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Chapter 18: Problem 41

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鈥檚 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.鈥� It takes into account all molecules, weighing the slower and faster ones.鈥� 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鈥檚 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 metal tank with volume 3.10 L will burst if the absolute pressure of the gas it contains exceeds 100 atm. (a) If 11.0 mol of an ideal gas is put into the tank at 23.0\(^\circ\)C, to what temperature can the gas be warmed before the tank ruptures? Ignore the thermal expansion of the tank. (b) Based on your answer to part (a), is it reasonable to ignore the thermal expansion of the tank? Explain.

Three moles of an ideal gas are in a rigid cubical box with sides of length 0.300 m. (a) What is the force that the gas exerts on each of the six sides of the box when the gas temperature is 20.0\(^\circ\)C? (b) What is the force when the temperature of the gas is increased to 100.0\(^\circ\)C?

A physics lecture room at 1.00 atm and 27.0\(^\circ\)C has a volume of 216 m\(^3\). (a) Use the ideal-gas law to estimate the number of air molecules in the room. Assume that all of the air is N\(_2\). Calculate (b) the particle density-that is, the number of N\(_2 \) molecules per cubic centimeter-and (c) the mass of the air in the room.

During a test dive in 1939, prior to being accepted by the U.S. Navy, the submarine \(Squalus\) sank at a point where the depth of water was 73.0 m. The temperature was 27.0\(^\circ\)C at the surface and 7.0\(^\circ\)C at the bottom. The density of seawater is 1030 kg/m\(^3\). (a) A diving bell was used to rescue 33 trapped crewmen from the \(Squalus\). The diving bell was in the form of a circular cylinder 2.30 m high, open at the bottom and closed at the top. When the diving bell was lowered to the bottom of the sea, to what height did water rise within the diving bell? (Hint: Ignore the relatively small variation in water pressure between the bottom of the bell and the surface of the water within the bell.) (b) At what gauge pressure must compressed air have been supplied to the bell while on the bottom to expel all the water from it?

(a) Calculate the specific heat at constant volume of water vapor, assuming the nonlinear triatomic molecule has three translational and three rotational degrees of freedom and that vibrational motion does not contribute. The molar mass of water is 18.0 g/mol. (b) The actual specific heat of water vapor at low pressures is about 2000 J/kg \(\cdot\) K. Compare this with your calculation and comment on the actual role of vibrational motion.
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