Two flasks \(A\) and \(B\) have equal volumes. \(A\) is maintained at \(300 \mathrm{~K}\) and \(B\) at \(600 \mathrm{~K}\), while \(A\) contains \(\mathrm{H}_{2}\) gas, \(B\) has an equal mass of \(\mathrm{CO}_{2}\) gas. Find the ratio of total K.E. of gases in flask \(A\) to that of \(B\). (a) \(1: 2\) (b) \(11: 1\) (c) \(33: 2\) (d) \(55: 7\)

The Kinetic Theory of Gases is a fundamental concept that explains the behavior of gases at the molecular level. According to this theory, gases are composed of a large number of small particles that are in constant random motion. The properties of a gas—pressure, volume, and temperature—arise from the collective movements of these particles.

One of the key implications of this theory is that the temperature of a gas is directly related to the average kinetic energy of its particles. This means that if you increase the temperature of a gas, you are essentially increasing the energy with which the particles are moving around. As a result, understanding the kinetic energy of gases helps scientists predict how a gas will behave under different temperature conditions. It's like understanding how energetic a crowd of people is—if they are more energetic, they are likely to be much more lively and active, much like the particles in a warmer gas.

Moving on to Avogadro's Law, it's like a golden rule for gases: Equal volumes of different gases, at the same temperature and pressure, contain an equal number of particles, or moles. Think of moles as a measure of the number of gas particles just like a dozen is a measure of eggs. No matter what gas you have, if the conditions of temperature and pressure are kept constant, one liter of gas will contain the same number of moles as one liter of any other gas.

This law is incredibly useful when comparing different gases under identical conditions. For example, if two balloons, one filled with helium and the other with nitrogen, are of the same size and are in the same room, Avogadro's Law tells us that they both contain the same number of gas molecules, regardless of the type of gas. So, when it comes to comparing the kinetic energies of different gases in equal volumes, as long as temperature and pressure are unchanged, this law simplifies the work by eliminating the need to consider the type of gas.

Finally, let's explore the relationship between temperature and kinetic energy. Temperature is a measure of the average kinetic energy of the particles in a substance. When we say that a gas has a certain temperature, we are essentially talking about how much kinetic energy its particles have on average.

For gases, this relationship is beautifully straight-forward: double the temperature (in Kelvin), and you double the average kinetic energy of the gas particles. So, in the context of our original problem, when flask A is at 300 K and flask B at 600 K, flask B's particles have, on average, twice the kinetic energy compared to those in flask A. But since the problem also tells us that each gas has equal mass, and thanks to Avogadro's Law, that means each gas also has the same number of moles. So, the gases' different natures don't complicate things too much because the factors of molar mass cancel out.

Understanding this relationship is crucial because it tells us how a gas will behave when heated or cooled. Just like you know you'll have more energy on a warm, sunny day compared to a cold one, a gas will have particles zipping around faster when the temperature rises and moving more sluggishly when it's cooler.