Math

QuestionHow many thorium atoms (240 pm radius) fit in a distance of 1.40 mm1.40 \mathrm{~mm}?

Studdy Solution

STEP 1

Assumptions1. The atomic radius of thorium is240 pm (picometers) .1 picometer (pm) is equal to 1×10121 \times10^{-12} meters3. The distance to be spanned is1.40 mm (millimeters) 4.1 millimeter (mm) is equal to 1×1031 \times10^{-3} meters5. The atoms of thorium are lined up side by side without any gap between them

STEP 2

First, we need to convert the atomic radius of thorium from picometers to millimeters. We can do this by using the conversion factor that1 pm equals 1×10121 \times10^{-12} meters and1 mm equals 1×101 \times10^{-} meters.
Atomicradius(inmm)=Atomicradius(inpm)×1×1012meters1pm×1mm1×10metersAtomic\, radius\, (in\, mm) = Atomic\, radius\, (in\, pm) \times \frac{1 \times10^{-12}\, meters}{1\, pm} \times \frac{1\, mm}{1 \times10^{-}\, meters}

STEP 3

Now, plug in the given value for the atomic radius of thorium to calculate the atomic radius in millimeters.
Atomicradius(inmm)=240pm×1×1012meters1pm×1mm1×103metersAtomic\, radius\, (in\, mm) =240\, pm \times \frac{1 \times10^{-12}\, meters}{1\, pm} \times \frac{1\, mm}{1 \times10^{-3}\, meters}

STEP 4

Calculate the atomic radius in millimeters.
Atomicradius(inmm)=240pm×1×1012meters1pm×1mm1×103meters=2.4×104mmAtomic\, radius\, (in\, mm) =240\, pm \times \frac{1 \times10^{-12}\, meters}{1\, pm} \times \frac{1\, mm}{1 \times10^{-3}\, meters} =2.4 \times10^{-4}\, mm

STEP 5

Since the atomic radius is half the diameter, we need to multiply the atomic radius by2 to get the diameter of one thorium atom.
iameterofonethoriumatom=2×Atomicradius(inmm)iameter\, of\, one\, thorium\, atom =2 \times Atomic\, radius\, (in\, mm)

STEP 6

Plug in the value for the atomic radius in millimeters to calculate the diameter of one thorium atom.
iameterofonethoriumatom=2×2.4×104mmiameter\, of\, one\, thorium\, atom =2 \times2.4 \times10^{-4}\, mm

STEP 7

Calculate the diameter of one thorium atom.
iameterofonethoriumatom=2×2.4×104mm=4.×104mmiameter\, of\, one\, thorium\, atom =2 \times2.4 \times10^{-4}\, mm =4. \times10^{-4}\, mm

STEP 8

Now that we have the diameter of one thorium atom, we can find the number of thorium atoms required to span a distance of1.40 mm. This can be done by dividing the total distance by the diameter of one thorium atom.
Numberofthoriumatoms=Totaldistance/DiameterofonethoriumatomNumber\, of\, thorium\, atoms = Total\, distance / Diameter\, of\, one\, thorium\, atom

STEP 9

Plug in the values for the total distance and the diameter of one thorium atom to calculate the number of thorium atoms.
Numberofthoriumatoms=.40mm/4.8×4mmNumber\, of\, thorium\, atoms =.40\, mm /4.8 \times^{-4}\, mm

STEP 10

Calculate the number of thorium atoms.
Numberofthoriumatoms=.40mm/4.8×104mm=2916666.67Number\, of\, thorium\, atoms =.40\, mm /4.8 \times10^{-4}\, mm =2916666.67Since we can't have a fraction of an atom, we round up to the nearest whole number. So, approximately2916667 thorium atoms would have to be lined up side by side to span a distance of.40 mm.

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