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In chemistry, the mole concept is a fundamental idea that helps us understand the amount of substance. When we talk about a mole, we're essentially talking about Avogadro's number, which is approximately \(6.022 \times 10^{23}\) particles. This could be atoms, molecules, ions, or electrons. In the exercise, we start with 1 mole of hydrogen gas (\(\mathrm{H}_{2}\) molecules), meaning there are \(6.022 \times 10^{23}\) hydrogen molecules in our sample.
This understanding is crucial because it allows us to calculate how many individual entities we have in a given mass. In the problem, when the hydrogen gas gets converted from molecules to atoms, our 1 mole of hydrogen molecules turns into 2 moles of hydrogen atoms as each hydrogen molecule has 2 atoms.
Hence, the mole concept helps us track the number of particles before and after the chemical change, ensuring that we can apply the Ideal Gas Law accurately.

Hydrogen gas is the simplest and most abundant element in the universe. It exists naturally as diatomic molecules (\(\mathrm{H}_{2}\)), meaning each molecule is made of two hydrogen atoms bonded together. Because of this, hydrogen gas is quite different from single hydrogen atoms in terms of behavior and properties.
When we apply heat to hydrogen gas, it can cause the molecules to dissociate into individual hydrogen atoms. In the context of the Ideal Gas Law, this change in molecular structure affects the pressure of the gas because, now, we have twice the number of gaseous particles in the same volume.
Hydrogen's light mass and reactive nature make it important not only in chemical reactions but also in calculations like those involving the Ideal Gas Law. Understanding how hydrogen gas behaves under different states, such as temperature changes, is essential for accurate predictions and outcomes when using gas laws.

Temperature is a measure of the average kinetic energy of particles. When the temperature of a gas increases, the kinetic energy of its molecules also increases, thereby causing the molecules to move more energetically. This is exactly what happens when the hydrogen gas in the problem is heated from 300 K to 3000 K.
A significant temperature increase not only makes the particles move faster but can also alter the physical state if enough energy is provided, as seen with hydrogen molecules dissociating into atoms. According to the Ideal Gas Law, \(PV = nRT\), if temperature (T) increases, pressure (P) will also increase if the volume (V) and number of moles (n) remain constant.
In our scenario, both the number of moles and the temperature dramatically increase, doubling the number of moles and increasing the temperature tenfold, which leads to a pressure increase proportional to these changes. Understanding this relationship is vital for predicting how a gas will react to energy changes, particularly in controlled environments.