A sealed container contains 1 mole of neon gas at STP. Estimate the number of neon atoms having speeds in the range from 200 m/s to 202 m/s. (Hint: Assume the probability of neon atoms having speeds between 200 m/s and 202 m/s is constant).
To estimate the number of neon atoms with speeds in the given range, we can use the Maxwell-Boltzmann distribution, which describes the distribution of speeds for particles in a gas at a given temperature. The probability that a particle has a speed within a particular range is proportional to the Boltzmann factor, exp(-E/kT), where E is the energy associated with the speed, k is the Boltzmann constant, and T is the temperature.
In this case, we assume that the probability of neon atoms having speeds between 200 m/s and 202 m/s is constant. To estimate the number of atoms within this speed range, we need to determine the total number of neon atoms in the container.
At STP (Standard Temperature and Pressure), 1 mole of any gas occupies 22.4 liters.
Step 1: Convert the volume to m³
Since 1 m³ = 1000 liters, we have:
Volume = 22.4 liters * (1 m³ / 1000 liters) = 0.0224 m³
Step 2: Convert the speed range to SI units
Speed range = 200 m/s to 202 m/s
Step 3: Calculate the total number of neon atoms
To calculate the total number of neon atoms, we can use the ideal gas law. The ideal gas law relates the number of moles of a gas to its volume, pressure, and temperature:
PV = nRT
At STP, the pressure is 1 atm and the temperature is 273.15 K (0°C). The ideal gas constant, R, is 0.0821 L·atm/(mol·K).
Plugging in the values, we have:
(1 atm) * (0.0224 m³) = n * (0.0821 L·atm/(mol·K)) * (273.15 K)
Solving for n (the number of moles), we find:
n = (1 atm * 0.0224 m³) / (0.0821 L·atm/(mol·K) * 273.15 K)
Step 4: Convert moles to atoms
Since 1 mole contains 6.022 x 10^23 atoms (Avogadro's number), we can multiply the number of moles by Avogadro's number to get the total number of neon atoms:
Number of atoms = n * (6.022 x 10^23 atoms/mol)
Step 5: Calculate the number of atoms within the given speed range
Since the probability of neon atoms having speeds between 200 m/s and 202 m/s is assumed to be constant, we can estimate the number of atoms in this range by multiplying the total number of atoms by the fraction of the range within the total speed range:
Number of atoms in range = Total number of atoms * (Fraction of range in total speed range)
The fraction of the range in the total speed range is (202 m/s - 200 m/s) / (total speed range). We can calculate the total speed range by subtracting the minimum speed (200 m/s) from the maximum speed (202 m/s).
Finally, we obtain the estimated number of neon atoms with speeds in the range from 200 m/s to 202 m/s.