Question

You have a sealed, flexible balloon filled with argon gas. The atmospheric pressure is \(1.00\) 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. b. Above what temperature would you heat the balloon so that it would float?

Short answer

A balloon filled with argon gas would float when heated because the buoyant force increases as the volume of the balloon expands due to the increase in internal temperature and pressure. To make the balloon float, the density of argon inside the balloon must be less than the density of the air outside. Using the Ideal Gas Law and the given molecular masses of argon, nitrogen, and oxygen, we find that the critical temperature for the balloon to float is above 298.48 K (or 25.33 °C).

Step-by-step solution

Understand Buoyancy

A balloon floats when the buoyant force it experiences is equal to or greater than its weight due to gravity. Buoyancy occurs because the pressure of a fluid (in our case, air) is greater at the bottom surface of an object than at the top surface. Mathematically, buoyant force = weight of the displaced air.

Check for Factors in Balloon Floating

The buoyant force is dependent on the volume of the displaced air and the difference in density between the air and the gas inside the balloon (argon in our case). The weight of the inflated balloon depends on the mass and the weight of argon gas inside it. When the buoyant force is equal to or greater than the weight of the inflated balloon, it will float.

Investigate How Heating Affects the Balloon

When heating the balloon, its volume and the number of moles of gas trapped inside remain constant. However, as the internal balloon's temperature increases, the internal pressure would also increase. As a result, the volume of the balloon would expand. Therefore, the balloon would displace a larger volume of air, increasing the buoyant force acting on it. Eventually, it will reach a point where the buoyant force is greater than or equal to the balloon's weight, causing it to float.

Relate buoyancy and the Ideal Gas Law

To find the temperature needed to make the balloon float, we need to compare the densities of argon (inside the balloon) and air (outside the balloon). Let's assume an initial situation when both are at same temperature: Using the Ideal Gas Law (PV = nRT): Density (inside the balloon) = \(\dfrac{mass_{argon}}{Volume_{balloon}} = \dfrac{n_{argon}M_{argon}}{n_{argon}RT/P_{atm}} = \dfrac{M_{argon}}{RT/P_{atm}}\) Density (outside the balloon) = \(\dfrac{0.790 * M_{N_2} + 0.210 * M_{O_2}}{RT/P_{atm}}\), where M_N2 and M_O2 are molecular masses of nitrogen and oxygen respectively.

Determine the Critical Temperature

To make the balloon float, we require Density (inside the balloon) \< Density (outside the balloon). We can now solve for the critical temperature, T: \(\dfrac{M_{argon}}{RT/P_{atm}} < \dfrac{0.790 * M_{N_2} + 0.210 * M_{O_2}}{RT/P_{atm}}\) \(M_{argon} < 0.790 * M_{N_2} + 0.210 * M_{O_2}\) T (in Kelvin) > \(\dfrac{M_{argon} * R * P_{atm}}{0.790 * M_{N_2} + 0.210 * M_{O_2}}\) Given the molecular masses of argon (39.95 g/mol), nitrogen (28.01 g/mol), and oxygen (32.00 g/mol), the atmospheric pressure of 1 atm, and the gas constant R (0.0821 L atm/mol K), we find: T (in Kelvin) > \(\dfrac{39.95 * 0.0821 * 1}{(0.790 * 28.01) + (0.210 * 32.00)} = 298.48 K\) Thus, the balloon would float if we heat it to a temperature above 298.48 K (or 25.33 °C).

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is crucial for grasping the principles behind the behavior of gases under various conditions. This law relates the pressure (P), volume (V), number of moles (n), temperature (T), and the ideal gas constant (R) through the equation: \[PV = nRT\]. The Ideal Gas Law allows us to predict how a gas will respond to changes in temperature, volume, or pressure, assuming that the gas behaves ideally.

In the context of our exercise with the argon-filled balloon, the law helps us analyze how heating the gas impacts its density and the corresponding buoyant force. For instance, when the temperature is increased while keeping the number of moles of argon and the pressure constant (as in the sealed balloon), the volume must increase according to the Ideal Gas Law. As the balloon heats up and expands, its density decreases because the same amount of mass occupies a larger volume, which is a central concept for predicting the balloon's capacity to float.

Buoyant Force

The buoyant force is the upward force exerted on an object by a fluid, such as a gas or liquid, that counteracts the object's weight. This force is what allows objects to float or rise in a fluid. According to Archimedes' principle, the buoyant force on an object in a fluid is equal to the weight of the fluid displaced by the object.

In our exercise, the buoyant force comes into play with the argon-filled balloon being submerged in atmospheric air. To determine whether the balloon will float, we compare the buoyant force to the weight of the balloon. If the buoyant force is equal to or greater than the weight, the balloon will float. As explained in the solution, when the balloon is heated, it will expand and displace more air, which increases the buoyant force acting on it. This occurs until the force is sufficient for the balloon to take off and rise.

Temperature and Gas Density

The relationship between temperature and gas density is inversely proportional when dealing with an ideal gas under constant pressure. As temperature increases, gas molecules move more rapidly and tend to occupy a greater volume. This spread of molecules results in a decreased density; fewer gas particles are present in any given volume.

For our heated balloon, this concept explains why the density of the argon gas inside decreases as the temperature rises. Initially, both the argon inside the balloon and the ambient air are assumed to be at the same temperature. However, once we start to heat the balloon, the argon's density will reduce, and thus the overall weight of the balloon relative to the air it displaces will decrease, leading to buoyancy. The exercise walkthrough identifies a critical temperature above which the density of the heated argon will become less than that of the surrounding air, allowing the balloon to float.