Estimate the uncertainty in the speed of an oxygen molecule if its position is known to be +/- 3 nm. The mass of an oxygen molecule is 5.31 x 10^26 kg.
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To estimate the uncertainty in the speed of an oxygen molecule, we can use the uncertainty principle, which states that there is a fundamental limit to how precisely we can simultaneously know both the position and momentum of a particle. Mathematically, it is given by:
Δx * Δp >= h/(4π)
Where Δx is the uncertainty in the position, Δp is the uncertainty in the momentum, and h is the Planck's constant (approximated as 6.626 x 10^-34 J·s).
In this case, the uncertainty in the position is given as +/- 3 nm, which can be converted to meters by dividing by 10^9: Δx = 3 x 10^-9 m.
To calculate the uncertainty in momentum, we need to first calculate the uncertainty in the velocity of the oxygen molecule using the formula:
Δv = Δp / m
Where Δv is the uncertainty in velocity and m is the mass of the oxygen molecule.
Given that the mass of an oxygen molecule is 5.31 x 10^26 kg:
Δv = Δp / (5.31 x 10^26 kg)
Now, we can rearrange the uncertainty principle formula to:
Δp >= h / (4π * Δx)
Plugging in the values, we have:
Δp >= (6.626 x 10^-34 J·s) / (4π * (3 x 10^-9 m))
Simplifying the equation, we find:
Δp >= 1.767 x 10^-25 J·s·m^(-1)
Finally, substituting this value back into the equation for Δv, we get:
Δv >= (1.767 x 10^-25 J·s·m^(-1)) / (5.31 x 10^26 kg)
Calculating the uncertainty in velocity, we find:
Δv >= 3.33 x 10^-52 m/s
Therefore, the uncertainty in the speed of an oxygen molecule would be approximately ± 3.33 x 10^-52 m/s.
To estimate the uncertainty in the speed of an oxygen molecule, we can make use of the uncertainty principle, which states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position and momentum, can be known simultaneously.
The uncertainty principle is expressed mathematically as:
Δx * Δp >= h/4π
Where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is the reduced Planck's constant (h/2π).
In this case, we are given the uncertainty in position as +/- 3 nm, which we can convert to meters: Δx = 6 × 10^-9 m.
To determine the uncertainty in momentum, we need to calculate the average momentum. Momentum (p) is defined as the product of mass (m) and velocity (v): p = m * v.
Given the mass of an oxygen molecule as 5.31 × 10^26 kg, we can use the uncertainty principle to find the uncertainty in momentum.
Δp * Δx = h/4π
Rearranging the equation, we get:
Δp = h/4πΔx
Substituting the values:
Δp = (6.626 × 10^-34 J*s) / (4π * 6 × 10^-9 m)
Calculating this expression, we find:
Δp ≈ 1.38 × 10^-24 kg * m/s
Now, to estimate the uncertainty in the speed of the oxygen molecule, we can use the equation:
v = p / m
Substituting the values:
v = (1.38 × 10^-24 kg * m/s) / (5.31 × 10^26 kg)
Calculating this expression, we find:
v ≈ 2.60 × 10^-51 m/s
Therefore, the estimated uncertainty in the speed of the oxygen molecule is approximately 2.60 × 10^-51 m/s.