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The concept of kinetic energy, which represents energy due to motion, plays a crucial role in the behavior of gases, especially at the molecular level. For a nitrogen molecule (N_2) inside a balloon, we are particularly interested in its translational kinetic energy. This form of kinetic energy is associated with the linear motion of molecules. The average translational kinetic energy of a molecule at a given temperature can be found using the formula \(\text{KE}_{\text{avg}} = \frac{3}{2}kT\), where \(k\) denotes the Boltzmann constant (1.38 \times 10^{-23} J/K) and \(T\) is the temperature in Kelvin. For example, when the temperature is 22^{\circ} \text{C}, converting to Kelvin gives 295.15 K. By substituting this into the energy formula, we get each nitrogen molecule's average translational kinetic energy as \(6.08 \times 10^{-21}\text{ J}\). This small value reflects the high-speed movement of very tiny molecules within the balloon, showing how temperature influences molecular motion.

Molar mass is an essential property in chemistry that helps us to relate the mass of a substance to the amount of matter it contains, using moles as a measuring unit. For gases like nitrogen (N_2), molar mass is the mass of one mole of its molecules. In the exercise, N_2 has a molar mass of 28.0 \, g/mol. From this, the mass of a single N_2 molecule can be determined using Avogadro's number (6.022 \times 10^{23}\text{ molecules/mol}), since it tells us the number of molecules in one mole. The calculation follows as \(\frac{28.0 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{molecules/mol}}\), resulting in approximately \(4.65 \times 10^{-23} \, \text{g}\). Converting to kilograms, it becomes \(4.65 \times 10^{-26} \, \text{kg}\), which is quite minuscule, reflecting the small but significant mass of individual molecules.

Understanding Avogadro's number is crucial when dealing with quantities of matter at the molecular scale. Named after the scientist Amedeo Avogadro, this constant (6.022 \times 10^{23} \text{ molecules/mol}) represents the number of particles in exactly one mole of a substance. It enables chemists to convert between the amount of substance (in moles) and the number of units each contains, such as atoms or molecules.In our exercise, after determining the number of moles (3.52) of nitrogen gas using the ideal gas law, Avogadro's number allows us to calculate the number of N_2 molecules inside the balloon. Multiplying the moles by Avogadro's number gives approximately \(2.12 \times 10^{24} \text{ molecules}\). This conversion demonstrates Avogadro's number's essential role in bridging the macroscopic and molecular worlds.

Translational motion pertains to the movement of molecules through space, contributing significantly to the macroscopic properties of gases, such as pressure and temperature. For molecules inside a balloon, such as nitrogen (N_2) molecules, they constantly move in collaboration, colliding with each other and the balloon's walls. This motion leads to the phenomenon we recognize as gas pressure.In this exercise, understanding the translational motion of each nitrogen molecule helped to determine the average and total translational kinetic energy. Total energy can be found by recognizing that each molecule carries a kinetic energy component. By multiplying the number of molecules (2.12 \times 10^{24}) by the average translational kinetic energy (6.08 \times 10^{-21} \text{ J}), it results in a total of \(1.28 \times 10^{4} \text{ J}\). This showcases the energy held by all molecules in the balloon, juxtaposing their minute individual energies with the total effect in aggregate.