Question

An atom of xenon has a radius rXe = 108. Pm and an average speed in the gas phase at 25°C of 137. ⁢/ms. Suppose the speed of a xenon atom at 25°C has been measured to within. 10%. Calculate the smallest possible length of box inside of which the atom could be known to be located with certainty. Write your answer as a multiple of rXe and round it to 2 significant figures. For example, if the smallest box the atom could be in turns out to be 42. 0 times the radius of an atom of xenon, you would enter "42. RXe" as your answer

Answer 1

To calculate the smallest possible length of the box inside of which the atom could be known to be located with certainty, we need to determine the range of possible positions of the atom. We can calculate the maximum and minimum speeds and kinetic energies of the xenon atom using the measured speed with 10% uncertainty. Finally, we can use the uncertainty principle to calculate the smallest possible length of the box.

Step-by-step explanation:

To calculate the smallest possible length of the box inside of which the atom could be known to be located with certainty, we need to determine the range of possible positions of the atom. Since the speed of the xenon atom at 25°C has been measured to within 10%, we can calculate the maximum and minimum speeds. The maximum speed would be 137 + 13.7 = 150.7 m/s, and the minimum speed would be 137 - 13.7 = 123.3 m/s.

Using the formula for kinetic energy (KE = 1/2 mv²), we can find the maximum and minimum kinetic energies of the xenon atom. Let's assume the mass of the xenon atom is mXe. The maximum kinetic energy would be KE_max = 1/2 mXe (150.7 m/s)², and the minimum kinetic energy would be KE_min = 1/2 mXe (123.3 m/s)².

The smallest possible length of the box can be calculated using the uncertainty principle, which states that the product of the uncertainty in position and the uncertainty in momentum is greater than or equal to h/4π, where h is the Planck's constant (6.63 x 10^-34 J·s). In this case, the uncertainty in momentum is equal to the uncertainty in kinetic energy.

Thus, the range of possible values for the box length is given by Δx ≥ h/(4π*Δp), where Δp = KE_max - KE_min. Plugging in the values, we have Δx ≥ (6.63 x 10^-34 J·s) / (4π*(KE_max - KE_min)).

Finally, we can calculate the smallest possible length of the box as a multiple of rXe by dividing Δx by rXe and rounding it to 2 significant figures. This gives us the answer.

Answer 2

To calculate the smallest possible length of box inside of which the atom could be known to be located with certainty, we can use the uncertainty principle. The uncertainty principle states that the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of a particle is greater than or equal to a constant value, h/4π.

Step-by-step explanation:

To calculate the smallest possible length of box inside of which the atom could be known to be located with certainty, we can use the uncertainty principle. The uncertainty principle states that the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) of a particle is greater than or equal to a constant value, h/4π. In this case, we have the uncertainty in position as the smallest possible length of the box and the uncertainty in momentum as the mass of the xenon atom multiplied by its average speed.

Using the formula Δx * Δp ≥ h/4π, we can rearrange it to solve for Δx: Δx ≥ h/4π(Δp). Substituting the given values, we have Δx ≥ (6.63 × 10^-34 J s)/(4π)(0.1)(108 Pm)(137 m/s), which gives us Δx ≥ 1.05x10^-18 Pm.

Therefore, the smallest possible length of the box inside of which the atom could be known to be located with certainty is 1.05x10^-18 times the radius of an atom of xenon.