The Ideal Gas Laws

The volume and temperature of this tire stay the same as air is added. However, the pressure in the tire increases as the amount of air present increases. 

The number of moles is a fourth variable that can be added to pressure, volume, and temperature as a way to describe a gas sample. Recall that as the other gas laws were presented, care was taken to state that the relationships hold true for a fixed mass or a given amount of a gas sample. Changing the number of gas particles present will affect at least one of the other three variables. 

Because pressure, volume, temperature, and the number of moles present are all interrelated, it would be helpful if one equation could describe their relationship. Remember that the combined gas law relates volume, temperature, and pressure of a sample of gas. 

For a specific sample of gas, you can see that this relationship of pressure, volume, and temperature is always the same. You could say that 

The ideal gas constant In the ideal gas equation, the value of R depends on the units used for pressure. Table 14-1 shows the numerical value of R for different units of pressure. 

The gas laws discussed in the previous section are limited, because they only allow us to examine the relationship between two variables at a time. Fortunately, all four laws can be combined into one general law called 

In this equation, R is called the ideal gas law constant. Its value depends on the units used, but assuming pressure is measured in atmospheres, volume in liters, and temperature in Kelvins its value is 0.082 atm-L/mol-K. Other forms of the ideal gas law are 

The four gas laws in the previous section are all special cases of the ideal gas law. We can use the ideal gas law to calculate the volume one mole of gas occupies at standard conditions. Standard conditions are 0°C (273 K) and 1 atm pressure. The volume at these conditions is known as a standard molar volume. Plugging the numbers into the ideal gas law equation gives a value of 22.4 liters for the standard molar volume. 

The ideal gas law is the combination of the four properties used in the measurement of a gas— pressure (P), volume (V), temperature (T), and amount of a gas (n)—to give a single expression, which is written as 

Rearranging the ideal gas law equation shows that the four gas properties equal the gas law constant, R. 

Use the ideal gas law equation to solve for P, V, T, or n of a gas when given three of the four values in the Ideal gas law equation. Calculate density, molar mass, or volume of a gas in a chemical reaction. 

The ideal gas law is a useful expression when you are given the quantities for any three of the four properties of a gas. Although real gases show some deviations in behavior, the ideal gas law closely approximates the behavior of real gases at typical conditions. In working problems using the ideal gas law, the units of each variable must match the units in the R you select. 

Dinitrogen oxide, N2O, which is used in dentistry, is an anesthetic also called laughing gas. What is the pressure, in atmospheres, of 0.350 mol of N2O at 22 °C in a 5.00-L container  

Experiment 19 Gas Laws 3 Molar Mass of a Volatile Liquid 

We have considered three laws that describe the behavior of gases as revealed by experimental observations  

These relationships, which show how the volume of a gas depends on pressure, temperature, and number of moles of gas present, can be combined as follows  

Unless Otherwise noted, all art on this page is Cengage Learning 2014. 

The ideal gas law applies best at pressures smaller than 1 atm. 

Each of the gas laws focuses on the effect that changes in one variable have on gas volume  

We can combine these individual effects into one relationship, called the ideal gas law (or ideal gas equation)  

We can obtain a value of R by measuring the volume, temperature, and pressure of a given amount of gas and substituting the values into the ideal gas law. For example, using standard conditions for the gas variables, we have 

This numerical value of corresponds to the gas variables P, V, and T expressed in these units. R has a different numerical value when different units are used. For example, later in this chapter, R has the value 8..314. l/mol-K (J stands for joule, the SI unit of energy). 

We use rearrangements of the ideal gas law such as this one to solve gas law problems, as you ll see next. The point to remember is that there is no need to memorize the individual gas laws. 

All gases closely resemble each other in the dependence of volume on amount, temperature, and pressure. 

Click Coached Problems for a self-study module and a simulation on the ideal gas law. 

Relation of gas volume (V) to number of moles (n) and temperature (7) at constant pressure (/ ). The volume of a gas at constant pressure is directly proportional to (a) the number of moles of gas and (b) the absolute temperature. The volume-temperature plot must be extrapolated to reach zero because most gases liquefy at low temperatures well above 0 K. 

Relation of gas volume (V) to pressure ) at constant temperature (T). The volume of a fixed quantity of gas at constant temperature is inversely proportional to the pressure. In this case, the volume decreases from 6 L to 1 L when the pressure increases from 1 atm to 6 atm. 

The quantity Jc3, like k2 and k2, is a constant This is the equation of an inverse proportionality. The fact that volume is inversely proportional to pressure was first established in 1660 by Robert Boyle (1627-1691), an Irish experimental scientist The equation above is one form of Boyle s law. 

The three equations relating the volume, pressure, temperature, and amount of a gas can be combined into a single equation. Because V is directly proportional to both n and T, 

We can evaluate the constant kik2ki) in this equation by taking advantage of Avoga-dro s law, which states that equal volumes of all gases at the same temperature and pressure contain the same number of moles. For this law to hold, the constant must be the same for all gases. Ordinarily it is represented by the symbol R. Both sides of the equation are multiplied by P to give the ideal gas law 

Inflated balloons are pushed into the very cold liquid (T = 77 K). 

State the ideal gas law equation, and tell what each term means. 

ANALYZING DATA Nitrous oxide is sometimes used as a source of oxygen gas  

What volume of each product will be formed from 2.22 L N2O At STP, what is the density of the product gases when they are mixed  

1 diffusion for gases depend on 1 the velocities of their 1 molecules., Effusion 

The relationships that we have discussed so far can be combined into a single law that encompasses all of them. So far, we have shown that  

Combining these three expressions, we find that V is proportional to nT/P  

This equation is the ideal gas law, and a hypothetical gas that exactly follows this law is an ideal gas. The value of R, the ideal gas constant, is the same for all gases and has the value  

The ideal gas law contains within it the simple gas laws that we have discussed. For example, recall that Boyle s law states that V oc l/p when the amount of gas (n) and the temperature of the gas (T) are kept constant. We can rearrange the ideal gas law as follows  

Then put the variables that are constant, along with R, in parentheses  

The two precursors of the ideal gas law were Boyle s (see Chapter 2) and Charles laws. Boyle found that the volume of a given mass of gas is inversely proportional to the absolute pressure if the temperature is kept constant  

Charles found that the volume of a given mass of gas varies directly with the absolute temperature at constant pressure  

Boyle s and Charles laws may be combined into a single equation in which neither temperature nor pressure need be held constant  

For Equation (3.20) to apply, the mass of gas must be constant as the conditions change from (Pi, Ti) to (P2, Ta)- This equation indicates that for a given mass of a specific gas, PV/Thas a constant value. Since, at the same temperature and pressure, volume and mass must be directly proportional, this statement may be extended to 

Note that the volume terms above may be replaced by volume rate (or volumetric flow rate), q or Q. 

The ideal gas equation is not much of a stretch from what you have already seen with the combined gas law. If you recall, when the combined gas law was first presented, it was written in the form  

This constant can be replaced by a value, known as the ideal gas constant, R, to yield the equation  

Equation 8.8 can be rearranged to look more like the equation as you have learned it  

With this equation, it is possible to determine any property of a gas, if provided with the other three properties. An important thing to remember is the units of the ideal gas constant. Because R has units of L atm mol-1 K-1, it is important that you have volumes in liters, pressure in atmospheres, and temperatures in kelvins. The problems generally are set up that way, but you should still be alert. 

Sample Determine the volume of 4.50 moles of an unknown gas that exerts 7.50 atm pressure at 350 K. 

We can re-express Avogadro s law as follows the molar gas volume at a given temperature and pressure is a specific constant independent of the nature of the 

In the previous section, we discussed the empirical gas laws. Here we will show that these laws can be combined into one equation, called the ideal gas equation. Earlier we combined Boyle s law and Charles s law into the equation 

This constant is independent of the temperature and pressure but does depend on the amount of gas. For one mole, the constant will have a specific value, which we will denote as R. The molar volume, V, is 

According to Avogadro s law, the molar volume at a specific value of T and P is a constant independent of the nature of the gas, and this implies that R is a constant independent of the gas. The molar gas constant, R, is the constant of proportionality relating the molar volume of a gas to T/P. Values of R in various units are given in Table 5.5. 

In 1998, the Nobel committee awarded its prize in physiology or medicine to three scientists for the astounding discovery that nitric oxide gas, NO, functions as the signaling agent between biological cells in a wide variety of chemical processes. Until this discovery, biochemists had thought that the major chemical reactions in a cell always involved very large molecules. Now they discovered that a simple gas, NO, could have a central role in cell chemistry. 

Knowledge of any three of these properties is enough to define completely the condition of the gas, because the fourth property can be determined by using the ideal gas law. 

It is important to recognize that the ideal gas law is based on experimental measurements of the properties of gases. A gas that obeys this equation is said to behave ideally. That is, this equation defines the behavior of an Ideal gas. Most gases obey this equation closely at pressures of approximately 1 atm or lower, when the temperature is approximately 0 °C or higher. You should assume ideal gas behavior when working problems involving gases in this text. 

A sample of hydrogen gas, H2, has a volume of 8.56 L at a temperature of 0 °C and a pressure of 1.5 atm. Calculate the number of moles of H2 present in this gas sample. Assume that the gas behaves ideally. 

The ideal gas law can be used to solve a variety of problems. Example 13.8 demonstrates one type, where you are asked to find one property characterizing the condition of a gas given the other three properties. 

So far, we have empirically deduced several relationships between properties of gases. From Boyle s law, 

A proportionality constant called R converts this proportionality to an equation  

Situations frequently arise in which a gas undergoes a change that takes it from some initial condition (described by Pi, Vi, Ti, and n ) to a final condition (described by Pi, Vi, Ti, and ni). Because R is a constant. 

A weather balloon filled with helium (He) has a volume of 1.0 X 10 L at 1.00 atm and 30°C. It rises to an altitude at which the pressure is 0.60 atm and the temperature is —20°C. What is the volume of the balloon then Assume that the balloon stretches in such a way that the pressure inside stays close to the pressure outside. 

Because the amount of helium does not change, we can set i equal to 2 and cancel it out of Equation 9.8, giving 

I Relate the amount of gas present to its pressure, temperature, and volume using the ideal gas law. 

Review Vocabulary mole an SI base unit used to measure the amount of a substance the amount of a pure substance that contains 6.02 X 10 representative particles 

Real-World Reading Link You know that adding air to a tire causes the pressure in the tire to increase. But did you know that the recommended pressure for car tires is specified for cold tires As tires roll over the road, friction causes their temperatures to increase. This also causes the pressure to increase. 

For example, suppose you want to find the number of moles in a sample of gas that has a volume of 3.72 L at STP. Use the molar volume to convert from volume to moles. 

This is an extension of the combined gas law. In the combined gas law, we saw the relationship between pressure, volume, and temperature. The ideal gas law can be expressed mathematically as follows  

P - pressure V - volume n -number of moles R - molar gas constant T - temperature 

Ideal gas molecules travel randomly in straight lines at dijfer-ent speeds. 

R (molar gas constant) has values 0.082 L.atm/(K.mol), or 8.31 J/(K.mol), and of course the difference in values is due to the fact that the gas constant is expressed here in two different units. 

A horizontal cylinder (a) is closed at one end by a piston that moves freely left or right depending on the pressure exerted by the enclosed gas. The gas consists of 10 two-atom molecules. A reaction occurs in which 5 of the molecules separate into one-atom particles. In cylinder (b), sketch the position to which the piston would move as a result of the reaction. Pressure and temperature remain constant throughout the process. (Hint How many total particles would be present after the reaction Include them in your sketch.) 

Having accumulated several different proportionalities between volume and the three other measurable properties of a gas, we now look for a single equation that ties them all together. We have seen from Charles s Law that V T, from Boyle s Law that 

V oc —, and from Avogadro s Law that V n. If volume is proportional to three different quantities, it is logical to assume that it is proportional to their product, 1 

Rearranging gives the ideal gas equation in its most common form  

The relationship symbolized in this equation is called the ideal gas law. If you are a science or engineering major, you will probably encounter Equation 14.3 in at least 

All the gas laws described so far apply only to a given sample of gas. If a gas is produced during a chemical reaction or some of the gas under study escapes during processing, these gas laws are not used because the gas sample has only one temperature and one pressure. The ideal gas law works (at least approximately) for any sample of gas. Consider a given sample of gas, for which 

If we increase the number of moles of gas at constant pressure and temperature, the volume must also increase. Thus, we can conclude that the constant k can be regarded as a product of two constants, one of which represents the number of moles of gas. We then get 

0821 L-atm/(moTK) (Note that there is a zero after the decimal point.) 

In the simplest ideal gas law problems, values for three of the four variables are given and you are asked to calculate the value of the fourth. As usual with the gas laws, the temperature must be given as an absolute temperature, in kelvins. The units of P and V are most conveniently given in atmospheres and liters, respectively, because the units of R with the value given above are in terms of these units. If other units are given for pressure or volume, convert each to atmospheres or liters, respectively. 

The volume of 1.00 mol of gas at STP is called the molar volume of a gas. This value should be memorized. 

These values are all equivalent. Use the one that matches the pressure units you are using. 

What pressure in atmospheres will 18.6 mol of methane exert when it is compressed in a 12.00-L tank at a temperature of 45°C  

As always, change the temperature to kelvins before doing anything else. 

Notice that this pressure makes sense because a large amount of gas is being squeezed into a much smaller space. 

In the mid-nineteenth century, scientists combined Boyle s law, Charles s law, and Avogadro s law into a single law called the ideal gas law. Usually, the ideal gas law is stated in the following form  

Copyright 2010 Cengage Learning, Inc. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. 

The ideal gas law is an equation of state for a gas, where the state of the gas is its condition at a given time. A particular state of a gas is described by its pressure, volume, temperature, and number of moles. Knowledge of any three of these properties is enough to completely define the state of a gas, since the fourth property can then be determined from the equation for the ideal gas law. 

The reaction of zinc with hydrochioric acid to produce bubbies of hydrogen gas. 

---

The above equation is valid at low pressures where the assumptions hold. However, at typical reservoir temperatures and pressures, the assumptions are no longer valid, and the behaviour of hydrocarbon reservoir gases deviate from the ideal gas law. In practice, it is convenient to represent the behaviour of these real gases by introducing a correction factor known as the gas deviation factor, (also called the dimensionless compressibility factor, or z-factor) into the ideal gas law ... 

The ideal gas law equation of state thus leads to a linear or Henry s law isotherm. A natural modification adds a co-area term ... 

In 1873, van der Waals [2] first used these ideas to account for the deviation of real gases from the ideal gas law P V= RT in which P, Tand T are the pressure, molar volume and temperature of the gas and R is the gas constant. Fie argried that the incompressible molecules occupied a volume b leaving only the volume V- b free for the molecules to move in. Fie further argried that the attractive forces between the molecules reduced the pressure they exerted on the container by a/V thus the pressure appropriate for the gas law isP + a/V rather than P. These ideas led him to the van der Waals equation of state ... 

To make the difTerences between the two kinds of walls clearer, consider the situation where botir are ideal gases, each satisfying the ideal-gas law pV= nRT. If the two were separated by a diatliennic wall, one would... 

Substances at high dilution, e.g. a gas at low pressure or a solute in dilute solution, show simple behaviour. The ideal-gas law and Henry s law for dilute solutions antedate the development of the fonualism of classical themiodynamics. Earlier sections in this article have shown how these experimental laws lead to simple dieniiodynamic equations, but these results are added to therniodynaniics they are not part of the fonualism. Simple molecular theories, even if they are not always recognized as statistical mechanics, e.g. the kinetic theory of gases , make the experimental results seem trivially obvious. 

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

The parameters a and b are characteristic of the substance, and represent corrections to the ideal gas law dne to the attractive (dispersion) interactions between the atoms and the volnme they occupy dne to their repulsive cores. We will discnss van der Waals equation in some detail as a typical example of a mean-field theory. 

So far, so good. The situation is really no different, say, than the ideal gas law, in which the gas constant is numerically different and has different units depending on the units chosen for p and V, The unit change in Example 10.1 is analogous to changing the gas constant from liter-atmospheres to calories it is apparent that one system is physically more meaningful than another in specific problems. Several considerations interfere with this straightforward parallel, however, and cause confusion ... 

Other conventions for treating equiUbrium exist and, in fact, a rigorous thermodynamic treatment differs in important ways. Eor reactions in the gas phase, partial pressures of components are related to molar concentrations, and an equilibrium constant i, expressed directiy in terms of pressures, is convenient. If the ideal gas law appHes, the partial pressure is related to the molar concentration by a factor of RT, the gas constant times temperature, raised to the power of the reaction coefficients. 

The foregoing discussion has dealt with nonideahties in the Hquid phase under conditions where the vapor phase mixes ideally and where pressure-temperature effects do not result in deviations from the ideal gas law. Such conditions are by far the most common in commercial distillation practice. However, it is appropriate here to set forth the completely rigorous thermodynamic expression for the Rvalue ... 

At pressures less than 2 MPa (20 bar) and temperatures greater than 273 K, PC 1.0. When the vapor obeys the ideal gas law, 2 = 1.0 then for ideal vapor solutions and for conditions such that PC = 1.0, equation 19 reduces to equation 6. 

Limiting L ws. Simple laws that tend to describe a narrow range of behavior of real fluids and substances, and which contain few, if any, adjustable parameters are called limiting laws. Models of this type include the ideal gas law equation of state and the Lewis-RandaH fugacity rule (10). 

Statistical mechanics provides physical significance to the virial coefficients (18). For the expansion in 1/ the term BjV arises because of interactions between pairs of molecules (eq. 11), the term C/ k, because of three-molecule interactions, etc. Because two-body interactions are much more common than higher order interactions, tmncated forms of the virial expansion are typically used. If no interactions existed, the virial coefficients would be 2ero and the virial expansion would reduce to the ideal gas law Z = 1). 

Ideal Gas Behavior, In 1787 it was demonstrated that the volume of a gas varies directly with temperature if the pressure remains constant. Other investigations determined complementary correlating relations from which the perfect or ideal gas law was drawn (1 3). Expressed mathematically, the ideal gas law is... 

The ideal gas law is an example of a correlating expression that comes direcdy from experimental observations, but has theoretical significance. Despite its simplicity, the ideal gas law is an excellent estimation tool. Often, it is the first approximation in systems involving real gases of all types. Unfortunately for Hquids and soHds no laws of such general utility are available. 

Reduced Properties. One of the first attempts at achieving an accurate analytical model to describe fluid behavior was the van der Waals equation, in which corrections to the ideal gas law take the form of constants a and b to account for molecular interactions and the finite volume of gas molecules, respectively. 

Correlation Methods Vapor densities are not correlated as functions of temperature alone, as pressure and temperature are both important. At high temperatures and very low pressures, the ideal gas law can be applied whde at moderate temperature and low pressure, vapor density is usually correlated by the virial equation. Both methods will be discussed later. 

For simple molecules at temperatures above the critical and at pressures no more than a few atmospheres, the ideal gas law, Eq. (2-66), may be used to estimate vapor density. 

The equation is rendered integrable by application of the stoichiometry of the reaction, the ideal gas law, and, for instance, the power law for rate of reaction. Some details are shown in Table 7-9. 

A key limitation of sizing Eq. (8-109) is the limitation to incompressible flmds. For gases and vapors, density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal-gas-law model. Deviations from ideal behavior are corrected for, to first order, with nommity values of compressibihty factor Z. (See Sec. 2, Thvsical and Chemical Data, for definitions and data for common fluids.) For compressible fluids... 

In terms of the fractional conversion and the ideal gas law, the rate equation becomes... 

Worst-case atmospheric conditions occur to maximize (C). This occurs with minimum dispersion coefficients and minimum wind speed u within a stability class. By inspection of Figs. 26-54 and 26-55 and Table 26-28, this occurs with F-stability and u = 2 m/s. At 300 m = 0.3 km, from Figs. 26-54 and 26-55, <3 = 11m and <3 = 5 m. The concentration in ppm is converted to kg/m by application of the ideal gas law. A pressure of 1 atm and temperature of 298 K are assumed. 

The ambieut air density is computed from the ideal gas law aud gives a result of 1.22 kg/ud. Thus... 

Ideal gas obeys the equation of state PV = MRT or P/p = MRT, where P denotes the pressure, V the volume, p the density, M the mass, T the temperature of the gas, and R the gas constant per unit mass independent of pressure and temperature. In most cases the ideal gas laws are sufficient to describe the flow within 5% of actual conditions. When the perfect gas laws do not apply, the gas compressibility factor Z can be introduced ... 

The Lapple charts for compressible fluid flow are a good example for this operation. Assumptions of the gas obeying the ideal gas law, a horizontal pipe, and constant friction factor over the pipe length were used. Compressible flow analysis is normally used where pressure drop produces a change in density of more than 10%. 

In our third step, we convert the compositions to a kg-moles-per-second basis. For this example (as well as many common industry cases), the ideal gas law can be used n = PV/RT, where n = number of moles... 

Now rearrange the ideal gas law to convert to volumetric flow rate ... 

Specific volume is determined by application of the ideal gas law. One pound mole of air occupies a volume of 359 cubic feet at standard conditions, hence ... 

A simplified estimate can be made by first converting the flow at actual conditions to the flow at standard conditions (i.e., at 70 F and 1 atm). The calculation basis for the linear velocity assumes a roughness coefficient of 0.0005 and a kinematic viscosity for air of 1.62 x lO fF/sec. From the ideal gas law, the following expression is developed ... 

Since non-ideal gases do not obey the ideal gas law (i.e., PV = nRT), corrections for nonideality must be made using an equation of state such as the Van der Waals or Redlich-Kwong equations. This process involves complex analytical expressions. Another method for a nonideal gas situation is the use of the compressibility factor Z, where Z equals PV/nRT. Of the analytical methods available for calculation of Z, the most compact one is obtained from the Redlich-Kwong equation of state. The working equations are listed below ... 

Of course, you should be familiar with this equation (the Ideal Gas Law), where n is the molar concentration of solute, R is the universal gas law constant, and T is absolute temperature in °K. The permeate flow can be calculated from ... 

---