, and amount of a gas are described and how these relationships can be combined to give a general expression that describes the behavior of a gas. of a gas is proportional to the number of moles and the temperature and inversely proportional to the pressure. This expression can also be written as , which is represented by the letter \(\). Inserting into Equation \(\ref{10.4.2}\) gives . is defined as a hypothetical gaseous substance whose behavior is independent of attractive and repulsive forces and can be completely described by the . In reality, there is no such thing as an , but an is a useful conceptual model that allows us to understand how gases respond to changing conditions. As we shall see, under many conditions, most real gases exhibit behavior that closely approximates that of an . The can therefore be used to predict the behavior of real gases under most conditions. The does not work well at very low temperatures or very high pressures, where deviations from ideal behavior are most commonly observed. , however, we need to know the value of the . Its form depends on the units used for the other quantities in the expression. If V is expressed in liters (L), P in atmospheres (atm), T in kelvins (K), and n in moles (mol), then = 0.08206 \dfrac{\rm L\cdot atm}{\rm K\cdot mol} \label{10.4.5} \] can also have units of J/(K•mol): = 8.3145 \dfrac{\rm J}{\rm K\cdot mol}\label{10.4.6} \] of 1.000 mol of an under standard conditions using the variant of the given in Equation \(\ref{10.4.4}\): of 1 mol of an is and , approximately equivalent to the of three basketballs. The molar volumes of several real gases at 0°C and 1 atm are given in Table 10.3, which shows that the deviations from behavior are quite small. Thus the does a good job of approximating the behavior of real gases at 0°C and 1 atm. The relationships described in Section 10.3 as Boyle’s, Charles’s, and Avogadro’s laws are simply special cases of the in which two of the four parameters (P, V, T, and n) are held fixed. allows us to calculate the value of the fourth variable for a gaseous sample if we know the values of any three of the four variables (, , , and ). It also allows us to predict the of a sample of a gas (i.e., its final temperature, pressure, , and amount) following any changes in conditions if the parameters (, , , and ) are specified for an Some applications are illustrated in the following examples. The approach used throughout is always to start with the same equation—the —and then determine which quantities are given and which need to be calculated. Let’s begin with simple cases in which we are given three of the four parameters needed for a complete physical description of a gaseous sample. to calculate the effect of in any of the specified conditions on any of the other parameters, as shown in Example \(\PageIndex{5}\). equation must be applied twice - to an initial condition and a final condition. This is: =\dfrac{P_iV_i}{n_iT_i} \hspace{1cm} =\dfrac{P_fV_f}{n_fT_f} \nonumber \] \). Therefore, we have: noted by Boyle ( = constant) and the relationship between and amount observed by Avogadro (/ = constant). We will not do so, however, because it is more important to note that the historically important gas laws are only special cases of the in which two quantities are varied while the other two remain fixed. The method used in Example \(\PageIndex{1}\) can be applied in such case, as we demonstrate in Example \(\PageIndex{2}\) (which also shows why heating a closed container of a gas, such as a butane lighter cartridge or an aerosol can, may cause an explosion). can also be used to calculate molar masses of gases from experimentally measured gas densities. To see how this is possible, we first rearrange the to obtain (mol/L). The number of moles of a substance equals its mass (\(m\), in grams) divided by its (\(M\), in grams per mole): of an unknown gas by measuring its density at a known temperature and pressure. This method is particularly useful in identifying a gas that has been produced in a reaction, and it is not difficult to carry out. A flask or bulb of known is carefully dried, evacuated, sealed, and weighed empty. It is then filled with a sample of a gas at a known temperature and pressure and reweighed. The difference in mass between the two readings is the mass of the gas. The of the flask is usually determined by weighing the flask when empty and when filled with a liquid of known density such as water. The use of density measurements to calculate molar masses is illustrated in Example \(\PageIndex{6}\). is derived from empirical relationships among the pressure, the , the temperature, and the number of moles of a gas; it can be used to calculate any of the four properties if the other three are known. : \(PV = nRT\), = 0.08206 \dfrac{\rm L\cdot atm}{\rm K\cdot mol}=8.3145 \dfrac{\rm J}{\rm K\cdot mol}\) , the temperature, the pressure, and the amount of a gas can be combined into the , = . The proportionality constant, , is called the and has the value 0.08206 (L•atm)/(K•mol), 8.3145 J/(K•mol), or 1.9872 cal/(K•mol), depending on the units used. The describes the behavior of an , a hypothetical substance whose behavior can be explained quantitatively by the and the . is 0°C and 1 atm. The of 1 mol of an at STP is 22.41 L, the . All of the empirical gas relationships are special cases of the in which two of the four parameters are held constant. The allows us to calculate the value of the fourth quantity (, , , or ) needed to describe a gaseous sample when the others are known and also predict the value of these quantities following a change in conditions if the original conditions (values of , , , and ) are known. The can also be used to calculate the density of a gas if its is known or, conversely, the of an unknown gas sample if its density is measured.