Question

An atom of helium has a radius = and an average speed in the gas phase at of . . Suppose the speed of a helium atom at has been measured to within . 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 and round it to significant figures. For example, if the smallest box the atom could be in turns out to be times the radius of an atom of helium, you would enter "" as your answer.

Answer 1

Answer:

$1.2\times 10^3 rNe$

Explanation:

Given that

Speed of neon = 350 m/s

Un-certainity in speed= (0.01 ÷ 100) × 350

= 0.035 m/s

As per heisenberg uncertainty principle

$\triangle X\times m \triangle \ v\geq \frac{h}{4\pi }$  ....... (i)

substituting the values in equation (i)

$\triangle X = 4.49 \times 10^-^8 m$

In terms of rNe i.e 38 pm = $38\times 10^-^1^2$

$\triangle X = \frac{4.49\times 10^-^8}{38\times 10^-^1^2}$

$= 0.118 \times 10^4 \times (rNe)$

$= 1.18\times 10^3 rN$

$= 1.2 \times 10^3 rNe$

Therefore the smallest possible length of the box inside in which the atom could be known for locating with certainty is $1.2\times 10^3 rNe$

Answer 2

$1.2*10^3 (Ne_r)$

Given that:

• Speed of neon = 350 m/s

• Uncertainty in speed can be calculated as:

$\frac{0.01}{100} *350=0.035 ms^{-1}$  m/s

Heisenberg uncertainty principle:

It states that the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory.

Heisenberg uncertainty principle can be stated as:

Δx Δp ≥ $\frac{h}{4\pi}$

Or

Δx mΔv  ≥ $\frac{h}{4\pi}$ .......... (i)

On substituting the values in equation (i)

Δx =$4.49 * 10^{-8} ms^{-1}$

In terms of $Ne_r$(radius of Neon) i.e. $38\text{pm}= 38 * 10^{12}\text{m}$

Δx = $\frac{4.49*10^{-8}}{38*10^{12}}$

$=0.0118*10^4* (Ne_r)\\\\=1.18*10^3 (Ne_r)\\\\=1.2*10^3 (Ne_r)$

Therefore the smallest possible length of the box inside in which the atom could be known for locating with certainty is $1.2*10^3 (Ne_r)$

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