Problem 87 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

At what temperature will the total translational kinetic energy of \(0.30\) mole of He gas be the same as the total translational kinetic energy of \(0.40 \mathrm{~mol}\) of \(\overline{\mathrm{Ar}}\) at \(400 \mathrm{~K} ?\) (a) \(533 \mathrm{~K}\) (b) \(400 \mathrm{~K}\) (c) \(300 \mathrm{~K}\) (d) \(266 \mathrm{~K}\)

Step-by-Step Solution

Answer

533 K

Step 1: Understand the Kinetic Theory of Gases

According to the kinetic theory of gases, the total translational kinetic energy (KE) of a gas can be calculated using the equation KE = (3/2) * n * k * T, where n is the amount in moles, k is the Boltzmann constant, and T is the temperature in Kelvin.

Step 2: Write the Equation for Helium Kinetic Energy

For helium gas, the translational kinetic energy is given by KE_He = (3/2) * n_He * k * T_He, where n_He is the number of moles of helium and T_He is its temperature.

Step 3: Write the Equation for Argon Kinetic Energy

For argon gas, the kinetic energy is given by KE_Ar = (3/2) * n_Ar * k * T_Ar, where n_Ar is the number of moles of argon and T_Ar is its temperature (400 K in this case).

Step 4: Set the Kinetic Energies Equal

Since we want to find the temperature where their kinetic energies are the same, we set them equal: (3/2) * n_He * k * T_He = (3/2) * n_Ar * k * T_Ar.

Step 5: Plug in the Given Values

Substitute the given values into the equation: (3/2) * 0.30 * k * T_He = (3/2) * 0.40 * k * 400 K.

Step 6: Simplify and Solve for T_He

Since (3/2) and k are constants and appear on both sides of the equation, they can be cancelled out. This simplifies to 0.30 * T_He = 0.40 * 400. Solving for T_He, we get T_He = (0.40 * 400) / 0.30.

Step 7: Calculate the Temperature for Helium

T_He = (0.40 * 400) / 0.30 results in T_He = 533.33 K, which can be rounded to 533 K.

Key Concepts

Kinetic Theory of Gases

The kinetic theory of gases is a fundamental principle that provides insight into the behavior of gases at the molecular level. It makes several key assumptions: gas particles are in constant, random motion; the size of the particles is negligible compared to the distances between them; collisions between particles and with the walls of the container are perfectly elastic; and no intermolecular forces are acting between the particles.

According to this theory, the temperature of a gas is directly proportional to the average kinetic energy of its particles. This relationship helps in understanding how temperature affects gas properties like pressure and volume, and it is also crucial in calculating the kinetic energy of gases. For educational exercises, it often leads to the equation for translational kinetic energy: \[ KE = \frac{3}{2} n k T \] where \( KE \) is the kinetic energy, \( n \) is the number of moles, \( k \) is the Boltzmann constant, and \( T \) is the temperature in Kelvin. This equation is the basis for understanding how molecular motion translates into macroscopic gas properties.

Molar Mass in Gas Law Calculations

Incorporating molar mass into gas law calculations allows us to convert between the mass of gas and the number of moles present, using the formula: \[ n = \frac{m}{M} \] where \( n \) is the number of moles, \( m \) is the mass of the gas, and \( M \) is the molar mass (mass per mole) of the gas.

Molar mass is a critical parameter when utilizing the gas laws because it serves as a conversion factor that relates the macroscopic properties of gases (like mass) with their microscopic properties (like the number of particles). In the context of kinetic energy, the molar mass doesn't explicitly appear in the equation, but it's implicit in the concept of moles (\( n \)), allowing us to quantify and compare the kinetic energies of different gases given their masses and temperatures.

Boltzmann Constant

The Boltzmann constant (\( k \)) is a fundamental constant in physics that links the macroscopic and microscopic worlds, particularly in the field of thermodynamics. It relates the average kinetic energy of particles in a gas with the temperature of the gas. The value of the Boltzmann constant is \[ k = 1.380649 \times 10^{-23} \frac{J}{K} \].

This constant is named after the Austrian physicist Ludwig Boltzmann who made significant contributions to the field of statistical mechanics. In the calculations for kinetic energy, the Boltzmann constant serves as a proportionality factor, allowing us to use the relatively macroscopic measurement of temperature to calculate the microscopic kinetic energy of individual gas particles. It is essential for students to comprehend the use of the Boltzmann constant in equations involving the kinetic theory of gases, as it underpins the relationship between temperature and kinetic energy.

Chemistry Thermodynamics

Chemistry thermodynamics is a branch of science that studies the relationships between heat, work, temperature, and energy in chemical processes. Translational kinetic energy of gases, which we calculated in the exercise, plays a key role in understanding these processes. Thermodynamics covers various laws, including the first law, which relates to the conservation of energy, stating that energy can neither be created nor destroyed.

The study of thermodynamics is vital in examining how energy changes and transfers during chemical reactions and phase changes. It also helps predict whether processes will occur spontaneously. The ability to calculate the translational kinetic energy of gases complements the understanding of the first law of thermodynamics by providing a quantifiable measure of energy at the molecular level for gases.