Applications of the Ideal Gas Equation

6-4 Applications of the Ideal Gas Equation 209 Applying the General Gas Equation 

Identify the quantities in the general gas equation that remain constant. Cancel out these quantities and solve 

We can base our check on a qualitative, intuitive understanding of what happens when a gas is heated in a closed container. Its pressure increases (possibly to the extent that the container bursts). If, by error, we had used the ratio of temperatures 273 K/373 K, the final pressure would have been less than 1.00 bar— an impossible result. 

PRACTICE EXAMPLE A A 1.00 mL sample of N2(g) at 36.2 °C and 2.14 atm is heated to 37.8 °C, and the pressure changed to 1.02 atm. What volume does the gas occupy at this final temperature and pressure  

Although the ideal gas equation can always be used as it was presented in equation (6.11), it is useful to recast it into slightly different forms for some applications. We will consider two such applications in this section determination of molar masses and gas densities. 

Using some simple algebraic manipulation, we can solve for variables other than those that appear explicitly in the ideal gas equation. For example, if we know the molar mass of a gas (g/mol), we can determine its density at a given temperature and pressure. Recall from Section 11.1 that the density of a gas is generally expressed in units of g/L. We can rearrange the ideal gas equation to solve for mol/L  

If we then multiply both sides by the molar mass. M, we get 

Another way to arrive at Equation 11.7 is to substitute m/M for n in the ideal gas equation and rearrange to solve for m/V (density)  

Conversely, if we know the density of a gas, we can determine its molar mass  

Similarly, the molar mass of a volatile liquid can be determined by placing a small volume of it in the bottom of a flask, the mass and volume of which are known. The flask is then immersed in a hot-water bath, causing the volatile liquid to completely evaporate and its vapor to fill the flask. Because the flask is open, some of the excess vapor escapes. When no more vapor escapes, the flask is capped and removed from the water bath. The flask is then weighed to determine the mass of the vapor. (At this point, some or all of the vapor has condensed but the mass remains the same.) The density of the vapor is determined by dividing the mass of the vapor by the volume of the flask. Equation 11.8 is then used to calculate the molar mass of the volatile liquid. 

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We see again that Henry s law is a limiting law valid when the molar volume of the gas is large with respect to the partial molar volume of the solute. When the pressure is used as the dependent variable, the applicability of the ideal gas equation of state is also required. 

Consider some examples of the application of the ideal gas equation. 

The Ideal-gas equation, PV = iiRT, is the equation of state for an Ideal gas. The term R in this equation is the gas constant We can use the ideal-gas equation to calculate variations in one variable when one or more of the others are changed. Most gases at pressures less than 10 atm and temperatures near 273 K and above obey the ideal-gas equation reasonably well The conditions of 273 K (0 °C) and I atm are known as the standard temperature and pressure (STP). In all applications of the ideal-gas equation we must remember to convert temperatures to the absolute-temperature scale (the Kelvin scale). 

To extract a VMR from Equation 4.1, it is necessary to include the total number density, Ni, of the gas in the drift tube, which can be determined from the pressure and the temperature. For example, if the drift tube pressure is 2 mbar and the temperature is 40°C, then application of the ideal gas equation gives Ni = pNJRT = 4.69 x 10 m = 4.69 X 10 cm to three significant figures. The VMR for substance M is obtained from the following adaptation of Equation 4.1 ... 

This is a relatively straightforward application of the ideal gas equation. We are given an amount of gas (in grams), a pressure (in kPa), and a temperature (in °C). Before using the ideal gas equation, we must express the amount in moles and the temperature in Kelvin. Include units throughout the calculation to ensure that the final result has acceptable units. 

We found the solution to this problem through the application of the ideal gas equation and our understanding of vapor pressure. Note that in the first two steps, we considered the two extremes, with the first being just liquid water and the second all vapor. 

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