Problem 54 - College Physics

Question

What is the average kinetic energy per molecule in a monatomic gas at (a) \(10^{\circ} \mathrm{C}\) and (b) \(90^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

(a) \(5.87 \times 10^{-21} \text{ J}\), (b) \(7.53 \times 10^{-21} \text{ J}\).

Step 1: Recall the formula for average kinetic energy

The average kinetic energy per molecule of a monatomic ideal gas is given by the formula: \[ KE_{avg} = \frac{3}{2} kT \] where \( k \) is the Boltzmann constant \( (1.38 \times 10^{-23} \text{ J/K}) \) and \( T \) is the absolute temperature in Kelvin.

Step 2: Convert Celsius to Kelvin

To convert the temperature from Celsius to Kelvin, use the formula: \[ T(K) = T(^{\circ}C) + 273.15 \] For (a) \( 10^{\circ}C \): \( T = 10 + 273.15 = 283.15 \text{ K} \). For (b) \( 90^{\circ}C \): \( T = 90 + 273.15 = 363.15 \text{ K} \).

Step 3: Calculate KE for 10°C

Substitute \( T = 283.15 \text{ K} \) into the formula for average kinetic energy: \[ KE_{avg} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 283.15 \approx 5.87 \times 10^{-21} \text{ J} \]

Step 4: Calculate KE for 90°C

Substitute \( T = 363.15 \text{ K} \) into the formula for average kinetic energy: \[ KE_{avg} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 363.15 \approx 7.53 \times 10^{-21} \text{ J} \]

Key Concepts

Monatomic Gas

A monatomic gas is a type of gas in which each particle is made up of a single atom. Unlike diatomic or polyatomic gases, where molecules consist of two or more atoms bonded together, monatomic gases exist as individual atoms. Common examples of monatomic gases include noble gases like helium (He), neon (Ne), and argon (Ar). These gases are chemically inert due to their filled electron shells.

Monatomic gases obey the ideal gas law closely, mainly because of their simplicity. In physics, they are used extensively to model ideal conditions in thermodynamic calculations. One key feature of monatomic gases is their simple energy structure where translational kinetic energy is the only form of mechanical energy they possess.

This simplicity also means they experience fewer intermolecular forces at high temperatures. This characteristic makes calculating properties like average kinetic energy straightforward.

Temperature Conversion

Temperature conversion is an essential step when working with thermodynamic equations, especially when transitioning between different temperature scales. One of the most common conversions is from degrees Celsius to Kelvin.

To perform this conversion, you simply add 273.15 to the Celsius temperature. This is because the Kelvin scale is an absolute temperature scale starting at absolute zero, where molecular motion stops. This conversion ensures that temperatures are positive and compatible within physical equations.

• Example: A temperature of 10° C converts to: 10 + 273.15 = 283.15 K.
• For 90° C, the conversion is 90 + 273.15 = 363.15 K.

Using the Kelvin scale helps in physics equations because it directly correlates with the energy scale, important for calculations of heat and thermodynamics, like the average kinetic energy of molecules.

Boltzmann Constant

The Boltzmann constant, denoted as \( k \), is a fundamental constant in physics that helps relate the average kinetic energy of particles in a gas to the temperature of the gas in Kelvin. The value of the Boltzmann constant is approximately 1.38 \times 10^{-23} J/K.

This constant plays a crucial role in statistical mechanics, providing a bridge between macroscopic and microscopic physics. It connects the thermodynamic properties of systems to the motions of particles.

• It appears in the formula for the average kinetic energy of molecules: \[ KE_{avg} = \frac{3}{2} kT \]
• Here, \( T \) represents the temperature in Kelvin.

By using the Boltzmann constant in this formula, we can calculate the average kinetic energy of particles in a monatomic ideal gas accurately, allowing us to understand how microscopic behaviors impact macroscopic phenomena. It illustrates how temperature is fundamentally linked to energy at the particle level.