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.

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

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

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.

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 = \(6.022 \times 10^{23} \times P \times 2.00\)

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 = \(6.022 \times 10^{23} \times P \times 2.00\)