An ideal gas is contained in a cylinder with a volume of \(5.0 \times\) \(10^{2} \mathrm{~mL}\) at a temperature of \(30 .^{\circ} \mathrm{C}\) and a pressure of 710 . torr. The gas is then compressed to a volume of \(25 \mathrm{~mL}\), and the temperature is raised to \(820 .{ }^{\circ} \mathrm{C}\). What is the new pressure of the gas?

Understanding how to calculate gas pressure is fundamental when dealing with problems related to the Ideal Gas Law. The pressure of a gas is a measure of the force that the gas molecules exert on the walls of their container. When working with the Ideal Gas Law, pressure can be represented using different units, such as atmospheres (atm), torr, or pascals (Pa). In the exercise, converting the initial pressure from torr to atm was necessary to use the Ideal Gas Law correctly, as the law involves consistent units.

When compressing a gas or changing its temperature, like in our scenario, the pressure will change according to the Ideal Gas Law, given by the formula:
\[ PV = nRT \]
Where P stands for pressure, V for volume, n for the number of moles, R for the gas constant, and T for temperature in Kelvin. By keeping the amount of gas and the gas constant fixed, the formula simplifies, helping us find the new pressure after changes in volume and temperature.

Temperature plays a crucial role in calculations involving gases. However, it's important to use the correct scale—Kelvin (K)—when applying the Ideal Gas Law. The Kelvin scale is an absolute temperature scale, meaning it starts at absolute zero (the theoretical point where particles have minimal thermal motion).

To convert Celsius to Kelvin, which was necessary in our exercise, you add 273.15 to the Celsius temperature:
\[ T(K) = T(^\textcirc C) + 273.15 \]
So, to convert the initial temperature from 30°C and the final temperature from 820°C to Kelvin, we simply added 273.15 to each, resulting in 303.15 K and 1093.15 K, respectively. Accurate conversion is key to solving chemical problems and especially essential when temperatures are extreme, such as in the given problem.

Chemical problem solving often involves a systematic approach to applying chemical laws and principles. The Ideal Gas Law is frequently used to solve problems about gases, like in the given textbook exercise where a gas undergoes compression and a temperature change.

By breaking the problem into smaller, manageable steps, we can systematically find a solution. The steps usually involve identifying the known quantities, converting those quantities to the proper units, applying the relevant chemical laws or formulas, and finally, performing the calculations to obtain the result.

In our exercise, identifying the initial and final states of the gas and then using the Ideal Gas Law allowed us to create a relationship between these states to find the unknown final pressure, demonstrating a practical method of chemical problem solving.