In order for a gas-filled balloon to rise in air, the...

In order for a gas-filled balloon to rise in air, the density of the gas in the balloon must be less than that of air. (a) Consider air to have a molar mass of $28.96 \mathrm{g} / \mathrm{mol}$; determine the density of air at $25^{\circ} \mathrm{C}$ and $1.00 \mathrm{atm}$ in $\mathrm{g} / \mathrm{L}$. (b) Show by calculation that a balloon filled with carbon dioxide at $25^{\circ} \mathrm{C}$ and 1 atm could not be expected to rise in air at $25^{\circ} \mathrm{C}$.

In order for a gas-filled balloon to rise in air, the density of the gas in the balloon must be less than that of air.
(a) Consider air to have a molar mass of $28.96 \mathrm{g} / \mathrm{mol}$; determine the density of air at $25^{\circ} \mathrm{C}$ and $1.00 \mathrm{atm}$ in $\mathrm{g} / \mathrm{L}$.
(b) Show by calculation that a balloon filled with carbon dioxide at $25^{\circ} \mathrm{C}$ and 1 atm could not be expected to rise in air at $25^{\circ} \mathrm{C}$.

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental equation (PV = nRT) that relates the pressure, volume, temperature, and number of moles of a gas. It is essential for calculating gas density by rearranging it to yield density as a function of pressure, molar mass, the gas constant, and temperature. This concept underpins the analysis of how gases behave under different conditions.

Molar Mass

Molar mass is the mass of one mole of a substance and is a critical factor in determining the density of a gas when used in the modified ideal gas law equation. In gas density calculations, the molar mass connects the microscopic properties of a molecule with the macroscopic property of density, making it indispensable for comparing different gases.

Gas Density Calculations

Calculating the density of a gas involves using the ideal gas law to derive the expression for density, typically written as (pressure multiplied by molar mass divided by the product of the gas constant and temperature). This approach allows for the comparison of densities of different gases under the same conditions, which is crucial when determining whether one gas is lighter than another.

Buoyancy and Lift Principles

For an object such as a gas-filled balloon to rise in air, the density of the gas inside must be lower than that of the surrounding air. This concept arises from the principle of buoyancy, where a net upward force exists if the weight of the displaced fluid exceeds the weight of the object, making it a key consideration in determining the conditions required for lift.

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