CHEMISTRY THE CENTRAL SCIENCE
10.4 THE IDEAL-GAS EQUATION
All three laws we just examined were obtained by holding two of the four variables P, V, T, and n constant and seeing how the remaining two variables affect each other. We can express each law as a proportionality relationship. Using the symbol ∝ for “is proportional to,” we have
We can combine these relationships into a general gas law:
and if we call the proportionality constant R, we obtain an equality:
which we can rearrange to
which is the ideal-gas equation (also called the ideal-gas law). An ideal gas is a hypothetical gas whose pressure, volume, and temperature relationships are described completely by the ideal-gas equation.
In deriving the ideal-gas equation, we assume (a) that the molecules of an ideal gas do not interact with one another and (b) that the combined volume of the molecules is much smaller than the volume the gas occupies; for this reason, we consider the molecules as taking up no space in the container. In many cases, the small error introduced by these assumptions is acceptable. If more accurate calculations are needed, we can correct for the assumptions if we know something about the attraction molecules have for one another and if we know the diameter of the molecules.
The term R in the ideal-gas equation is the gas constant. The value and units of R depend on the units of P, V, n, and T. The value for T in the ideal-gas equation must always be the absolute temperature (in kelvins instead of degrees Celsius). The quantity of gas, n, is normally expressed in moles. The units chosen for pressure and volume are most often atmospheres and liters, respectively. However, other units can be used. In most countries other than the United States, the pascal is most commonly used for pressure. TABLE 10.2 shows the numerical value for R in various units. In working with the ideal-gas equation, you must choose the form of R in which the units agree with the units of P, V, n, and T given in the problem. In this chapter we will most often use R = 0.08206 L-atm/mol-K because pressure is most often given in atmospheres.
TABLE 10.2 • Numerical Values of the Gas Constant R in Various Units
Suppose we have 1.000 mol of an ideal gas at 1.000 atm and 0.00 °C (273.15 K). According to the ideal-gas equation, the volume of the gas is
The conditions 0 °C and 1 atm are referred to as standard temperature and pressure (STP). The volume occupied by 1 mol of ideal gas at STP, 22.41 L, is known as the molar volume of an ideal gas at STP.
GIVE IT SOME THOUGHT
How many molecules are in 22.41 L of an ideal gas at STP?
Suggest an explanation for the “ideal” nature of helium compared to the other gases.
FIGURE 10.11 Comparison of molar volumes at STP.
The ideal-gas equation accounts adequately for the properties of most gases under a variety of circumstances. The equation is not exactly correct, however, for any real gas. Thus, the measured volume for given values of P, n, and T might differ from the volume calculated from PV = nRT (FIGURE 10.11). Although real gases do not always behave ideally, their behavior differs so little from ideal behavior that we can ignore any deviations for all but the most accurate work.
SAMPLE EXERCISE 10.4 Using the Ideal-Gas Equation
Calcium carbonate, CaCO3(s), the principal compound in limestone, decomposes upon heating to CaO(s) and CO2(g). A sample of CaCO3 is decomposed, and the carbon dioxide is collected in a 250-mL flask. After decomposition is complete, the gas has a pressure of 1.3 atm at a temperature of 31 °C. How many moles of CO2 gas were generated?
Analyze We are given the volume (250 mL), pressure (1.3 atm), and temperature (31 °C) of a sample of CO2 gas and asked to calculate the number of moles of CO2 in the sample.
Plan Because we are given V, P, and T, we can solve the ideal-gas equation for the unknown quantity, n.
Solve In analyzing and solving gas law problems, it is helpful to tabulate the information given in the problems and then to convert the values to units that are consistent with those for R (0.08206 L-atm/mol-K). In this case the given values are
V = 250 mL = 0.250 L
P = 1.3 atm
T = 31 °C = (31 + 273) K = 304 K
Remember: Absolute temperature must always be used when the ideal-gas equation is solved.
We now rearrange the ideal-gas equation (Equation 10.5) to solve for n
Check Appropriate units cancel, thus ensuring that we have properly rearranged the ideal-gas equation and have converted to the correct units.
Tennis balls are usually filled with either air or N2 gas to a pressure above atmospheric pressure to increase their bounce. If a tennis ball has a volume of 144cm3 and contains 0.33 g of N2 gas, what is the pressure inside the ball at 24 °C?
Answer: 2.0 atm
STRATEGIES IN CHEMISTRY
CALCULATIONS INVOLVING MANY VARIABLES
In this chapter we encounter a variety of problems based on the ideal-gas equation, which contains four variables—P, V, n, and T—and one constant, R. Depending on the type of problem, we might need to solve for any of the four variables. To extract the necessary information from problems involving more than one variable, we suggest the following steps:
1. Tabulate information. Read the problems carefully to determine which variable is the unknown and which variables have numeric values given. Every time you encounter a numerical value, jot it down. In many cases constructing a table of the given information will be useful.
2. Convert to consistent units. Make certain that quantities are converted to the proper units. In using the ideal-gas equation, for example, we usually use the value of R that has units of L-atm/mol-K. If you are given a pressure in torr, you will need to convert it to atmospheres before using this value of R in your calculations.
3. If a single equation relates the variables, solve the equation for the unknown. For the ideal-gas equation, these algebraic rearrangements will all be used at one time or another:
4. Use dimensional analysis. Carry the units through your calculation. Using dimensional analysis enables you to check that you have solved an equation correctly. If the units in the equation cancel to give the units of the desired variable, you have probably used the equation correctly.
Sometimes you will not be given explicit values for several variables, making it look like a problem cannot be solved. In these cases, however, you will be given information that can be used to determine the needed variables. For example, suppose you are using the ideal-gas equation to calculate a pressure in a problem that gives a value for T but not for n or V. However, the problem states that “the sample contains 0.15 mol of gas per liter.” We can turn this statement into the expression
Solving the ideal-gas equation for pressure yields
Thus, we can solve the equation even though we are not given values for n and V.
As we have continuously stressed, the most important thing you can do to become proficient at solving chemistry problems is to do the practice exercises and end-of-chapter exercises. By using systematic procedures, such as those described here, you should be able to minimize difficulties in solving problems involving many variables.
Relating the Ideal-Gas Equation and the Gas Laws
The gas laws we discussed in Section 10.3 are special cases of the ideal-gas equation. For example, when n and T are held constant, the product nRT contains three constants and so must itself be a constant:
Thus, we have Boyle's law. We see that if n and T are constant, the values of P and V can change, but the product PV must remain constant.
We can use Boyle's law to determine how the volume of a gas changes when its pressure changes. For example, if a cylinder fitted with a movable piston holds 50.0 L of O2 gas at 18.5 atm and 21 °C, what volume will the gas occupy if the temperature is maintained at 21 °C while the pressure is reduced to 1.00 atm? Because the product PV is a constant when a gas is held at constant n and T, we know that
where P1 and V1 are initial values and P2 and V2 are final values. Dividing both sides of this equation by P2 gives the final volume, V2:
The answer is reasonable because a gas expands as its pressure decreases.
In a similar way, we can start with the ideal-gas equation and derive relationships between any other two variables, V and T (Charles's law), n and V (Avogadro's law), or P and T.
SAMPLE EXERCISE 10.5 Calculating the Effect of Temperature Changes on Pressure
The gas pressure in an aerosol can is 1.5 atm at 25 °C. Assuming that the gas obeys the ideal-gas equation, what is the pressure when the can is heated to 450 °C?
Analyze We are given the initial pressure (1.5 atm) and temperature (25 °C) of the gas and asked for the pressure at a higher temperature (450 °C).
Plan The volume and number of moles of gas do not change, so we must use a relationship connecting pressure and temperature. Converting temperature to the Kelvin scale and tabulating the given information, we have
Solve To determine how P and T are related, we start with the ideal-gas equation and isolate the quantities that do not change (n, V, and R) on one side and the variables (P and T) on the other side.
Because the quotient P/T is a constant, we can write
(where the subscripts 1 and 2 represent the initial and final states, respectively). Rearranging to solve for P2 and substituting the given data give
Check This answer is intuitively reasonable—increasing the temperature of a gas increases its pressure.
Comment It is evident from this example why aerosol cans carry a warning not to incinerate.
The pressure in a natural-gas tank is maintained at 2.20 atm. On a day when the temperature is –15 °C, the volume of gas in the tank is 3.25 × 103 m3. What is the volume of the same quantity of gas on a day when the temperature is 31 °C?
Answer: 3.83 × 103 m3
We are often faced with the situation in which P, V, and T all change for a fixed number of moles of gas. Because n is constant in this situation, the ideal-gas equation gives
If we represent the initial and final conditions by subscripts 1 and 2, respectively, we can write
This equation is often called the combined gas law.
SAMPLE EXERCISE 10.6 Calculating the Effect of Changing P and T on Gas Volume
An inflated balloon has a volume of 6.0 L at sea level (1.0 atm) and is allowed to ascend until the pressure is 0.45 atm. During ascent, the temperature of the gas falls from 22 °C to –21 °C. Calculate the volume of the balloon at its final altitude.
Analyze We need to determine a new volume for a gas sample when both pressure and temperature change.
Plan Let's again proceed by converting temperatures to kelvins and tabulating our information.
Because n is constant, we can use Equation 10.8.
Solve Rearranging Equation 10.8 to solve for V2 gives
Check The result appears reasonable. Notice that the calculation involves multiplying the initial volume by a ratio of pressures and a ratio of temperatures. Intuitively, we expect decreasing pressure to cause the volume to increase. Similarly, decreasing temperature should cause the volume to decrease. Because the pressure difference is more dramatic than the temperature difference, we expect the effect of the pressure change to predominate in determining the final volume, as it does.
A 0.50-mol sample of oxygen gas is confined at 0 °C and 1.0 atm in a cylinder with a movable piston. The piston compresses the gas so that the final volume is half the initial volume and the final pressure is 2.2 atm. What is the final temperature of the gas in degrees Celsius?
Answer: 27 °C