Problem 115 - Chemistry and Chemical Reactivity

Question

If the absolute temperature of a gas doubles, by how much does the rms speed of the gaseous molecules increase?

Step-by-Step Solution

Answer

The rms speed increases by a factor of \( \sqrt{2} \).

Step 1: Understanding the Relationship

The root mean square (rms) speed of gas molecules is given by the formula \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature in Kelvin (absolute temperature), and \( m \) is the mass of a single molecule of the gas. This shows that the rms speed is proportional to the square root of the temperature.

Step 2: Doubling the Temperature

If the absolute temperature \( T \) of the gas doubles, it becomes \( 2T \). Therefore, we need to find the new rms speed \( v'_{rms} \) when \( T \) is replaced with \( 2T \).

Step 3: Calculating New RMS Speed

Using the formula \( v'_{rms} = \sqrt{\frac{3k(2T)}{m}} = \sqrt{2} \times \sqrt{\frac{3kT}{m}} \), we can see that the new rms speed is \( \sqrt{2} \) times the original rms speed \( v_{rms} \).

Step 4: Comparing RMS Speeds

Since \( v'_{rms} = \sqrt{2} \times v_{rms} \), the rms speed increases by a factor of \( \sqrt{2} \) when the absolute temperature doubles.

Key Concepts

Absolute Temperature

Absolute temperature is a fundamental concept in thermodynamics that helps us understand how temperature affects the behavior of gases. It is measured in Kelvin (K), starting from absolute zero, the theoretical point where molecular motion ceases entirely. At this lowest limit, molecular kinetic energy is at a minimum.

Understanding absolute temperature is crucial for interpreting the behavior of gas molecules. Temperature changes can influence the energy and speed of these molecules. Crucially, in the Kelvin scale, zero Kelvin is used as the baseline to measure how energetic gas molecules are at any given temperature.

• Absolute temperature is always positive.
• Measured in Kelvin, not degrees Celsius or Fahrenheit.
• Directly proportional to the kinetic energy of the molecules in a gas.

When working with gas laws and principles like root mean square speed, it's essential to use absolute temperature for accurate calculations and predictions.

Boltzmann Constant

The Boltzmann constant, denoted by \( k \), plays a pivotal role in connecting the kinetic energy of particles in a gas to its absolute temperature. It is a fundamental constant in physics, named after Ludwig Boltzmann, an Austrian physicist known for his work in statistical mechanics.

The value of the Boltzmann constant is approximately \( 1.38 \times 10^{-23} \) joules per Kelvin (J/K). It provides a bridge between macroscopic and microscopic physical quantities.

• Helps relate temperature with energy on a molecular level.
• Used in calculating kinetic energy in gas law equations.
• Aids in deriving the rms speed of gas molecules.

Thus, the Boltzmann constant enables us to understand how gas molecules' energy distribution changes with temperature. When using formulas like the root mean square speed equation, \( k \) is vital in accurately representing the microscopic dynamics of gas molecules.

Root Mean Square Speed

Root mean square speed is a measure of the speed of gas molecules within a given sample. It provides an average speed that takes into account the energy distribution of all the molecules in that sample.

In physics, the root mean square speed \( v_{rms} \) is calculated using the formula:\[v_{rms} = \sqrt{\frac{3kT}{m}}\]where \( T \) is the absolute temperature in Kelvin, \( k \) is the Boltzmann constant, and \( m \) is the mass of a single molecule.

• Gives an average speed taking into account thermal energy.
• Proportional to the square root of the absolute temperature.
• Shows how speeds of molecules increase as temperature rises.

When the temperature of a gas doubles, the rms speed increases by a factor of \( \sqrt{2} \). Root mean square speed is essential for characterizing the motion of molecules and predicting how changes in temperature affect their speed.

Gas Molecules

Gas molecules are the tiny particles that make up a gas. These molecules are in constant motion, colliding with each other and the walls of their container, which results in what we see as gas pressure and temperature.

The behavior of gas molecules is largely influenced by their speed, which depends on their kinetic energy and, consequently, the absolute temperature of the gas.

• Molecules move faster at higher temperatures.
• A gas's pressure and volume can change with temperature due to molecular motion.
• Individual speeds vary greatly, but the average is given by concepts like rms speed.

Understanding the dynamics of gas molecules allows us to apply principles of thermodynamics and kinetic theory to predict how a gas will behave under different conditions, such as changes in temperature, pressure, or volume. As temperature increases, molecules gain energy, move faster, and impact rms speed, further highlighting the interconnectedness of these core concepts.