Question

For diatomic carbon dioxide gas (CO${}_2$, molar mass 44.0 g/mol) at $T$ = 300 K, calculate (a) the most probable speed $\upsilon {}_m{}_p$; (b) the average speed $\upsilon {}_a{}_v$; (c) the root-mean-square speed $\upsilon {}_r{}_m{}_s$.

Short answer

For CO${}_2$ at 300 K, calculate: ${\upsilon }_{mp}\approx 395\text{ m/s}$, ${\upsilon }_{av}\approx 420\text{ m/s}$, ${\upsilon }_{rms}\approx 430\text{ m/s}$.

Step-by-step solution

Define the Problem

We need to calculate three types of speeds (most probable, average, and root-mean-square) for CO${}_2$ gas at temperature $T=300$ K. These speeds are related to the kinetic theory of gases and depend on the temperature and molar mass of the gas.

Calculate the Most Probable Speed

The most probable speed, ${\upsilon }_{mp}$, can be calculated using the formula: $${\upsilon }_{mp}=\sqrt{\frac{2kT}{m}}$$where $k$ is the Boltzmann constant ($1.38\times {10}^{-23}\text{ J/K}$), $T=300$ K is the temperature, and $m$ is the mass of one molecule of CO${}_2$. The molar mass of CO${}_2$ is 44.0 g/mol, so $m=\frac{44.0}{1000\times 6.022\times {10}^{23}}\approx 7.30\times {10}^{-26}\text{ kg}$.Using these values, calculate ${\upsilon }_{mp}$.

Calculate the Average Speed

The average speed, ${\upsilon }_{av}$, is given by:$${\upsilon }_{av}=\sqrt{\frac{8kT}{\pi m}}$$Substitute $k=1.38\times {10}^{-23}\text{ J/K}$, $T=300$ K, and $m\approx 7.30\times {10}^{-26}\text{ kg}$ into the formula and compute ${\upsilon }_{av}$.

Calculate the Root-Mean-Square Speed

The root-mean-square speed, ${\upsilon }_{rms}$, can be calculated using:$${\upsilon }_{rms}=\sqrt{\frac{3kT}{m}}$$Using $k=1.38\times {10}^{-23}\text{ J/K}$, $T=300$ K, and $m\approx 7.30\times {10}^{-26}\text{ kg}$, find ${\upsilon }_{rms}$.

Perform Calculations

Compute the values for ${\upsilon }_{mp}$, ${\upsilon }_{av}$, and ${\upsilon }_{rms}$ using the formulas and substitutions performed in the previous steps. Ensure correct unit conversion and arithmetic operations.

Key concepts

Molecular Speeds

In the kinetic theory of gases, we explore different ways to measure how fast gas molecules move. This concept of molecular speeds helps us understand how individual gas particles behave. Molecular speeds are essential because they influence how gases interact with each other. For example, faster molecules will collide more and may escape a container faster.

• Most Probable Speed: The speed at which most gas molecules are moving.
• Average Speed: The mean speed of all the molecules, which balances out slower and faster molecules.
• Root-Mean-Square Speed: A measure that squares the speed of each molecule, finds the average, and then takes the square root.

Each of these speeds provides different insights into the behavior of gas molecules at a given temperature.

Boltzmann Constant

The Boltzmann constant is a fundamental constant in physics, denoted by the symbol, $k$, and has a value of $1.38\times {10}^{-23}\text{ J/K}$. It plays a crucial role in the kinetic theory of gases, linking the macroscopic and microscopic worlds. It helps us translate temperature into energy when studying gases.Whenever we calculate molecular speeds such as the most probable, average, or root-mean-square speed, the Boltzmann constant is part of the formula. By doing so, it scales the temperature to the kinetic energy of gas molecules. This allows us to move from purely theoretical calculations to practical, observable results.

Root-Mean-Square Speed

The root-mean-square (RMS) speed is a statistical measure of the speed of particles in a gas that gives insight into their energy. Mathematically, it is given by the formula:$${\upsilon }_{rms}=\sqrt{\frac{3kT}{m}}$$where $k$ is the Boltzmann constant, $T$ is the temperature, and $m$ is the mass of a single molecule of the gas.RMS speed considers each molecule’s speed squared, averages these values, and takes the square root of the result. This gives us an effective measure of speed that considers both the direction and magnitude of molecular velocities. Knowing the RMS speed helps us predict gas behaviors, such as diffusion rates and kinetic energies.

Average Speed

The average speed of gas molecules is another important component of molecular speeds in the kinetic theory of gases.This average speed is calculated using the formula:$${\upsilon }_{av}=\sqrt{\frac{8kT}{\pi m}}$$The formula shows that the average speed is related to the temperature and mass of the gas molecules, similar to the other types of molecular speeds.• It takes into account all molecules, weighing the slower and faster ones.• While different from the root-mean-square speed, it provides a more intuitive measure of molecular motion.Understanding average speed is useful when evaluating how gases mix or transport energy through collisions.

Most Probable Speed

The most probable speed is the speed of a gas molecule that is most likely observed in a large sample of a gas.It is calculated by:$${\upsilon }_{mp}=\sqrt{\frac{2kT}{m}}$$Here’s why this speed matters:

• It's the peak of the speed distribution curve for gases, meaning most molecules have this speed.
• The formula highlights that the most probable speed depends on temperature and molecular mass.
• The most probable speed is often lower than both the average and RMS speeds.

Knowing the most probable speed helps in predicting how gases behave under different conditions, such as pressure changes in a balloon.