A molecule of roughly spherical shape has a mass of 6.10 x 10-25 kg and a diameter of 0.70 nm. The uncertainty in the measured position of the molecule is equal to the molecular diameter. What is the minimum uncertainty in the speed of this molecule? (h = 6.626 x 10^34 J • s) The issue is there seems to be a similar question that is asking for just the minimum speed not the minimum uncertainty in the speed and those are clearly different because I found two answer keys with those different questions and different answers. For minimum speed it should be .2 m/s For minimum uncertainty in speed choices are A) 78 m/s B) 7.8 m/s C) .78 m/s <-(might be correct D) .078 m/s E) .0078 m/s 

Expert Answers

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This problem is similar to the previous. The uncertainty principle states that both position and velocity of a particle cannot be measured exactly. Mathematically,

`Delta p*Delta x gt= bar h/2,`

where `p` is for impulse (momentum), which is equal to `m*v,` mass by velocity, and `x` is the position. Correspondingly, `Delta p` means uncertainty of measuring momentum and `Delta x`  means uncertainty of measuring position. Obviously `Delta p = m*Delta v.`

`bar h` is the so-called reduced Planck's constant, `h/(2 pi).` Therefore the minimum uncertainty in the speed is

`Delta v = h/(4pi)*1/(m*Delta x).`

All quantities are given, so the numerical result is

`Delta v approx (6.626*10^(-34))/(4*3.14)*1/(6.10*10^(-25)*0.7*10^(-9)) =(6.626)/(4*3.14*6.10*0.7) approx0.1235 (m/s).`

I took into account that nano- means `10^(-9).`

To obtain the result you want one should "forget" to divide by `2pi,` then it would be about 0.7759 m/s, C. But I'm almost sure about the values.

The second question, about the minimum speed, requires a separate consideration.

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