A tank of \(\mathrm{O}_{2}\) gas is stored in a cool basement \(\left(T=10.0^{\circ} \mathrm{C}\right)\). Its pressure gauge reads \(50.5 \mathrm{lb} / \mathrm{in} .^{2} .\) Diver Dan then places the tank in his truck, where the sun warms it to \(20.0^{\circ} \mathrm{C}\). What does the pressure gauge read at this elevated temperature?

The temperature-pressure relationship in gases is a fundamental concept in chemistry and physics, describing how the temperature of a gas affects its pressure. This relationship is crucial in understanding how gas behaves under different temperatures.

According to Gay-Lussac's law, a gas's pressure and temperature are directly proportional to each other when its volume is held constant. This means that if you increase the temperature of a gas, its pressure will increase if the volume doesn't change. The law is mathematically expressed as \( P_1/T_1 = P_2/T_2 \) where \( P_1 \) and \( P_2 \) represent the initial and final pressures, and \( T_1 \) and \( T_2 \) represent the initial and final temperatures in Kelvin.

In the case of our diving tank example, when the tank was warmed from 10.0°C to 20.0°C, Gay-Lussac's law allows us to predict how much the pressure would increase without actually measuring the pressure again. By maintaining a consistent unit of temperature (Kelvin in this case), we can accurately determine the pressure change.

Working with scientific data often requires converting units to perform calculations correctly, as seen in the tank exercise. Unit conversion means changing a measurement from one unit to another.

For instance, temperature needs to be converted from Celsius to Kelvin because Kelvin is the SI unit for temperature and is used in gas law equations. The formula to convert Celsius to Kelvin is \( T(K) = T(°C) + 273.15 \).

Similarly, when dealing with pressure, it's often necessary to convert units like pounds per square inch (lb/in²) to Pascals (Pa) to align with the International System of Units (SI). In our exercise, this conversion allows us to use Gay-Lussac's law, which relies on SI units. The specific conversion factor from lb/in² to Pa is about 6894.76. Conversions like these ensure that we can apply scientific laws and equations universally, regardless of the original measurement units.

Gas laws are a set of fundamental principles that predict the behavior of gases under various conditions of pressure, volume, and temperature. The laws governing these typically include Boyle's Law, Charles's Law, Gay-Lussac's Law, and the Combined Gas Law.

Boyle's Law states that the pressure of a gas is inversely proportional to its volume when temperature is constant. Charles's Law outlines how a gas expands as the temperature increases when pressure is constant. Gay-Lussac's Law, as seen in this exercise, focuses on the direct relationship between temperature and pressure. The Combined Gas Law combines the three into a single equation that helps solve problems when the state of a gas changes in more than one way.

Understanding how to apply these laws is essential for problem solving in chemistry and physics. It allows you to predict how a gas will respond to changes in temperature, pressure, and volume, which is critical in numerous scientific and industrial processes.