Question

A gas-filled weather balloon with a volume of \(65.0 \mathrm{~L}\) is released at sea-level conditions of 745 torr and \(25^{\circ} \mathrm{C}\). The balloon can expand to a maximum volume of \(835 \mathrm{~L}\). When the balloon rises to an altitude at which the temperature is \(-5^{\circ} \mathrm{C}\) and the pressure is 0.066 atm, will it have expanded to its maximum volume?

Short answer

Yes, the balloon will expand to its maximum volume.

Step-by-step solution

Convert all units to standard units

Convert the initial and final temperatures from Celsius to Kelvin. Initial temperature: \[T_1 = 25^{\circ} \mathrm{C} + 273.15 = 298.15 \mathrm{~K}\] Final temperature: \[T_2 = -5^{\circ} \mathrm{C} + 273.15 = 268.15 \mathrm{~K}\] Convert the initial pressure from torr to atm. \[P_1 = 745 \text{torr} \times \frac{1 \text{atm}}{760 \text{torr}} = 0.980 \text{atm}\]

Apply the Ideal Gas Law

Use the Ideal Gas Law to find the final volume. Given P\(_1\), V\(_1\), T\(_1\), P\(_2\), and T\(_2\), use the combined gas law: \[ \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \] Rearrange to solve for \(V_2\): \[ V_2 = \frac{P_1 V_1 T_2}{T_1 P_2} \]

Substitute known values into the equation

Plug in the values from the previous steps: \[ V_2 = \frac{(0.980 \text{atm}) (65.0 \text{L}) (268.15 \text{K})}{(298.15 \text{K}) (0.066 \text{atm})} \] Calculate the right-hand side to determine \(V_2\).

Calculate final volume

Perform the calculation: \[ V_2 \approx \frac{(0.980) (65.0) (268.15)}{(298.15) (0.066)} \approx \frac{17099.375}{19.6761} \approx 869 \mathrm{~L} \]

Compare to maximum volume

Compare the calculated final volume \(V_2\) to the balloon's maximum volume. \(869 \mathrm{~L}\) is greater than \(835 \mathrm{~L}\). Thus, the balloon will have expanded to its maximum volume.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry that relates the pressure, volume, temperature, and the amount of gas. It is represented by the equation \( PV = nRT \). Here, P stands for pressure, V for volume, n for the number of moles of gas, R is the gas constant (approximately 0.0821 L.atm/K.mol for common units), and T is the temperature in Kelvin. This law assumes that gas molecules move randomly and that their collisions are perfectly elastic without any intermolecular forces. It's very useful for simplifying and solving many practical problems involving gases.

Gas Laws in Chemistry

Understanding gas laws is crucial in chemistry because they describe how gases behave under different conditions. These include Boyle's Law (pressure-volume relationship), Charles's Law (volume-temperature relationship), and Avogadro's Law (volume-amount relationship). By combining these individual laws, we get the Combined Gas Law and Ideal Gas Law which help predict the behavior of gases when any of the properties change. Knowing these rules helps chemists and engineers in areas like weather forecasting, designing airbags, and even in scuba diving.

Unit Conversion in Chemistry

Properly converting units is essential in solving gas law problems. Temperatures must be in Kelvin, as the gas laws are formulated based on absolute temperature. Conversion from Celsius to Kelvin is done by adding 273.15 to the Celsius temperature. Pressure can be measured in various units like torr or atmospheres (atm). To convert from torr to atm, you use the relation \(1 \text{ atm} = 760 \text{ torr}\). Ensuring all units are consistent helps in avoiding errors and achieving accurate results.

Combined Gas Law

The Combined Gas Law combines Boyle's, Charles's, and Gay-Lussac’s laws into a single equation: \( \frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2} \). This equation allows us to calculate the unknown property of a gas when the other properties change. It’s particularly useful when the amount of gas remains constant while the pressure, volume, or temperature change. By rearranging the formula, you can solve for any of the variables. For instance, to find the final volume \(V_2\), rearrange it to \( V_2 = \frac{P_1 V_1 T_2}{T_1 P_2} \).

Pressure-Temperature-Volume Relationship

The relationship between pressure, temperature, and volume of a gas is pivotal in understanding gas behavior. Boyle's Law says that the volume of a gas is inversely proportional to its pressure when temperature is constant. Charles's Law indicates that volume is directly proportional to temperature at constant pressure. Gay-Lussac's Law points out that pressure is directly proportional to temperature at constant volume. The Combined Gas Law and Ideal Gas Law encapsulate these relationships, showing how all three properties interact. Practically, these principles help us understand phenomena like how balloons expand as they rise and temperatures drop.