Question

If the absolute temperature of a gas doubles, by how much does the average speed of the gaseous molecules increase? (See General ChemistryNow Screen \(12.9 .\) )

Short answer

The average speed of gaseous molecules increases by a factor of \(\sqrt{2}\) when the absolute temperature doubles.

Step-by-step solution

Understanding the Relationship

The average speed of gaseous molecules in a gas is related to the temperature by the formula \( v = \sqrt{\frac{3kT}{m}} \), where \( v \) is the average speed, \( k \) is the Boltzmann constant, \( T \) is the absolute temperature, and \( m \) is the mass of a molecule. Our task is to determine how the average speed changes when \( T \) doubles.

Doubling the Temperature

If the absolute temperature \( T \) doubles, we represent the new temperature as \( T' = 2T \). This means our new expression for speed becomes \( v' = \sqrt{\frac{3k(2T)}{m}} \).

Simplifying the New Speed Expression

Rewrite the expression for the new speed as \( v' = \sqrt{2} \times \sqrt{\frac{3kT}{m}} \). Notice that \( \sqrt{\frac{3kT}{m}} \) is the original speed \( v \), such that \( v' = \sqrt{2} \times v \).

Concluding the Change in Speed

The result \( v' = \sqrt{2} \times v \) indicates that the average speed of the gaseous molecules increases by a factor of \( \sqrt{2} \) when the temperature doubles.

Key concepts

Boltzmann constant

The Boltzmann constant, denoted as \( k \), plays a crucial role in the kinetic molecular theory of gases. It provides a bridge between microscopic and macroscopic physical quantities. In the formula for the average speed of gaseous molecules, \( v = \sqrt{\frac{3kT}{m}} \), the Boltzmann constant is a key ingredient. This constant has a value of approximately \( 1.38 \times 10^{-23} \text{ J/K} \). It relates the individual energy levels of molecules to the temperature of the gas.

The significance of the Boltzmann constant can be understood from its ability to convert temperature, measured in Kelvin, into energy at the molecular level, in joules. This is fundamental because temperature is a macroscopic quantity, while energy is more easily associated with individual molecules.

Thanks to the Boltzmann constant, we can directly link thermal energy to molecular movements. In essence, this constant allows us to predict how particles in a gas behave at different temperatures, which is vital for understanding chemical reactions and the behaviors of gases in different environments.

absolute temperature

Absolute temperature is measured in Kelvin and is a crucial concept in thermodynamics and kinetic molecular theory. It provides an absolute scale to measure thermal energy. When discussing the speed of gaseous molecules, absolute temperature is essential because it directly affects how these molecules move.

Unlike Celsius or Fahrenheit, the Kelvin scale begins at absolute zero (0 K), the theoretical point where molecular motion ceases. Thus, when we talk about doubling the absolute temperature, as in the problem, we're considering an increase from, say \( T \) to \( 2T \). This doubling directly affects molecular speeds, as seen in the formula \( v = \sqrt{\frac{3kT}{m}} \).

With absolute temperature, it's easier to understand how temperature impacts molecular behavior. Each increment in Kelvin equates to an increase in kinetic energy, ensuring that as temperature rises, molecules move faster, interact more, and potentially escape into other environments. This insight is invaluable for predicting PVT (pressure-volume-temperature) relationships and interpreting experimental data in various scientific fields.

average molecular speed

In the realm of gases, the average molecular speed gives insight into how fast molecules are moving on average. This concept is part of the broader kinetic molecular theory, which provides a quantitative explanation of how molecular activity correlates with temperature and other state variables.

In the formula \( v = \sqrt{\frac{3kT}{m}} \), \( v \) represents the average speed, while \( T \) is the absolute temperature and \( m \) is the molecular mass. When the absolute temperature \( T \) doubles, the average speed increases by a factor of \( \sqrt{2} \) because speed is proportional to the square root of the temperature. Hence, if temperature increases, the molecular speed increases, thus increasing the energy and, consequently, the pressure the gas exerts.

Understanding average molecular speed helps explain why gases expand when heated, how sound travels, and why different gases diffuse at different rates. It's a foundational concept that helps predict real-world phenomena related to weather patterns, industrial processes, and even everyday occurrences like cooking or refrigeration.