Problem 57 - College Physics

Question

(a) What is the average kinetic energy per molecule of a monatomic gas at a temperature of \(25^{\circ} \mathrm{C} ?\) (b) What is the rms speed of the molecules if the gas is helium? (A helium molecule consists of a single atom of mass \(6.65 \times 10^{-27} \mathrm{~kg}\).

Step-by-Step Solution

Answer

(a) \(6.17 \times 10^{-21}\) J; (b) 1253 m/s.

Step 1: Convert Temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin. The formula to convert is: \( T(K) = T(^{\circ}C) + 273.15 \). For 25°C, \( T = 25 + 273.15 = 298.15 \) K.

Step 2: Calculate Average Kinetic Energy

Use the formula for average kinetic energy per molecule of a monatomic gas: \( KE_{avg} = \frac{3}{2} k T \) where \( k = 1.38 \times 10^{-23} \, \text{J/K} \). Substituting the values gives: \( KE_{avg} = \frac{3}{2} \times 1.38 \times 10^{-23} \times 298.15 \approx 6.17 \times 10^{-21} \) J.

Step 3: Calculate RMS Speed

The formula for the root-mean-square (rms) speed of gas molecules is \( v_{rms} = \sqrt{\frac{3kT}{m}} \). Substituting the known values, \( k = 1.38 \times 10^{-23} \, \text{J/K} \), \( T = 298.15 \) K, and \( m = 6.65 \times 10^{-27} \, \text{kg} \), the calculation is: \( v_{rms} = \sqrt{\frac{3 \times 1.38 \times 10^{-23} \times 298.15}{6.65 \times 10^{-27}}} \approx 1253 \) m/s.

Key Concepts

Monatomic Gas

A monatomic gas consists of individual atoms that are not bound to each other. This means in a monatomic gas, each molecule is made up of a single atom. Examples include helium, neon, and argon. When dealing with gases like helium, it’s important to understand that they have some distinct characteristics:

• They exert pressure through countless collisions against the walls of their container.
• Despite being a single type of atom, they follow the ideal gas laws well under many conditions.
• The behavior of monatomic gases is simpler to model than that of polyatomic gases due to the absence of intermolecular forces.

Understanding these characteristics helps us grasp how such gases behave under different temperature and pressure conditions.

RMS Speed

The root-mean-square (rms) speed of gas molecules provides a measure of how fast the molecules are moving in a gas sample. It is critical in understanding the kinetic theory of gases, where:

• RMS speed (\(v_{rms}\)) connects directly to the temperature and molecular mass of the gas.
• It gives insight into gas properties like diffusion rates and pressure exerted by the gas.

The formula to compute the rms speed is \(v_{rms} = \sqrt{\frac{3kT}{m}}\), where \(k\) is Boltzmann's constant, \(T\) is the temperature in Kelvin, and \(m\) is the mass of one molecule.Hence, the higher the temperature, the higher the rms speed, and vice versa.

Temperature Conversion

Temperature in scientific contexts is often measured in Kelvin rather than Celsius. Kelvin is the SI unit for temperature and is essential for calculations involving physical properties:

• Conversion from Celsius to Kelvin requires adding 273.15 to the Celsius temperature.
• This conversion ensures temperatures are always positive, simplifying the math involved.

For example, a temperature of 25°C can be converted as follows:\(T(K) = 25 + 273.15 = 298.15\) K.Utilizing Kelvin is crucial when applying formulas involving energy and gas laws, yielding accurate and consistent results.

Helium Molecules

Helium is a noble gas, and its molecules are unique in that they are monatomic. Each helium molecule is comprised simply of a single helium atom.Some specific properties relevant to helium in gas behavior are:

• Each helium atom has a mass of \(6.65 \times 10^{-27}\) kg.
• Helium is inert, meaning it doesn't easily react with other substances.

The lightweight nature of helium atoms contributes to their high rms speed, particularly when heated. Studying helium gives insight into fundamental gas behaviors without the complexity introduced by molecular interactions.