Question

For a gas, the rms speed at \(800 \mathrm{~K}\) is (A) Four times the value at \(200 \mathrm{~K}\) (B) Twice the value at \(200 \mathrm{~K}\) (C) Half the value at \(200 \mathrm{~K}\) (D) same as at \(200 \mathrm{~K}\)

Short answer

The root-mean-square speed at 800 K is twice the value at 200 K. The correct choice is (B) Twice the value at 200 K.

Step-by-step solution

Write down the rms speed formula

The formula for the root-mean-square (rms) speed of a gas is given by: \(v_\text{rms} = \sqrt{\frac{3RT}{M}}\) where \(v_\text{rms}\) is the root-mean-square speed, \(R\) is the universal gas constant, \(T\) is the temperature of the gas, and \(M\) is the molar mass of the gas.

Determine the relationship between the rms speeds

Divide the rms speed formula at 800 K by the rms speed formula at 200 K to find the relationship between the two rms speeds: \(\frac{v_\text{rms, 800 K}}{v_\text{rms, 200 K}} = \frac{\sqrt{\frac{3R(800)}{M}}}{\sqrt{\frac{3R(200)}{M}}}\)

Simplify the equation

Notice that we can cancel out the factors of \(3R\) and \(M\) in the numerator and denominator: \(\frac{v_\text{rms, 800 K}}{v_\text{rms, 200 K}} = \frac{\sqrt{800}}{\sqrt{200}}\) Next, simplify the expression by taking the square root of 800 and 200: \(\frac{v_\text{rms, 800 K}}{v_\text{rms, 200 K}} = \frac{20\sqrt{2}}{10\sqrt{2}}\) Now, we can cancel out the factor of \(\sqrt{2}\) in the numerator and denominator: \(\frac{v_\text{rms, 800 K}}{v_\text{rms, 200 K}} = \frac{20}{10}\)

Compute the final answer

Divide 20 by 10 to find the final answer: \(\frac{v_\text{rms, 800 K}}{v_\text{rms, 200 K}} = 2\) The root-mean-square speed at 800 K is twice the value at 200 K. The correct choice is (B) Twice the value at 200 K.

Key concepts

Ideal Gas Law

The ideal gas law is a fundamental equation used to describe the behavior of gases under different conditions. It is expressed as \( PV = nRT \), where:

• \( P \): Pressure of the gas
• \( V \): Volume of the gas
• \( n \): Number of moles of gas
• \( R \): Universal gas constant
• \( T \): Absolute temperature in Kelvins

This formula assumes that the gas is "ideal," meaning that it perfectly follows the principles without any intermolecular forces or volume.
Ideal gases help us predict and calculate behaviors such as pressure, volume, and temperature changes. When you change one property while keeping others constant, this law helps in understanding how the gas will react.

Temperature Impact on Gas

Temperature plays a crucial role in determining the speed and energy of gas molecules. When the temperature of a gas increases, its molecules move faster because temperature is directly related to the kinetic energy of molecules.
The relationship between temperature and kinetic energy can be observed using the root-mean-square speed formula. As temperature increases, the rms speed of gas molecules also increases.
For instance, in our example, when the temperature increased from 200 K to 800 K, the root-mean-square speed doubled, demonstrating how temperature directly impacts the motion of gas molecules. This change occurs because increasing temperature provides more energy to the molecules, enabling them to overcome intermolecular forces and move more quickly.

Universal Gas Constant

The universal gas constant \( R \) is an essential part of various gas-related equations, such as the ideal gas law. It represents how energy is transferred between molecules in ideal conditions and is consistent for all ideal gases.
The value of \( R \) is approximately 8.31 J/(mol K), which helps relate pressure, volume, temperature, and the number of moles in a gas. In the root-mean-square speed formula, \( R \) enables us to find the speed of molecules based on temperature and molar mass.
Consequently, \( R \) serves as a bridge between different properties of gases, allowing us to connect and calculate these properties efficiently. Understanding \( R \) not only helps in solving problems about gas behaviors but also enhances our comprehension of molecular motion and energy transfer.

Molar Mass of Gas

Molar mass, denoted by \( M \), represents the mass of one mole of a gas, usually measured in g/mol. It is a key factor in various calculations of gas properties, impacting both speed and diffusion rates.
In the root-mean-square speed formula \( v_\text{rms} = \sqrt{\frac{3RT}{M}} \), molar mass influences molecular speed. A higher molar mass results in a lower rms speed because heavier molecules move more sluggishly than lighter ones at the same temperature.
Understanding the molar mass is crucial for predicting how fast molecules can move and react, as well as for calculating the concentrations and reactions in different gas mixtures. It plays an important role in applications ranging from industrial gas production to biological respiration processes.