Question

Consider three identical flasks with differ- ent gases: Flask A: \(\mathrm{CO}\) at 760 torr and \(273 \mathrm{~K}\) Flask \(\mathrm{B}: \mathrm{N}_{2}\) at 250 torr and \(273 \mathrm{~K}\) Flask \(\mathrm{C}: \mathrm{H}_{2}\) at 100 torr and \(273 \mathrm{~K}\) In which flask will the molecules have the greatest average kinetic energy per mole? (a) \(\overline{\mathrm{A}}\) (b) \(\underline{B}\) (c) \(\mathrm{C}\) (d) same in all

Short answer

The molecules in all the flasks will have the same average kinetic energy per mole since they are all at the same temperature of 273 K, so the correct answer is (d) the same in all.

Step-by-step solution

Understand the Kinetic Molecular Theory

The Kinetic Molecular Theory of Gases states that the average kinetic energy of a gas molecule depends only on the temperature of the gas, not on the type of molecule or the pressure. This can be represented by the formula: \( \text{KE}_{\text{avg}} = \frac{3}{2} k T \), where \( k \) is the Boltzmann constant and \( T \) is the temperature of the gas in kelvin.

Compare Temperatures of the Gases

Since all the gases are at the same temperature of \(273 K\), they should have the same average kinetic energy per mole according to the Kinetic Molecular Theory of Gases.

Choose the Correct Answer

Given that the average kinetic energy per mole for a gas is determined by its temperature, and all three flasks are at the same temperature of \(273 K\), the molecules in all the flasks will have the same average kinetic energy per mole. Therefore, the answer is (d) the same in all.

Key concepts

Kinetic Molecular Theory

The Kinetic Molecular Theory (KMT) is a fundamental principle in understanding the behavior of gases in thermodynamics and physical chemistry. It provides a microscopic explanation of their macroscopic properties such as pressure, temperature, and volume. According to KMT, a gas is composed of a large number of small particles, often molecules or atoms, which are in constant and random motion.

The main postulates of the Kinetic Molecular Theory include that gas particles are in constant, straight-line motion except when they collide with each other or the walls of their container. These collisions are perfectly elastic, meaning that there is no net loss of kinetic energy. The theory also claims that the particles are much smaller than the distance between them and that there are no forces of attraction or repulsion between the particles except during collisions.

A critical aspect of KMT is that the average kinetic energy of gas particles is directly proportional to the absolute temperature of the gas. This means that, irrespective of the gas type or pressure, if two gases are at the same temperature, their particles have the same average kinetic energy. This provides a simple explanation for our textbook exercise, ensuring a strong foundation for students to understand why the average kinetic energy per mole is the same for all three flasks.

Kelvin Temperature Scale

The Kelvin temperature scale is an absolute temperature scale starting at absolute zero, the point at which all molecular motion stops. It's vital for scientific calculations because it provides a direct measure of thermal energy. Unlike the Celsius and Fahrenheit scales, where temperatures can have negative values, the Kelvin scale sets its zero point at the lowest possible temperature.

Absolute zero, or 0 K, is equivalent to -273.15°C, and it's impossible for temperatures to go lower. The Kelvin scale is crucial within the context of the Kinetic Molecular Theory because the kinetic energy of the particles in a substance is directly related to its absolute temperature in Kelvin. This means that any increase in Kelvin temperature corresponds to an increase in the average kinetic energy of the particles of the substance.

In our exercise involving gases in flasks at 273 K, understanding the Kelvin scale is essential as it enables students to comprehend why the molecules in all flasks must possess the same degree of kinetic energy, considering they are at the same Kelvin temperature. This fundamental insight helps clarify the relationship between temperature and energy in gaseous systems.

Boltzmann Constant

The Boltzmann constant, denoted as k or sometimes k_B, is a fundamental physical constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. Its value is approximately 1.38 x 10^-23 J/K (joules per Kelvin). The constant is named after the Austrian physicist Ludwig Boltzmann, who made significant contributions to statistical mechanics.

The equation \( \text{KE}_{\text{avg}} = \frac{3}{2} k T \) visually encapsulates the role of the Boltzmann constant in defining the average kinetic energy of particles in a gas. Here, T represents the absolute temperature in Kelvin, and the factor 3/2 arises from the three degrees of freedom (motion in the x, y, and z directions) of particles in a gas.

Understanding the Boltzmann constant is especially important in our educational exercise because it bridges the conceptual gap between the macroscopic measure of temperature and the microscopic kinetic energies of particles. In essence, it translates the temperature of the gas into the energy scale, thereby providing a critical component for the comprehension of gas behavior as outlined by the Kinetic Molecular Theory.