A helium balloon occupies \(8.0 \mathrm{L}\) at \(20^{\circ} \mathrm{C}\) and 1.0 -atm pressure. The balloon rises to an altitude where the air pressure is 0.65 atm and the temperature is \(-10^{\circ} \mathrm{C} .\) What is its volume when it reaches equilibrium at the new altitude?

Understanding gas laws is fundamental in predicting how gases will behave under various conditions. These laws help us grasp concepts such as pressure, volume, and temperature, and how they are interrelated when it comes to gases.

The most basic gas laws were determined by scientists such as Boyle, Charles, and Gay-Lussac, who each discovered a piece of the puzzle. Boyle's law states that pressure and volume are inversely proportional at a constant temperature, while Charles's law says that volume and temperature are directly proportional at a constant pressure. Gay-Lussac's law adds that pressure and temperature are directly proportional at a constant volume.

However, the Combined Gas Law combines these individual laws into one equation that covers all pressure-volume-temperature relationship scenarios, where none of the three variables remain constant. This law is represented as \( P1 \times V1 / T1 = P2 \times V2 / T2 \), where P stands for pressure, V for volume, T for temperature, and the subscripts 1 and 2 represent the initial and final states of the gas, respectively. This equation is particularly useful for problems like our example with the helium balloon, where all three conditions are changing simultaneously.

The pressure-volume-temperature relationship is a key concept in understanding how gases respond to changes in their environment. This relationship is illustrated by the aforementioned gas laws and encompasses how gas volume may increase with temperature if pressure remains constant (Charles's Law), or how gas volume might decrease with increasing pressure if temperature is constant (Boyle's Law), and so on.

For any given sample of gas, if the pressure is increased, the volume tends to decrease, unless the temperature is also increased to compensate. Similarly, if a gas is heated up, its volume tends to increase unless the gas is compressed to counteract the expansion.

In practical situations like the behavior of a helium balloon ascending to higher altitudes, we can predict its volume change by knowing the relationship between pressure, volume, and temperature. Importantly, it is assumed the gas behaves ideally, meaning it follows the gas laws precisely without any deviation due to intermolecular forces or the volume occupied by the gas molecules themselves.

The Kelvin scale is crucial for scientific work, particularly when dealing with gas laws because it avoids negative temperature value issues and provides a true representation of the kinetic energy of particles. In Kelvin, the lowest possible temperature (absolute zero) is where all molecular movement stops, set at 0 K.

To convert Celsius to Kelvin, which is necessary for using the gas laws correctly, you add 273.15 to the Celsius temperature. Thus, \( 0^{\text{\(\text{o}\)}}C \rightarrow 273.15 K \) and so on for any other temperature. For the helium balloon problem, the conversion is straightforward: \( 20^{\text{\(\text{o}\)}}C \rightarrow 293.15 K \) and \( -10^{\text{\(\text{o}\)}}C \rightarrow 263.15 K \), representing the initial and final temperatures respectively.

This conversion ensures that the gas law equations are dimensionally consistent and the relationships between variables are accurately represented since they are all expressed in absolute terms, allowing for proper calculation and comparison.