A sample of a gas occupies a volume of \(25.6 \mathrm{~L}\) at \(19^{\circ} \mathrm{C}\) and 678 torr. What will be the new volume if temperature increases to \(35^{\circ} \mathrm{C}\) and pressure decreases to 595 torr?

Understanding how the volume of a gas changes under different conditions is crucial in many scientific calculations. The concept of gas volume calculation can seem daunting at first, but by using the right formulas and a systematic approach, it becomes more manageable. As highlighted in the textbook example, the volume of a gas can change with variations in both temperature and pressure.

To handle these calculations, it's imperative to grasp the use of the Combined Gas Law, which provides a direct method to ascertain the new volume of a gas when its pressure or temperature is altered. The law is expressed mathematically as: \( P1 * V1 / T1 = P2 * V2 / T2 \).

Following an organized step-by-step approach, such as the one demonstrated in the textbook solution, simplifies the process. By plugging the known initial conditions and the changed variables into the formula, and rearranging to solve for the unknown volume, students can systematically resolve the required new volume of the gas.

Accurate temperature measurement is a cornerstone of gas calculations since gas properties are highly temperature-dependent. The exercise demonstrates that converting temperature from Celsius to Kelvin is essential before using the Combined Gas Law because temperature in Kelvin maintains a proportional relationship with volume and pressure of an ideal gas.

To convert Celsius to Kelvin, one must add 273.15 to the Celsius temperature. This is because the Kelvin scale starts at absolute zero, the theoretical point where particles have minimal thermal motion, which corresponds to \( -273.15^\circ \mathrm{C} \).For example, the solution outlined starts with an original temperature of \(19^\circ \mathrm{C}\) which converts to \(292.15 \mathrm{K}\) when 273.15 is added. Such conversions are imperative as using Celsius directly in gas law calculations would provide incorrect results since the Celsius scale does not start at zero energy unlike the Kelvin scale.

The relationship between pressure, volume, and temperature of a gas is fundamental to understanding gas behavior under different conditions. This relationship is precisely articulated by the Combined Gas Law, which is derived from Boyle's Law, Charles's Law, and Gay-Lussac's Law, consolidating them into a single formula.

The law states that for a fixed amount of gas, the ratio of the product of pressure and volume to the temperature is constant. As such, if the pressure of a gas is increased, and the temperature remains constant, the volume will decrease - this example illustrates Boyle's Law. On the other hand, if the temperature rises while the pressure is kept constant, the volume will increase, which is in accordance with Charles's Law. Lastly, Gay-Lussac's Law describes that the pressure of a gas will increase as the temperature rises if the volume remains unchanged.

Through the example provided, we see that these principles are applied to calculate how the initial volume of a gas changes with a decrease in pressure and an increase in temperature. This embodies the intricate dance between pressure, volume, and temperature, which can only be accurately computed by understanding and applying the Combined Gas Law.