Question 5.F.1: A balloon is to be used to float instruments for observation......
A balloon is to be used to float instruments for observation into the upper atmosphere. At sea level, this balloon occupies a volume of 95 L at a temperature of 20°C. If helium gas is used to fill the balloon, how would you determine the maximum height that this balloon and its instruments will attain?
Strategy This is the type of question that the engineers who design apparatus such as this must answer. The law of nature here is that objects that are less dense float on those that are denser. The balloon will float upward until it reaches a height where its density is equal to that of the atmosphere. With this understanding, we can begin to define the variables we need to understand in this problem and note the relationships between those variables that would allow us to estimate an answer to such a problem.
First identify what is known about the variables in the problem. In this case we have a gas that will be at low pressures, so we can assume it will behave as an ideal gas. We note that we will release this balloon from sea level where the pressure can be estimated as 1 atm. Therefore, we can establish the initial pressure, volume, and temperature. We also know the molar mass of helium.
The key relationship is provided by the ideal gas law, which allows us to calculate density. To see this, think about n/V as the number of moles per unit volume. This is equal to the mass density divided by the molar mass. Replacing n/V with that relationship gives us a gas law in terms of density:
P=\frac{\rho R T}{M M},\mathrm{or}\ \rho=\frac{M M\cdot P}{R T}
Looking at this set of equations, the importance of temperature and pressure is apparent.
Because the key condition is the point at which the density of the helium in the balloon is equal to that of the atmosphere we must look up or determine the density of the atmosphere as a function of altitude. The source of this information might also provide the temperature and pressure of the atmosphere as a function of altitude.
Knowing T and P provides the density of helium at various altitudes via the equation above. Having looked up the density of the atmosphere as a function of altitude, we must simply find the altitude at which this calculation equals the looked up value to estimate the maximum height that can be achieved by the balloon. We must recognize, however, that the mass of the instruments must be considered if we wish an accurate answer to this type of problem.