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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

Expert verified
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

01

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.
02

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 \( \upsilon_{mp} \).
03

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 \( \upsilon_{av} \).
04

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 \( \upsilon_{rms} \).
05

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

A hot-air balloon stays aloft because hot air at atmospheric pressure is less dense than cooler air at the same pressure. If the volume of the balloon is 500.0 m\(^3\) and the surrounding air is at 15.0\(^\circ\)C, what must the temperature of the air in the balloon be for it to lift a total load of 290 kg (in addition to the mass of the hot air)? The density of air at 15.0\(^\circ\)C and atmospheric pressure is 1.23 kg/m\(^3\).

You have several identical balloons. You experimentally determine that a balloon will break if its volume exceeds 0.900 L. The pressure of the gas inside the balloon equals air pressure (1.00 atm). (a) If the air inside the balloon is at a constant 22.0\(^\circ\)C and behaves as an ideal gas, what mass of air can you blow into one of the balloons before it bursts? (b) Repeat part (a) if the gas is helium rather than air.

A balloon of volume 750 m\(^3\) is to be filled with hydrogen at atmospheric pressure (1.01 \(\times\) 10\(^5\) Pa). (a) If the hydrogen is stored in cylinders with volumes of 1.90 m\(^3\) at a gauge pressure of 1.20 \(\times\) 10\(^6\) Pa, how many cylinders are required? Assume that the temperature of the hydrogen remains constant. (b) What is the total weight (in addition to the weight of the gas) that can be supported by the balloon if both the gas in the balloon and the surrounding air are at 15.0\(^\circ\)C? The molar mass of hydrogen (H\(_2\)) is 2.02 g/mol. The density of air at 15.0\(^\circ\)C and atmospheric pressure is 1.23 kg/m\(^3\). See Chapter 12 for a discussion of buoyancy. (c) What weight could be supported if the balloon were filled with helium (molar mass 4.00 g/mol) instead of hydrogen, again at 15.0\(^\circ\)C?

If a certain amount of ideal gas occupies a volume \(V\) at STP on earth, what would be its volume (in terms of \(V\)) on Venus, where the temperature is 1003\(^\circ\)C and the pressure is 92 atm?

The rate of \(effusion\)-that is, leakage of a gas through tiny cracks-is proportional to \(v_{rms}\) . If tiny cracks exist in the material that's used to seal the space between two glass panes, how many times greater is the rate of \(He\) leakage out of the space between the panes than the rate of \(Xe\) leakage at the same temperature? (a) 370 times; (b) 19 times; (c) 6 times; (d) no greater-the \(He\) leakage rate is the same as for \(Xe\).

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