{"id":4704,"date":"2024-10-05T17:39:06","date_gmt":"2024-10-05T17:39:06","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=4704"},"modified":"2024-10-05T17:39:07","modified_gmt":"2024-10-05T17:39:07","slug":"ideal-gases","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/10\/05\/ideal-gases\/","title":{"rendered":"Ideal Gases"},"content":{"rendered":"\n

You have no doubt been exposed to the concept of the ideal gas in chemistry and physics. Why go over ideal gases again? At least two reasons exist. First, the experimental and theoretical properties of ideal gases are far simpler than the corresponding properties of liquids and solids. Second, use of the ideal gas concept is of considerable industrial importance.<\/p>\n\n\n\n

In this section we explain how the ideal gas law can be used to calculate the pressure, temperature, volume, or number of moles in a quantity of gas, and we define the partial pressure of a gas in a mixture of gases. We also discuss how to calculate the specific gravity and density of a gas. Then we apply the concepts to solving material balances.<\/p>\n\n\n\n

7.1.1. Ideal Gas<\/h4>\n\n\n\n

The most famous and widely used equation that relates\u00a0p<\/em>,\u00a0V<\/em>,\u00a0n<\/em>, and\u00a0T<\/em>\u00a0for a gas is the\u00a0ideal gas law:<\/strong><\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

where p<\/em> is the absolute pressure<\/strong> of the gas
         V<\/em> is the total volume occupied by the gas
         n<\/em> is the number of moles of the gas
         R is the ideal (universal) gas constant in appropriate units
         T<\/em> is the absolute temperature<\/strong> of the gas<\/p>\n\n\n\n

You can find values of R in various units inside the front cover of this book. Sometimes the ideal gas law is written as<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Note that in Equation (7.1a)\u00a0\"Image\"\u00a0is the\u00a0specific molar<\/em>\u00a0volume (volume per mole,\u00a0V<\/em>\/n<\/em>) of the gas. When gas volumes are involved in a problem,\u00a0\"Image\"\u00a0will be the volume per mole and not the volume per mass. The inverse of\u00a0\"Image\"\u00a0is the molar density, moles per volume.\u00a0Figure 7.2\u00a0illustrates the surface generated by Equation (7.1a) in terms of the three properties\u00a0p<\/em>,\u00a0\"Image\", and\u00a0T<\/em>. Look at the projections of the surface in\u00a0Figure 7.2\u00a0onto the two-parameter planes. The interpretation is as follows:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Figure 7.2. Representation of the ideal gas law in three dimensions as a surface<\/p>\n\n\n\n

1.<\/strong>\u00a0The projection to the upper left onto the\u00a0p<\/em>\u00a0\u2013\u00a0T<\/em>\u00a0plane shows straight lines for constant values of\u00a0\"Image\". Why? Equation (7.1a) for constant specific volume reduces to\u00a0p<\/em>\u00a0= (constant)(T<\/em>), the equation of a straight line that passes through the origin.<\/p>\n\n\n\n

2.<\/strong>\u00a0The projection to the upper right onto the\u00a0p<\/em>\u00a0\u2013\u00a0\"Image\"\u00a0plane shows curves for values of constant\u00a0T<\/em>. What kinds of curves are they? For constant\u00a0T<\/em>,\u00a0Equation (7.1a) becomes\u00a0p<\/em>\"Image\"\u00a0= constant, namely, a hyperbola.<\/p>\n\n\n\n

3.<\/strong>\u00a0The projection downward onto the\u00a0T<\/em>\u00a0\u2013\u00a0\"Image\"\u00a0plane again shows straight lines. Why? Equation (7.1a) for constant\u00a0p<\/em>\u00a0is\u00a0\"Image\"\u00a0= (constant)(T).<\/p>\n\n\n\n

For an ideal gas, Equation (7.1) can be applied to a pure component or to a mixture.<\/p>\n\n\n\n

What are the conditions for a gas to behave as predicted by the ideal gas law? The major ones for a gas to be ideal are as follows:<\/p>\n\n\n\n

1.<\/strong> The molecules do not occupy any space; they are infinitesimally small.<\/p>\n\n\n\n

2.<\/strong> No attractive forces exist between the molecules so the molecules move completely independently of each other.<\/p>\n\n\n\n

3.<\/strong> The gas molecules move in random, straight-line motion and the collisions between the molecules, and between the molecules and the walls of the container, are perfectly elastic.<\/p>\n\n\n\n

Gases at low pressure and\/or high temperature meet these conditions. Solids, liquids, and gases at high density\u2014that is, high pressure and\/or low temperature\u2014do not. From a practical viewpoint, within reasonable error, you can treat air, oxygen, nitrogen, hydrogen, carbon dioxide, methane, and even water vapor, under most of the ordinary conditions you encounter, as ideal gases.<\/p>\n\n\n\n

Several equivalent standard states known as\u00a0standard conditions<\/strong>\u00a0(S.C., or S.T.P., an acronym for \u201cstandard temperature and pressure\u201d) of temperature and pressure have been specified for gases by custom. Refer to\u00a0Table 7.1. Note that the standard conditions for the SI, Universal Scientific, and American Engineering systems are exactly the same conditions, but in different units. On the other hand, the natural gas industry uses a different reference temperature (15\u00b0F) but the same reference pressure (1 atm).<\/p>\n\n\n\n

<\/a>Table 7.1. Common Standard Conditions for the Ideal Gas<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

You can insert the values at S.C. into the ideal gas equation to calculate R in any units you want.<\/p>\n\n\n\n

For example, what is R for 1 g mol of ideal gas with a volume in cubic centimeters, pressure in atmospheres, and temperature in kelvin?<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

The fact that a substance cannot exist as a gas at 0\u00b0C and 1 atm is immaterial. Thus, as we shall see later, water vapor at 0\u00b0C cannot exist at a pressure greater than its vapor pressure of 0.61 kPa (0.18 in. Hg) without condensation occurring. However, you can calculate the imaginary volume at standard conditions, and it is just as useful a quantity in the calculation of volume-mole relationships as though it could exist. In what follows, the symbol V<\/em> will stand for total volume and the symbol \"Image\" for volume per mole.<\/p>\n\n\n\n

Because the SI, Universal Scientific, and AE standard conditions all refer to the same point in the p<\/em>, \"Image\", and T<\/em> space, you can use the values in Table 7.1<\/a> with their units to change from one system of units to another. If you memorize the standard conditions, you will find it easy to work with mixtures of units from different systems.<\/p>\n\n\n\n

The following example illustrates how you can use the standard conditions to convert mass or moles to volume. After reading it, see if you can explain to someone else how to convert volume to moles or mass.<\/p>\n\n\n\n

\n\n\n\n

<\/a>Example 7.1. Use of Standard Conditions to Calculate Volume from Mass<\/p>\n\n\n\n

Calculate the volume, in cubic meters, occupied by 40 kg of CO2<\/sub> at standard conditions, assuming CO2<\/sub> acts as an ideal gas.<\/p>\n\n\n\n

Solution<\/p>\n\n\n\n

Basis: 40 kg of CO2<\/sub><\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Notice in this problem that the information that 22.42 m3<\/sup>\u00a0of gas at S.C. = 1 kg mol of gas is applied to transform a known number of moles into an equivalent number of cubic meters. An alternate way to calculate the volume at standard conditions is to use Equation (7.1). Incidentally, whenever you report a volumetric value,\u00a0you must establish the conditions of temperature and pressure at which the volumetric measure exists<\/strong>, since the term m3<\/sup>\u00a0or ft3<\/sup>, standing alone, is really not any particular\u00a0quantity<\/em>\u00a0of material.<\/p>\n\n\n\n

\n\n\n\n

In many processes going from an initial state to a final state, you use the ratio of the ideal gas laws in the respective states and thus eliminate R as follows (the subscript 1 designates the initial state, and the subscript 2 designates the final state):<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

or<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Note that Equation (7.2) involves ratios of the same variable. This result has the convenient feature that the pressures may be expressed in any system of units you choose, such as kilopascals, inches of Hg, millimeters of Hg, atmospheres, and so on, as long as the same units are used for both conditions of pressure (do not forget that the pressure must be\u00a0absolute<\/em>\u00a0pressure in both cases). Similarly, the ratio of the\u00a0absolute<\/em>\u00a0temperatures and ratio of the volumes result in dimensionless ratios. Note how the\u00a0ideal gas constant\u00a0R is eliminated in taking the ratios.<\/p>\n\n\n\n

Let\u2019s see how you can apply the ideal gas law to problems in the form of both Equation (7.2) and Equation (7.1).<\/p>\n\n\n\n

\n\n\n\n

<\/a>Example 7.2. Application of the Ideal Gas Law to Calculate a Volume<\/p>\n\n\n\n

Calculate the volume occupied by 88 lb of CO2<\/sub> at 15\u00b0C and a pressure of 32.2 ft of water.<\/p>\n\n\n\n

Solution<\/p>\n\n\n\n

Examine\u00a0Figure E7.2. To use Equation (7.2) the initial volume has to first be calculated as shown in\u00a0Example 7.1.<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Figure E7.2<\/p>\n\n\n\n

Then the final volume can be calculated via Equation (7.2) in which both R and (n<\/em>1<\/sub>\/n<\/em>2<\/sub>) cancel.\u00a0Table 7.1\u00a0does not list the pressure in feet of H2<\/sub>O at S.C. Where do you get the value? Look in\u00a0Chapter 2\u00a0at the discussion of pressure, or calculate it (\u03c1gh<\/em>\u00a0=\u00a0p<\/em>).<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Assume that the given pressure is absolute pressure.<\/p>\n\n\n\n

At S.C. (state 1)        At p<\/em> and T<\/em> (state 2)
p<\/em> = 33.91 ft H2<\/sub>O       p<\/em> = 32.2 ft H2<\/sub>O
T<\/em> = 273 K                T<\/em> = 273 + 15 = 288 K<\/p>\n\n\n\n

Basis: 88 lb of CO2<\/sub><\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

You can mentally check your calculations by saying to yourself: The temperature goes up from 0\u00b0C at S.C. to 15\u00b0C at the final state; hence, the volume must increase from S.C., and the temperature ratio must be greater than unity. Similarly, you can say: The pressure goes down from S.C. to the final state, so the volume must increase from S.C.; hence the pressure ratio must be greater than unity.<\/p>\n\n\n\n

The same result can be obtained by using Equation (7.1). First obtain the value of R in the same units as the variables\u00a0p<\/em>,\u00a0\"Image\", and\u00a0T<\/em>. Look it up or calculate the value from\u00a0p<\/em>,\u00a0\"Image\", and\u00a0T<\/em>\u00a0at S.C.:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

At S.C.: p<\/em> = 33.91 ft H2<\/sub>O            \"Image\" = 359 ft3<\/sup>\/lb mol              T<\/em> = 273 K<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

Now, using Equation (7.1), insert the given values, and perform the necessary calculations.<\/p>\n\n\n\n

Basis: 88 lb of CO2<\/sub><\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

If you inspect the two solutions, you will observe that in both the same numbers appear, and that the results are identical.<\/p>\n\n\n\n

\n\n\n\n

To calculate the volumetric flow rate<\/strong> of a gas, \"Image\", such as in cubic meters or cubic feet per second, through a pipe, you divide the volume of the gas passing through the pipe in a time interval by the value of the time interval. To get the velocity<\/strong>, \"Image\" of the flow, you divide the volumetric flow rate by the area, A<\/em>, of the pipe<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

The (mass) density of a gas<\/strong> is defined as the mass per unit volume and can be expressed in various units, including kilograms per cubic meter, pounds per cubic foot, grams per liter, and so on. Inasmuch as the mass contained in a unit volume varies with the temperature and pressure, as we have previously mentioned, you should always be careful to specify these two conditions in calculating density. If not otherwise specified, the densities are presumed to be at S.C. As an example, what is the density of N2<\/sub> at 27\u00b0C and 100 kPa in SI units?<\/p>\n\n\n\n

Basis: 1 m3<\/sup> of N2<\/sub> at 27\u00b0C and 100 kPa<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n\n\n\n

of N2<\/sub> at 27\u00b0C (300 K) and 100 kPa<\/p>\n\n\n\n

In addition to the mass density, sometimes the \u201cdensity\u201d of a gas refers to the molar density, namely, moles per unit volume. How can you tell the difference if the same symbol is used for the density?<\/p>\n\n\n\n

The\u00a0specific gravity<\/strong>\u00a0of a gas is usually defined as the ratio of the density of the gas at a desired temperature and pressure to that of air (or any specified reference gas) at a certain temperature and pressure. The use of specific gravity occasionally may be confusing because of the sloppy manner in which the values of specific gravity are reported without citing\u00a0T<\/em>\u00a0and\u00a0p<\/em>, such as \u201cWhat is the specific gravity of methane?\u201d The answer to the question is not clear; hence, assume S.C. for both the gas and the reference gas:<\/p>\n\n\n\n

\"Image\"\/<\/figure>\n","protected":false},"excerpt":{"rendered":"

You have no doubt been exposed to the concept of the ideal gas in chemistry and physics. Why go over ideal gases again? At least two reasons exist. First, the experimental and theoretical properties of ideal gases are far simpler than the corresponding properties of liquids and solids. Second, use of the ideal gas concept […]<\/p>\n","protected":false},"author":1,"featured_media":4604,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[570],"tags":[],"class_list":["post-4704","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-ideal-and-real-gases"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/gases.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4704","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4704"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4704\/revisions"}],"predecessor-version":[{"id":4705,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4704\/revisions\/4705"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4604"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4704"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4704"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4704"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}