A balloon is filled to a volume of \(7.00 \times 10^{2} \mathrm{~mL}\) at a temperature of \(20.0^{\circ} \mathrm{C}\). The balloon is then cooled at constant pressure to a temperature of \(1.00 \times 10^{2} \mathrm{~K}\). What is the final volume of the balloon?

Understanding how gases behave under different conditions is essential for a wide range of scientific applications. The gas laws are a set of rules that predict how a gas will change when pressure, volume, and temperature are varied. One of the fundamental gas laws is Charles's Law, which states that the volume of a gas is directly proportional to its temperature when pressure is held constant.

This means that if you increase the temperature of a gas, its volume will increase as well, provided the amount of gas and the pressure remain constant. Similarly, if the temperature decreases, the volume of the gas will also decrease. Charles's Law is mathematically expressed as \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \), where \(V_1\) and \(V_2\) are the initial and final volumes, and \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin, respectively.

Understanding Charles's Law is not only crucial in scientific research but also has practical applications, such as explaining how hot air balloons rise and why car tires can burst in the heat if overinflated.

Working with gas laws often requires converting temperatures from one scale to another. In many scientific calculations, including those involving Charles's Law, temperature must be measured in Kelvin, which is the SI unit for thermodynamic temperature. There's a simple relationship between the Celsius and Kelvin scales:

\( K = ^{\circ}C + 273.15 \)

What's crucial to remember is that Kelvin is an absolute scale, meaning it starts at absolute zero, which is the theoretical temperature at which all kinetic motion of particles stops. This conversion is essential because using degrees Celsius could result in negative volume values in calculations, which are not physically meaningful for gases. For Fahrenheit to Kelvin, you would first convert Fahrenheit to Celsius and then to Kelvin using the following steps:

\( ^{\circ}C = (^{\circ}F - 32) \times \frac{5}{9} \)
\( K = ^{\circ}C + 273.15 \)

Accurate temperature conversion is key to correctly applying the gas laws and obtaining meaningful results in your experiments or calculations.

Charles's Law illustrates the volume-temperature relationship for gases, clarifying that this relationship is linear when plotted on a graph with volume on the y-axis and temperature on the x-axis. When visualizing this, imagine that as a balloon heats up, the gas particles inside move faster and take up more space, causing the balloon to expand and hence, increase its volume.

This linear correlation implies that the temperatures must be expressed in absolute terms, which, as mentioned earlier, uses the Kelvin scale where zero reflects the complete absence of thermal energy. To fully understand the implications of this relationship, one can perform thought experiments or real-life experiments, such as heating a fixed amount of gas in a sealed container and observing the volume increase.

The practical aspect of this knowledge is significant. For instance, engineers must consider these laws when designing engines and storage containers for gases. At the same time, weather forecasters may use them to predict changes in weather patterns caused by the expansion and contraction of gases in the atmosphere.