An atom of tungsten has a radius of 193. pm and the average orbital speed of the electrons in it is about ×1.6108/ms.Calculate the least possible uncertainty in a measurement of the speed of an electron in an atom of tungsten. Write your answer as a percentage of the average speed, and round it to 2 significant digits.%

To solve this problem, we need to use Heisenberg's Uncertainty Principle, which states that the more precisely the position of a particle is known, the less precisely its momentum can be known, and vice versa. The principle is often expressed in the form of an inequality:

Δx * Δp ≥ h/4π

where Δx is the uncertainty in position, Δp is the uncertainty in momentum, and h is Planck's constant.

In this case, we're given the radius of the atom (which we'll use as the uncertainty in position) and asked to find the uncertainty in speed (which is related to momentum by the equation p = mv, where m is mass and v is speed).

First, we need to convert the radius from picometers to meters:

193 pm = 193 * 10^-12 m

Next, we can rearrange the uncertainty principle to solve for Δp:

Δp = h/4πΔx

Substituting the given values:

Δp = (6.626 * 10^-34 J*s) / (4π * 193 * 10^-12 m)

Solving this gives us Δp ≈ 2.72 * 10^-25 kg*m/s.

Now we can use the equation p = mv to find Δv:

Δv = Δp / m

We'll use the mass of an electron, which is about 9.11 * 10^-31 kg:

Δv = (2.72 * 10^-25 kg*m/s) / (9.11 * 10^-31 kg)

Solving this gives us Δv ≈ 2.98 * 10^5 m/s.

Finally, we can express this as a percentage of the average speed:

(Δv / 1.6108 * 10^8 m/s) * 100% ≈ 0.185%

Rounding to two significant digits gives us 0.19%. So the least possible uncertainty in a measurement of the speed of an electron in an atom of tungsten is about 0.19% of the average speed.