Question

You have a sealed, flexible balloon filled with argon gas. The atmospheric pressure is \(1.00 \mathrm{~atm}\) and the temperature is \(25^{\circ} \mathrm{C}\). Assume that air has a mole fraction of nitrogen of \(0.790\), the rest being oxygen. a. Explain why the balloon would float when heated. Make sure to discuss which factors change and which remain constant, and why this matters. Be complete. b. Above what temperature would you heat the balloon so that it would float?

Short answer

When the sealed and flexible balloon filled with argon gas is heated, its temperature increases, causing the gas volume to increase and density to decrease. If the balloon's density becomes lower than the air's density, it experiences an upward buoyancy force and starts to float. Heating the argon gas above 325.82 K (52.82 °C) ensures the balloon's density becomes lower than air, thus causing it to float.

Step-by-step solution

a. Explanation of floating when heated

Since the balloon is sealed and flexible, it means that the amount of argon gas (n) inside the balloon and the pressure (P) will stay constant, while the volume (V) and temperature (T) can change. When the balloon is heated, its temperature increases, which leads to an increase in the gas' volume based on the ideal gas law, \(PV = nRT\), where R is the ideal gas constant. As the gas expands, the average density of the gas in the balloon decreases. Now to make the balloon float, the buoyancy force acting on it should be greater than its weight. The buoyancy force is determined by the weight of the displaced air. When the balloon's density becomes lower than that of the surrounding air, it experiences an upward force (the buoyancy force) and starts to float. As the temperature increases, the density of the gas in the balloon decreases by the increase in volume, while the density of the surrounding air remains almost constant. This decrease in density eventually causes the balloon to float. So, when we heat the balloon, it expands, the density decreases, and it starts to float when the buoyancy force is greater than its weight.

b. Determining the floating temperature

To find the temperature above which the balloon would float, we need to determine the temperature at which the average density of the argon gas inside the balloon becomes less than the average density of the surrounding air. The density of the air can be calculated by using the average molecular weight of air and the ideal gas law: \[ρ_{air} = \frac{P M_{air}}{RT} \] Since 79.0% of air is nitrogen and the rest is oxygen, we can find the average molecular weight of air: \[M_{air} = 0.790 \times M_{N_2} + 0.210 \times M_{O_2} = 0.790 \times 28.02 + 0.210 \times 32.00 = 28.85 \,g/mol\] We can calculate the density of the argon gas in the balloon at 25°C using the ideal gas law: \[ρ_{Ar_{25°C}} = \frac{P M_{Ar}}{R T_{25°C}} = \frac{(1.00 \, atm)(39.95 \,g/mol)}{(0.0821 \, L atm / mol K)(298 K)} = 1.641 \, g/L \] Now we need to find the temperature at which the density of argon gas becomes equal to the density of air. At this point, the floating begins. So, we can equate the densities: \[ρ_{air} = ρ_{Ar_{T}} \] \[\frac{1.00 \, atm \cdot 28.85 \, g/mol}{0.0821 \, L \, atm/mol K \cdot T} = \frac{1.00 \, atm \cdot 39.95 \, g/mol}{0.0821 \, L \, atm/mol K \times 298 \, K} \Rightarrow T = \frac{(39.95 \, g/mol)(298 \, K)}{28.85 \, g/mol} = 325.82 \, K\] In order to make the balloon float, we should heat the argon gas above this temperature: \[T_{float} > 325.82 K\] Therefore, the balloon will start to float when it is heated above 325.82 K (52.82 °C).

Key concepts

Argon Gas Properties

Argon is a noble gas, which means it is part of the group of inert gases that show complete non-reactivity due to their filled valence electron shell. It holds the atomic number 18 and is colorless, odorless, and tasteless in both its liquid and gaseous forms. This stability makes argon extremely useful in applications where materials should not react with the surrounding atmosphere, such as in welding.

When examining how argon behaves under different conditions, we apply the same principles that govern all ideal gases. For instance, argon's density and volume will change in response to alterations in temperature, assuming that the pressure and amount of gas remain constant. This property enables us to manipulate argon to achieve certain conditions, such as making it less dense than air to enable a balloon filled with argon to float - a direct application of buoyancy in gases.

Moreover, argon has a molecular weight of about 39.95 g/mol, which is pertinent when calculating its density using the ideal gas law. Despite being heavier than air, argon can still be made to float within air by adjusting its temperature, which inversely affects its density.

Ideal Gas Law

The cornerstone of understanding gas behavior is the Ideal Gas Law, encapsulated in the equation \(PV = nRT\), where P represents pressure, V volume, n the number of moles, R the ideal gas constant, and T the absolute temperature in Kelvin. This equation succinctly relates the physical properties of an ideal gas in a closed system. Although no gas is perfectly ideal, many gases under standard conditions of temperature and pressure behave similarly enough to an ideal gas that the law provides accurate predictions.

According to the law, if a gas, such as argon, is heated (increasing T), and if pressure and the amount of gas (n) remain constant, the volume (V) must increase. For buoyancy applications, heating a gas and causing it to expand reduces its density, which can change its behavior in an environment, like making it possible for a balloon filled with the gas to float. Understanding the ideal gas law is critical for predicting these outcomes and solving practical problems involving gases in our everyday world.

Molecular Weight of Air

Air is not a single gas but a mixture of various gases, primarily nitrogen (N₂) and oxygen (O₂), with several other trace gases. The molecular weight of air is a weighted average based on the percentage composition of these constituent gases. Typically, air is about 78% nitrogen and 21% oxygen, with the remaining 1% composed of other gases. The molecular weight is calculated using the formula \(M_{air} = M_{N₂} \times fraction_{N₂} + M_{O₂} \times fraction_{O₂}\).

For our example here, using 79% nitrogen and 21% oxygen, we find a calculated molecular weight for air of 28.85 g/mol. The precise value is crucial for calculations involving the density of air, particularly when using the ideal gas law to compare the density of air with that of other gases, such as argon. Understanding the molecular weight of air helps us grasp how changes in temperature, pressure, or volume of a gas will alter its buoyancy in relation to the surrounding atmosphere, a key concept for applications in meteorology, aviation, and various scientific research fields.