To what temperature must a gas initially at \(20.0^{\circ} \mathrm{C}\) be heated to double the volume and triple the pressure?

Understanding gas temperature is crucial when studying how gases behave under various conditions. Temperature is a measure of the average kinetic energy of the particles in a substance. In the context of gases, as temperature increases, the particles move faster. This is important because faster-moving particles will collide with greater force and more often with the walls of their container, which generally increases the pressure if the volume is constant. Likewise, under constant pressure, increasing temperature can cause the gas to expand, hence increasing its volume.

This behavior is described by the fundamental gas laws, which include Charles's Law, stating that volume is directly proportional to temperature (when pressure is constant), and Gay-Lussac's Law, highlighting that pressure is directly proportional to temperature (when volume is constant). The exercise given effectively puts these principles to the test, asking us to determine the new temperature needed to double a gas's volume and triple its pressure.

The intricate dance between volume and pressure in gases is governed by Boyle's Law, which tells us that volume is inversely proportional to pressure when temperature is held constant. This means if we increase the pressure exerted on a gas, its volume will decrease proportionally, and vice versa.

In the textbook exercise, we're asked to consider a scenario where both volume and pressure change. To navigate this, we look to the Combined Gas Law, which integrates Charles's, Gay-Lussac's, and Boyle's laws, allowing us to predict the state of a gas when multiple variables change. By doubling the volume and tripling the pressure, the Combined Gas Law helps us mathematically relate these changes to the final temperature, assuming we know the initial state of the gas.

The Kelvin temperature scale is pivotal in scientific calculations involving temperature, especially for gas laws. It is an absolute scale with 0 Kelvin representing absolute zero - the theoretical point where particles have minimum thermal motion.

The exercise's solution hinges on the use of the Kelvin scale, which is necessary because it provides a direct proportionality between volume, pressure, and temperature. Unlike the Celsius or Fahrenheit scales, the Kelvin scale ensures that a change in temperature by one Kelvin corresponds to the same change in energy, no matter what the starting temperature is. This is why we convert Celsius to Kelvin before using the Combined Gas Law equation by adding 273.15, resulting in a starting temperature of 293.15 K for the gas initially at 20.0°C.