An atom of rhodium (Rh) has a diameter of about 2.7×10-8cm.

Question

• What is the radius of a rhodium atom in meters (m)? Express your answer using two significant figures.
• How many Rh atoms would have to be placed side by side to span a distance of 2.5μm? Express your answer using two significant figures.
• If the atom is assumed to be a sphere, what is the volume in m³ of a single Rh atom? Express your answer using two significant figures.
   

Answer 1

An atom of rhodium (Rh) has a diameter of about 2.7×10^-8cm.

• What is the radius of a rhodium atom in meters (m)? Express your answer using two significant figures.

To answer this, one must know the relationship between radius and diameter of a sphere, how to convert from cm to m, and how to round to correct number of significant figures (sig figs).

 

Radius of a sphere is half of the diameter of a sphere;  radius = 0.5 (diameter)

radius = 0.5 (2.7 x 10^-8 cm) = 1.35 x 10^-8 cm

Use dimensional analysis to convert from cm to m

(1.35 x 10^-8 cm)   X    1 m/100 cm  = 1.35 x 10^-10 m

     

Answer must be expressed with two significant figures

Answer must be either 1.3 x 10^-10 m or 1.4 x 10^-10 m;

Rules for rounding numbers:

• if the digit to the right of the last sig fig is 0 to 4, drop the digit(s) to the right of the last sig fig
• if the digit to the right of the last significant figure is 5 to 9, increase the last sig fig by 1 and drop next digit(s)

Therefore 1.35 should be rounded to 1.4 (two sig figs) and the correct answer is radius = 1.4 x 10^-10 m

 

• How many Rh atoms would have to be placed side by side to span a distance of 2.5μm? Express your answer using two significant figures.

To answer this you convert the diameter given in cm to  μm, then use it as an equivalence to make a conversion factor for dimensional analysis, and round to two sig figs.

 

To convert from cm to μm, move the decimal 4 places to the right  (1 cm = 10 000 μm) or increase the sci notation exponent by 4

2.7 x 10^-8 cm  x  10 000 μm/1 cm  =  2.7 x 10^-4 μm;  therefore 1 Rh atom spans 2.7 x 10^-4 μm

        this is an equivalence:  1 Rh atom = 2.7 x 10^-4 μm

 

Use the equivalence to make a conversion factor for dimensional analysis

2.5μm   x   1 Rh atom/ 2.7 x 10^-4 μm  =   9.26 x 10³ Rh atoms

 

Expressed with two sig figs, the answer is 9.3 x 10³ Rh atoms span a distance of 2.5μm

 

• If the atom is assumed to be a sphere, what is the volume in m³ of a single Rh atom? Express your answer using two significant figures.

To answer this you must know the relationship between radius and volume for a sphere, and round to two sig figs

 

V  =   4/3 Π r³         where V is volume, Π is pi, r is radius

              

Substitute the radius value determined in the first part of this problem (r = 1.4 x 10^-10 m)

 

V =  4/3 Π (1.4 x 10^-10 m)³ 

            

V = 1.15 x 10^-29 m³ 

rounded to two sig figs, and the answer is 1.2 x 10^-29 m³