THE GAS LAWS

The gas laws developed by Robert Boyle, Jacques Charles and Joseph Gay-Lussac are based upon empirical observations and describe the behavior of a gas in macroscopic terms, that is, in terms of properties that a person can directly observe and experience. The kinetic theory of gases describes the behavior of molecules in a gas based on the mechanical movements of single molecules. A gas is defined as a collection of small particles (atom- and molecule-sized) with the mass w, (subscript i denotes a defined substance)  

With the definition of the gas density p as a product of particle density Cn and molecule mass the fundamental equation of the kinetic theory of gases follows  

Expressing the gas density q as the ratio between total mass m (m = N w ) and gas volume V the Boyle-Mariotte law then follows. represents the quantity of an energy, or more specifically the pressure-volume work of this gaseous system  

Empirically, it has been found that (J is the absolute temperature) pV 

This is the combined gas law that combines Charles s law (V — constant Twhere Cn, P = constant), Boyle s law pV = constant where c, T = constant) and Gay-Lussac s law (p T = constant where m, V = constant). Avogadro s law expresses that the ratio of a given gas volume to the amount of gas molecules within that volume is constant (where T,p = constant)  

The gas laws we will study in this chapter are the product of countless experiments on the physical properties of gases that were carried out over several centuries. Each of these generalizations regarding the macroscopic behavior of gaseous substances represents a milestone in the history of science. Together they have played a major role in the development of matty ideas in chemistry. 

Apparatus for studying the relationship between pressure and volume of a gas. (a) The levels of mercury are equal and the pressure of the gas is equal to the atmospheric pressure (760 mmHg). The gas volume is 100 mL. (b) Doubling the pressure by adding more mercury reduces the gas volume to 50 mL. (c) Tripling the pressure decreases the gas volume to one-third of the original volume. The temperature and amount of gas are kept constant. 

The mathematical expression showing an inverse relationship between pressure and volume is 

Charles s Law V= ( ) T is constant Heating or cooling a gas at constant volume 

Graphs showing variation of the volume of a gas with the pressure exerted on the gas, at constant temperature, (a) P versus V. Note that the volume of the gas doubles as the pressure is halved, (b) P versus 1/V. 

The gas laws apply to ideal gases, which are described by the kinetic theory in the following five statements. 

A sample of compressed methane has a volume of 648 mL at a pres-sixre of 503 kPa. To what pressixre would the methane have to be compressed in order to have a volume of 216 mL  

Examine the Boyle s law equation. You need to find P2, the new pressixre, so solve the equation for P2. 

Solving Problems A Chemistry Handbook Chemistry Matter and Change 137 

To determine whether your answer is reasonable, notice that the gas is being squeezed to a smaller volume, which requires that the pressure be increased. So, your answer is probably correct. 

Many of the gas law problems on the AP test are conceptual questions rather than calculations. Make sure you understand each of these topics at a conceptual level as well as at a mathematical level. 

One of the earliest discoveries of the behavior of gases came from Robert Boyle in 1661. Boyle discovered that the volume of a gas is inversely proportional to the pressure placed on it. That is, in a system where volume is allowed to change, an increase in pressure will decrease the volume. Likewise, if you think of a closed system, such as a cylinder with a piston above it, pushing down on the piston (i.e., decreasing the volume) will have the effect of increasing the pressure. This relationship can be expressed mathematically as Equation 8.3  

Remember, Kelvin temperatures must be used for this to be accurate. Charles s Law The next discoveries about gases came from Jacques Charles, who in 1787 discovered what is today known as Charles s law. This law states that for a fixed amount of gas at constant pressure, the relationship between volume and temperature is directly proportional. Written as a formula Charles s law, Equation 8.4, takes the form  

Since the outcome of the equation remains constant, you can use this equation to determine the behavior of a gas as conditions change (i.e., changing temperature, pressure, volume, or quantity of gas). To use the equation this way, it is helpful to rewrite it as Equation 8.8  

Sample If 1.50 moles of a gas at 300 K and under a pressure of 1.5 atm occupy 0.5 liters, at what temperature will 3.00 moles of the same gas occupy 1.5 liters and exert a pressure of 3.0 atm  

I Apply the gas laws to problems involving the pressure, temperature, and volume of a constant amount of gas. 

Boyle s law absolute zero Charles s law Gay-Lussac s law combined gas law 

Real-World Reading Link What might happen to the gas in a balloon if you decreased its volume by squeezing it You would feel increasing resistance as you squeeze and might see part of the balloon bulge. 

As the balloon example illustrates, the pressure of a gas and its volume are related. Robert Boyle (1627-1691), an Irish chemist, described this relationship between the pressure and the volume of a gas. 

For a given amount of gas held at constant temperature, the product of pressure and volume is a constant. 

Would it be easier to drink water with a straw on top or at the foot of Mt. Everest  

In the seventeenth century, Robert Boyle studied the behavior of gases systematically and quantitatively. In one series of studies, Boyle investigated the pressure-volume relationship of a gas sample. Typical data collected by Boyle are shown in Table 5.2. Note that as the pressure (P) is increased at constant temperature, the volume (V) occupied by a given amount of gas decreases. Compare the first data point with a pressure of 724 mmHg and a volume of 1.50 (in arbitrary unit) to the last data point with a pressure of 2250 mmHg and a volume of 0.58. Clearly there is an inverse relationship between pressure and volume of a gas at constant temperature. As the pressure is increased, the volume occupied by the gas dcCTcases. Conversely, if the applied pressure is decreased, the volume the gas occupies increases. This relationship is now known as Boyle s law, which states that the pressure of a fixed amount of gas at a constant temperature is inversely proportional to the volume of the gas. 

TABLE 5.2 Typical Pressure-Volume Relationship Obtained by Boyle 

You can t understand gases without understanding the movement of gas particles. Remember from your study of the kinetic-molecular theory in Chapter 13 that gas particles behave differently than those of liquids and solids. The kinetic theory provides a model that is used to explain the properties of solids, liquids, and gases in terms of particles that are always in motion and the forces that exist between them. The kinetic theory assumes the following concepts about gases are true. 

The kinetic theory reiates pressure and the number of coiii-sions per unit time for a gas. 

Q When the bicycie pump is puiied out as far as it wiii go, the pressure of the air inside the pump equais that of the atmosphere. 

Almost all the volume of a gas is empty space. Gases can be compressed by moving gas particles closer together because of this low density of particles. 

The nature of gases Actual gases don t obey all the assumptions made by the kinetic theory. But for many gases, their behavior approximates the behavior assumed by the kinetic theory. You will learn more about real gases and how they vary from these assumptions in Section 14.3. 

This inverse relationship between the volume of a gas and its pressure is a consequence of the compressibility of matter in the gas phase (recall that solids and liquids are not significantly compressible). 

Charles law gives the relationship between gas volume and temperature, and states that the volume of a fixed quantity of gas is directly proportional to the absolute temperature (°C + 273) at constant pressure. This law may be stated mathematically as 

The third of the three fundamentally important gas laws is Avt iadro s law, relating the volume of gas to its quantity in moles. This law states that at constant temperature and pressure, the volume of a gas is directly proportional to the number of molecules of gas, commonly expressed as moles. Mathematically, this relationship is 

The three gas laws just defined may be combined into a general gas law stating that the volume of a quantity of ideal gas is proportional to the number of moles of gas and its absolute temperature and inversely proportional to its pressure. Mathematically, this law is 

Designating the proportionality constant as the ideal gas constant, R, yields the ideal gas equation  

We use various devices to measure the pressures of enclosed gases. Tire gauges, for example, measure the pressure of air in automobile and bicycle tires. In laboratories we sometimes use a manometer, which operates on a principle similar to that of a barometer, as shown in Sample Exercise 10.2. 

On a certain day a laboratory barometer indicates that the atmospheric pressure is 764.7 torr. A sample of gas is placed in a flask attached to an open-end mercury manometer FIGURE 10.3), and a meter stick is used to measure the height of the mercury in the two arms of the U tube. The height of the mercury in the open-end arm is 136.4 mm, and the height in the arm in contact with the gas in the flask is 103.8 mm. What is the pressure of the gas in the flask (a) in atmospheres, (b) in kilopascals  

Analyze We are given the atmospheric pressure (764.7 torr) and the mercury heights in the two arms of the manometer and asked to determine the gas pressure in the flask. Recall that millimeters of mercury is a pressure unit. We know that the gas pressure from the flask must be greater than atmospheric pressure because the mercury level in the arm on the flask side (103.8 mm) is lower than the level in the arm open to the atmosphere (136.4 mm). Therefore, the gas from the flask is pushing mercury from the arm in contact with the flask into the arm open to the atmosphere. 

Check The calculated pressure is a bit more than 1 atm, which is about 101 kPa. This makes sense because we anticipated that the pressure in the flask would be greater than the atmospheric pressure (764.7 torr = 1.01 atm) acting on the manometer. 

Four variables are needed to define the physical condition, or state, of a gas temperature, pressure, volume, and amount of gas, usually expressed as number of moles. The equations that express the relationships among these four variables are known as the gas laws. Because volume is easily measured, the first gas laws to be studied expressed the effect of one of the variables on volume, with the remaining two variables held constant. 

At a constant temperature, the volume of a fixed amount of a perfect gas varies inversely with its pressure. 

Remember that water Boyle s at a constant temperature and that Prince Charles is under constant pressure to be king. 

A gas that contains molecules of infinitely small size, which, therefore, occupy no volume themselves, and which have no force of attraction between them. 

This equation is provided for you in the AP formulas. Although the gas law questions on the AP exam usually provide the temperature in kelvins, you should be on the lookout for degrees Celsius. 

Eor gases, this force can be a factor of their motion or their weight. Atmospheric pressure is caused by the weight of air particles that are attracted toward earth. Inside a sealed container, pressure is exerted by the collisions of particles on the sides of the container. By determining the force of those collisions on a given area, you can determine the pressure exerted by the particles. There are many units that describe pressure. The SI unit of pressure is the pascal. Pa. The kilopascal is a bit more practical as a unit, however, since a pascal is quite small. Other units of pressure include millimeters of Hg (mm Hg), torr, bar, and atmospheres. The relationship between the units is as follows  

You might have noticed that torr and mm Hg have the same value. On the AP exam, units of pressure are usually atm in the ideal gas problems R uses atm as the pressure unit) and mm Hg in partial pressure problems. 

Continuing to use a syringe as a container, the basic gas laws can be explained. These laws apply to what is referred to as an ideal or perfect gas. An ideal or perfect gas can be thought of as a gas that conforms to the kinetic molecular theory. In reality, gas molecules do have volume and exert forces on each other. Under normal conditions of temperature and pressure, though, the kinetic molecular theory explains the behavior of gases quite well. It is only when a gas is at very low temperatures and/or under extremely high pressure that a gas no longer behaves ideally. 

Pushing in on the plunger decreases the volume, which causes an increase in pressure due to the collision frequency increasing 

Remember, because pressure is force divided by area, if the area decreases the pressure increases. When we push in on the plunger, the pressure increases to some value above 1 atmosphere. The smaller the syringe s barrel volume becomes, the higher the pressure exerted by the air in syringe. The relationship between volume and pressure is known as Boyle s Law. Boyle s Law was first mentioned in Chapter 3 and is named after Robert Boyle. Simply stated, Boyle s Law says that the pressure and volume of an ideal gas are inversely related, as one goes up, the other goes down. Boyle s Law can be stated mathematically as 

So if the gas in the syringe was originally at 1 atmosphere and the volume was 50 mL, then PV would equal 50 atm-mL. If we pushed in on the plunger to decrease the volume to 25 mL, then the pressure would have to increase to 2 atmospheres for PV to remain constant. Remember, when we apply Boyle s Law, the only two variables that change are pressure and volume. We are assuming the temperature of the gas and the number of molecules of gas in the syringe remain constant. 

What happens when the pressure of a gas remains constant and the temperature and volume change Before looking at the relationship between temperature and volume, though, the concept of temperature must be understood. Temperature is one of those terms that is continually used, but rarely given much thought. As long as we can remember, we have had our temperature taken, seen and heard the daily temperature reported, and baked foods at various temperatures. Intuitively, we think of tempera- 

Boyle s law, which summarizes these observations, states that the volume of a fixed quantity of gas maintained at constant temperature is inversely proportional to the pressure. When two measurements are inversely proportional, one gets smaller as the other gets larger. Boyle s law can be expressed mathematically as 

The value of the constant depends on temperature and on the amount of gas in the sample. 

What is the total pressure on the gas after the 760 mm Hg has been added  

Boyle s law occupies a special place in the history of science because Boyle was the first to carry out experiments in which one variable was systematically changed to determine the effect on another variable. The data from the experiments were then employed to establish an empirical relationship—a law.  

In contrast to the condensed phases, all gases, even those with vastly different chemical compositions, exhibit remarkably similar physical behavior. Numerous experiments carried out in the seventeenth and eighteenth centuries showed that the physical state of a sample of gas can be described completely with just four parameters temperature (7, pressure (P). volume (V), and number of moles (n). Knowing any three of these parameters enables us to calculate the fourth. The relationships between these parameters are known as the gas laws. 

A conversion factor to replace mmHg with atm can be derived from 1 atm = 760 mmHg (exactly). 

You need to be aware of the Pascal and the kilopascal, but because neither is used widely in the United States, pressures will be given in atmospheres, mmHg, and torr in the following discussions. 

There are two ways of describing the pressure of a gas the pressure exerted by the gas or the pressure exerted on the gas. Think of a balloon filled with helium. The helium exerts a pressure on the inner wall of the balloon that is identical to the pressure exerted by the atmosphere against the outer surface. If they were not the same, the balloon would either expand or contract until they were equal. 

What is a vacuum Simply stated, a vacuum is the absence of pressure—the absence of gaseous particles. Very high vacuums are stated as very low pressures, on the order of 10 6 mmHg. 

When working problems involving gases, temperature is always in Kelvin, not Celsius. To convert a Celsius temperature to Kelvin  

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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 ... 

Almost everyone has a concept of pressure from weather reports of tlie pressure of the atmosphere around us. In this context, high pressure is a sign of good weather while very low pressures occur at the eyes of cyclones and hurricanes. In elementary discussions of mechanics, hydrostatics of fluids and the gas laws, most scientists leam to compute pressures in static systems as force per unit area, often treated as a scalar quantity. They also leam that unbalanced pressures cause fluids to flow. Winds are the flow of the atmosphere from regions of high to low... 

The composition of the vapour can easily be calculated as follows — Assuming that the gas laws are applicable, it follows that the number of molecules of each component in the vapour wdll be proportional to its partial pressure, i.e., to the vapour pressure of the pure liquid at that temperature. If and p are the vapour pressures of the two liquids A and B at the boiling point of the mixture, then the total pressure P is given by ... 

When a solid such as charcoal is exposed in a closed space to a gas or vapour at some definite pressure, the solid begins to adsorb the gas and (if the solid is suspended, for example, on a spring balance) by an increase in the weight of the solid and a decrease in the pressure of the gas. After a time the pressure becomes constant at the value p, say, and correspondingly the weight ceases to increase any further. The amount of gas thus adsorbed can be calculated from the fall in pressure by application of the gas laws if the volumes of the vessel and of the solid are known or it can be determined directly as the increase in weight of the solid in the case where the spring balance is used. 

Electrostatic Interaction. Similarly charged particles repel one another. The charges on a particle surface may be due to hydrolysis of surface groups or adsorption of ions from solution. The surface charge density can be converted to an effective surface potential, /, when the potential is <30 mV, using the foUowing equation, where -Np represents the Faraday constant and Ai the gas law constant. 

In many cases, heating or cooling of the gaseous effluent will be required before if enters the control device. The engineer must be thoroughly aware of the gas laws, thermodynamic properties, and reactions involved to secure a satisfactory design. For example, if a gas is cooled there may be condensation if the temperature drops below the dewpoint. If water is sprayed into... 

Tn normal work, "C is used in preference to the absolute temperature K. However, it is essential that K be used when working with the gas laws, radiation, and the coefficient of cubical expansion. The symbol for normal temperature is 0 followed by a suffix, while T always denotes absolute temperature. 

Note, most manufacturers tables or charts give SCFH capacities at 14.7 psia and 60°F, and these must be corrected by the gas laws to the artwa/volume at flowng conditions in order to represent the actual performance of the system. The tables or charts of the manufacturers read in SCFH for selected relief device setting and for tank pressure, expressed as air at SCFH (see Figures 7-37A and 37B). 

If this volume were needed to be expressed at 55°F and 1.5 psig, then using the gas laws ... 

Use can be made of this latter effect. The power element can be limit charged so that all the refrigerant within it has vaporized by a predetermined temperature (commonly 0°C). Above this point, the pressure within it will follow the gas laws ... 

Air at sea level exerts a static pressure, due to the weight of the atmosphere, of 1013.25 mhar. The density, or specific mass, at 20°C is 1.2 kg/m . Densities at other conditions of pressure and temperature can he calculated from the Gas Laws ... 

To obtain a two-point equation, write the gas law twice and divide to eliminate constants. 

With the aid of the gas laws we can calculate the relative concentrations of the components in moist air. Suppose that on a certain day the... 

It is often desirable to calculate the mass of sample from the gas laws. Pressure and temperature must then be known. 

At atmospheric pressure and at the temperature of melting ice, 32 gr. of oxygen occupy a volume of 22,412 c.c. (corrected for a slight deviation from the gas laws) ... 

It would appear at first sight necessary to define an ideal gas as one which strictly obeys all the gas laws. As a matter of fact we can prove that if it conforms to two conditions it will conform to all the conditions we shall take as defining an ideal gas. 

If we neglect v1 in comparison with vn, and assume the vapour obeys the gas laws, we have, for a mol ... 

Again we assume that F is negligible in comparison with V2, and that the saturated vapour obeys the gas laws.. 

At very low temperatures the pressures of the saturated vapours of liquids and solids are very small, and since the deviation of an actual gas from the laws of ideal gases becomes all the less the smaller is its density ( 70), we can safely assume that the vapour obeys the gas laws. 

Wtiat Are the Key Ideas/ We can predict the physical properties of any gas by using the set of equations known as the gas laws. These equations can be explained in terms of a model of a gas in which the molecules are in ceaseless random motion and so widely separated that they do not interact with one another. 

Summaries of the properties of gases, particularly the variation of pressure with volume and temperature, are known as the gas laws. The first reliable measurements of the properties of gases were made by the Anglo-Irish scientist Robert Boyle in 1662 when he examined the effect of pressure on volume. A century and a half later, a new pastime, hot-air ballooning, motivated two French scientists, Jacques Charles and Joseph-Louis Gay-Lussac, to formulate additional gas laws. Charles and... 

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