Aerosol cans carry clear warnings against incineration because of the high pressures that can develop upon heating. Suppose that a can contains a residual amount of gas at a pressure of \(755 \mathrm{mmHg}\) and a temperature of \(25^{\circ} \mathrm{C}\). What would the pressure be if the can were heated to \(1155^{\circ} \mathrm{C}\) ?

Understanding the relationship between gas pressure and temperature is fundamental to studying gases and their behaviors under various conditions. According to the ideal gas law, this relationship is direct; as temperature increases, if the volume remains constant, so does the pressure, and vice versa.

This concept is crucial when working with gases contained in rigid containers, such as aerosol cans. When you increase the temperature of a gas within a fixed volume, its molecules gain kinetic energy, moving faster and colliding with the container's walls more aggressively. This increased molecular activity results in a rise in pressure. Mathematically, this is expressed in the portion of the ideal gas law that equates the ratios of pressure and temperature of a gas before and after a change, under constant volume conditions: \(\frac{P1}{T1} = \frac{P2}{T2}\), where P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final values.

To solve problems like the one posed in the exercise, this formula allows us to predict what happens to the pressure of a gas when it is heated. It's important to work in Kelvin for these calculations because this scale starts at absolute zero, ensuring that our ratios make sense—an essential aspect when calculating changes in properties that rely on an absolute scale.

When assessing the behavior of gases under changing temperatures, it is essential to use the Kelvin scale. This absolute temperature scale is crucial in scientific calculations, particularly those involving thermal dynamics and gas laws.

Converting Celsius to Kelvin is a simple yet pivotal step in many thermodynamics problems. To convert degrees Celsius to Kelvin, add 273.15 to the Celsius temperature. This adjustment aligns the freezing point of water, 0 degrees Celsius, with 273.15 K on the Kelvin scale, and importantly, sets absolute zero at 0 K — the point where molecular motion theoretically stops.

Formula for Converting Celsius to Kelvin:

\(K = ^\text{o}C + 273.15\), where K represents the temperature in Kelvin and \(^\text{o}C\) is the temperature in degrees Celsius.

For instance, in the aerosol can example, the initial temperature of 25 degrees Celsius is converted to Kelvin by calculating 25 + 273.15 to obtain 298.15 K. This step is fundamental as working in Kelvin allows for correct utilization of the gas laws, which have been derived based on an absolute scale.

The warnings against incinerating aerosol cans are based on the principles of gas dynamics and the potential for hazardous consequences. As seen from the gas pressure-temperature relationship, heating an aerosol can result in a drastic increase in pressure, as the temperature of the gas inside the can substantially rises.

Considering the can as a rigid container, this means that the internal pressure can increase to the point where the can's structural integrity fails, leading to an explosion. This is dangerous not only because of the flying debris and loud noise but also due to potential harm from the can's contents, which could be flammable, toxic, or otherwise harmful.

Therefore, the hazards involved with heating aerosol cans to high temperatures, such as in the situation of incineration, highlight the importance of understanding and respecting the chemical principles at play. The risk of explosion underscores the necessity for strict adherence to safety guidelines regarding the use and disposal of aerosol products.