Problem 62 - University Physics with Modern Physics

Question

A sealed container contains 1.00 mole of neon gas at STP. Estimate the number of neon atoms having speeds in the range from \(200.00 \mathrm{~m} / \mathrm{s}\) to \(202.00 \mathrm{~m} / \mathrm{s}\). (Hint: Assume the probability of neon atoms having speeds between \(200.00 \mathrm{~m} / \mathrm{s}\) and \(202.00 \mathrm{~m} / \mathrm{s}\) is constant.

Step-by-Step Solution

Answer

Question: Estimate the number of neon atoms in 1 mole of gas with speeds in the range of 200.00 m/s to 202.00 m/s at STP, assuming a constant probability distribution. Answer: The estimation for the number of neon atoms in the given speed range is given by the formula: Number of atoms in the speed range = \(6.022 \times 10^{23} \times P \times 2.00\), where P is the constant probability for all atoms at those speeds. Without knowing the exact probability (P), we cannot provide a precise value.

Step 1: Calculate the total number of neon atoms

First, we need to find the total number of neon atoms in the container. We know that there is 1 mole of neon gas, so we will use Avogadro's number to determine the total number of atoms: Total number of neon atoms = 1 mole × Avogadro's number Total number of neon atoms = \(1 \times 6.022 \times 10^{23}\) atoms

Step 2: Determine the speed range

The problem gives us a speed range to consider, specifically that we want the number of atoms with speeds between 200.00 m/s and 202.00 m/s. Thus, the width of the speed range is: Speed range width = 202.00 m/s - 200.00 m/s Speed range width = 2.00 m/s

Step 3: Calculate the fraction of atoms in the given speed range

Since we're given the hint to assume that the probability distribution of speeds between 200.00 m/s and 202.00 m/s is constant, we can say that the fraction of atoms with speeds in this range is equal to this constant probability (which we will denote as "P") multiplied by the speed range width: Fraction of atoms in speed range = P × Speed range width The probability (P) is a constant that we are not given, but we may estimate using the range by assuming the probability is equal for all atoms at those speeds. Using this assumption, the fraction of atoms with speeds between 200.00 m/s and 202.00 m/s is simply equal to the width of the speed range divided by the total speed range across all molecules. However, estimating the total speed range of the neon gas can be difficult. Because of this, the exercise has simplified the problem so that we can still find the number of neon atoms just by the information we have.

Step 4: Estimate the number of neon atoms in the speed range

Now that we have the fraction of atoms in the speed range, we can calculate the number of atoms with speeds between 200.00 m/s and 202.00 m/s by taking the fraction of atoms in the speed range and multiplying it by the total number of neon atoms: Number of atoms in the speed range = Total number of neon atoms × Fraction of atoms in the speed range Number of atoms in the speed range = \(6.022 \times 10^{23} \times P \times 2.00\)

Step 5: Final estimation

Without knowing the exact probability (P), we cannot precisely calculate the number of neon atoms in the given speed range. However, the above formula can give an estimation if we assume a value for P. Keep in mind, since P is a constant probability for all atoms at those speeds, our estimation may not be perfect but is useful in understanding the impact of changing the number of atoms within the speed range. Given the information provided, the estimation of the number of neon atoms having speeds in the range of 200.00 m/s to 202.00 m/s would be: Number of atoms in the speed range = \(6.022 \times 10^{23} \times P \times 2.00\)

Key Concepts

Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann Distribution is a fundamental principle in the Kinetic Theory of Gases. It describes the spread of speeds among particles in a gas. In essence, it shows how many particles are moving at various speeds at a given temperature.

Imagine a large group of gas atoms spinning around in constant motion. They're not all moving at the same speed. Some atoms move slowly, while others zip around very quickly. Most atoms, though, have an average speed that's somewhere in between these two extremes.

The exact number of atoms at each speed is determined by the Maxwell-Boltzmann Distribution. This distribution can be pictured as a curve that peaks at a certain speed, called the "most probable speed". At this peak speed, the largest number of atoms are moving. On either side of this peak, the curve trails off, indicating fewer atoms moving at those slower and faster speeds.

• Atoms mostly move at the 'most probable speed.'
• Fewer atoms move at very slow or very fast speeds.
• The shape of the curve varies with temperature changes.

Understanding this distribution is crucial for estimating the number of atoms with specific speeds, just like in our exercise about neon atoms. Here, we presume that an equal number of atoms share speeds within a small range due to a hint given in the problem. However, usually, you'd need the full Maxwell-Boltzmann curve to find the exact probability of atom speeds.

Mole Concept

The Mole Concept is a way to measure and understand amounts of any substance. It's a cornerstone idea in chemistry that helps relate mass, number of molecules, and volume at a molecular level.

Think of a mole as a 'chemist's dozen'. Just as a dozen eggs equals twelve eggs, a mole equals a fixed number of particles, which is Avogadro's Number: approximately 6.022 x 10^{23}.

This gigantic number allows chemists to work with quantities on a human-sized scale. For gases, this means being able to handle volumes instead of worrying about individual atoms.

• 1 mole = 6.022 x 10^{23} particles.
• It helps connect macroscopic measurements with microscopic particles.
• Extremely useful in chemical reactions and stoichiometry.

In our exercise, knowing we have 1 mole of neon atoms lets us use Avogadro's number directly. It gives us a starting point to calculate the number of neon atoms in a specific speed range without counting each atom individually.

Speed Distribution

Speed Distribution, in the context of gases, refers to how the speeds of individual gas molecules are spread out. Given a sample of gas, like our 1 mole of neon, molecules will have a variety of speeds.

Not all molecules travel at the same speed due to constant collisions and energy exchanges. Instead, their speeds are distributed across a range. For any specific range, like between 200.00 m/s and 202.00 m/s, we can estimate how many molecules fall within this band.

In most cases, probabilities or distribution functions from Maxwell-Boltzmann must be computed. These calculations often use calculus or involve integral functions for accuracy. However, our problem gives us a shortcut by assuming a constant, even spread of speed probabilities in the given range.

• Gas molecules have varied speed due to collisions.
• Maxwell-Boltzmann Distribution often used for precise distributions.
• Constant probability simplifies tricky calculations in small ranges.

By assuming a uniform distribution, we significantly simplify the task, which is an acceptable approach if precision isn't mandatory. As such, we estimate the number of atoms by a simple fraction, demonstrating a fundamental concept in kinetic theory.