, which is read "is proportional to," we have
In the preceding section we examined three historically important gas laws. Each was obtained by holding two variables constant in order to see how the other variables affect each other. Using the symbol , which is read "is proportional to," we have
We can combine these relationships to make a more general gas law:
If we call the proportionality constant R, we obtain
Rearranging, we have this relationship in its more familiar form:
[10.5]
This equation is known as the ideal-gas equation. An ideal gas is a hypothetical gas whose pressure, volume, and temperature behavior is completely described by the ideal-gas equation.
The term R in the ideal-gas equation is called the gas constant. The value and units of R depend on the units of P, V, n, and T. Temperature must always be expressed as an absolute-temperature. The quantity of gas, n, is normally expressed in moles. The units chosen for pressure and volume are most often atm and liters, respectively. However, other units can be used. Table 10.2 shows the numerical value for R in various units. As we saw in the Closer Look box in Section 5.3, the product PV has the units of energy. Therefore, the units of R can include calories or joules. In working problems with the ideal-gas equation, the units of P, V, n, and T must agree with the units in the gas constant. In this chapter we will use the value R = 0.08206 L-atm/mol-K (four significant figures) or 0.0821 L-atm/mol-K (three significant figures) whenever we use the ideal-gas equation.
Suppose we have 1.000 mol of an ideal gas at 1.000 atm and 0.00°C (273.15 K). Then, from the ideal-gas equation the volume of the gas is:
The conditions 0°C and 1 atm are referred to as the standard temperature and pressure (STP). Many properties of gases are tabulated for these conditions. 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.
The ideal-gas equation does not always accurately describe real gases. For example, the measured volume, V, for given conditions of P, n, and T, might differ from the volume calculated from PV = nRT. Ordinarily, the difference between ideal and real behavior is so small that we may ignore it.
Calcium carbonate, CaCO3(s), decomposes upon heating to give CaO(s) and CO2(g). A sample of CaCO3 is decomposed, and the carbon dioxide is collected in a 250-mL flask. After the 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?
SOLUTION 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. Because we are given V, P, and T, we can solve the ideal gas equation for the unknown quantity, n.
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 (0.0821 L-atm/mol-K). In this case the given values are
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.
Notice that appropriate cancellation of units ensures that we have properly rearranged the ideal-gas equation and have converted to the correct units.
A flashbulb contains 2.4 × 10-4 mol of O2 gas at a pressure of 1.9 atm and a temperature of 19°C. What is the volume of the flashbulb in cubic centimeters? Answer: 3.0 cm3
The simple gas laws that we discussed in Section 10.3, such as Boyle's law, are special cases of the ideal-gas equation. For example, when the quantity of gas and the temperature are held constant, n and T have fixed values. Therefore, the product nRT is the product of three constants and must itself be a constant:
[10.6]
Thus, we have Boyle's law. We see that if n and T are constant, the individual 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 metal cylinder such as that in Figure 10.12 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
[10.7]
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:
Substituting the given quantities into this equation gives
The answer is reasonable because gases expand as their pressures are decreased.
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. The following sample exercise illustrates how these relationships can be derived and used.
The gas pressure in an aerosol can is 1.5 atm at 25°C. Assuming that the gas inside obeys the ideal-gas equation, what would the pressure be if the can were heated to 450°C?
SOLUTION We are given the pressure and temperature of the gas at 1.5 atm and 25°C and asked for the pressure at a higher temperature (450°C). The volume and number of moles of gas do not change. Converting temperature to the Kelvin scale and tabulating the given information, we have
In order to determine how P and T are related, we start with the ideal-gas equation and isolate the quantities that don't 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 gives
This answer is intuitively reasonable--increasing the temperature of a gas increases its pressure. It is evident from this example why aerosol cans carry a warning not to incinerate.
A large natural-gas storage tank is arranged so that the pressure is maintained at 2.20 atm. On a cold day in December when the temperature is -15°C (4°F), the volume of gas in the tank is 28,500 ft3. What is the volume of the same quantity of gas on a warm July day when the temperature is 31°C (88°F)? Answer: 33,600 ft3
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 under these circumstances, the ideal-gas equation gives
If we represent the initial and final conditions of pressure, temperature, and volume by subscripts 1 and 2, respectively, we can write:
[10.8]
An inflated balloon has a volume of 6.0 L at sea level (1.0 atm) and is allowed to ascend in altitude 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.
SOLUTION Let's again proceed by converting temperature to the Kelvin scale and tabulating the given information.
Because n is constant, we can use Equation 10.8. Rearranging to solve for V2 gives