixing for an ideal gas, calculate A5mix for a sample of 1.000 mol of this element. 

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