Problem 138 - General Chemistry: Principles and Modern Applications

Question

If the Kelvin temperature of a sample of ideal gas doubles (e.g., from 200 K to 400 K), what happens to the root-mean-square speed, \(u_{\mathrm{rms}}\) ? (a) \(u_{\mathrm{rms}}\) increases by a factor of \(\sqrt{2} ;\) (b) \(u_{\mathrm{rms}}\) increases by a factor of \(2 ;(\mathrm{c}) u_{\mathrm{rms}}\) decreases by a factor of 2 (d) \(u_{\mathrm{rms}}\) increases by a factor of \(4 ;\) (e) \(u_{\mathrm{rms}}\) decreases by a factor of 4.

Step-by-Step Solution

Answer

The root-mean-square speed, \(u_{\mathrm{rms}}\) increases by a factor of \( \sqrt{2} \).

Step 1: Identify the relationship between temperature and rms speed

We know that \(u_{rms} = \sqrt{ \frac {3kT} {m} }\). This equation shows us that the rms speed is proportional to the square root of the temperature.

Step 2: Calculate the change in rms speed due to temperature doubling

If we double the temperature i.e., \( T \rightarrow 2T \), the new rms speed \( u_{new} \) will be \( u_{new} = \sqrt{ \frac {3k(2T)} {m} } \). This can also be written as \(u_{new} = \sqrt{2} \cdot \sqrt{ \frac {3kT} {m} } \) = \( \sqrt{2} \cdot u_{rms} \). This shows us that the rms speed increases by a factor of \( \sqrt{2} \).

Step 3: Choose the correct option

Based on our calculations, the root mean square speed increases by a factor of \( \sqrt{2} \) which corresponds to option (a).

Key Concepts

Root-Mean-Square Speed

The root-mean-square speed, often referred to as rms speed, is a measure of the speed of particles in a gas. It reflects how fast, on average, gas molecules are moving. This concept is essential in understanding the behavior of gases from a microscopic perspective.
While individual gas molecules move at different speeds, the rms speed provides a useful average that simplifies calculations and predictions about gas behavior. It is given by the formula:

• \( u_{rms} = \sqrt{ \frac {3kT} {m} } \)

Where:

• \( u_{rms} \) is the root-mean-square speed.
• \( k \) is the Boltzmann constant, a fundamental physical constant that relates energy to temperature.
• \( T \) is the absolute temperature of the gas in Kelvin.
• \( m \) is the mass of a gas molecule.

Understanding the rms speed helps us grasp how temperature affects molecular motion, which in turn influences properties like pressure and volume in gas systems.

Kelvin Temperature

Kelvin temperature is a scale that is essential when dealing with gases. Unlike Celsius or Fahrenheit, the Kelvin scale starts at absolute zero, the theoretical point where particles have no kinetic energy at all.
The Kelvin scale is crucial because it maintains a direct relationship with energy. In chemical and physical calculations, using Kelvin avoids the complications that negative temperatures or arbitrary freezing points might introduce on other scales.
When dealing with gases, particularly, Kelvin helps make sense of the proportional relationship between temperature and molecular speed, as the formula for rms speed directly uses Kelvin. This ensures that zero on the scale accurately represents no thermal energy, making it a preferred unit in scientific equations involving temperature.

Proportional Relationship

In physics, understanding proportional relationships helps us predict how changes in one quantity might affect another. In the context of the ideal gas law, the root-mean-square speed of gas molecules is directly proportional to the square root of the temperature. This is evident from the equation:

• \( u_{rms} \propto \sqrt{T} \)

Simplified, this means if you increase the temperature, the rms speed increases, too. However, because of the square root relationship, doubling the temperature of a gas does not double the rms speed.
Instead, it increases by a factor of \( \sqrt{2} \). This spatial understanding of proportionality helps us make accurate predictions and understand the behavior of gases under different thermal conditions without performing exhaustive calculations each time. In practical terms, it lets us see the direct link between heat energy input and molecular motion.

Temperature Doubling

Doubling the temperature of a gas sample can be quite significant, especially in terms of molecular kinetics. A common question arises, what happens to various properties such as rms speed when the Kelvin temperature of an ideal gas doubles?
Upon doubling the temperature (

• \( T \rightarrow 2T \)

), the new rms speed follows the relationship:

• \( u_{new} = \sqrt{2} \cdot u_{rms} \)

This illustrates the idea that increasing the Kelvin temperature by a factor of two results in an increase of the rms speed by a factor of \( \sqrt{2} \). This relationship underscores the non-linear way in which molecular speeds react to temperature changes.
By understanding this, students can better appreciate why temperature adjustments have the effects they do and apply this understanding to solve problems related to kinetic molecular energies in gases.