Go to some English-metric unit conversions

**Problem #1:** How many palladium atoms would it take to encircle the Earth at its equator? (The atomic diameter of Pd is 140 pm; the equatorial diameter of the Earth is 40,075.02 km.)

**Problem #2:** In America, a car's gasoline efficiency is measured in miles/gallon. In Europe, it is measured in km/L. If your car's gas mileage is 40.0 mi/gal, how many liters of gasoline would you need to buy to complete a 142 km trip? Use the following conversions: 1 km = 0.6214 mi and 1 gal = 3.7854 L

**Problem #3:** Convert 2.1 Å to nanometers

**Solution:**

1) Convert Å to meter:

1 Å = 10^{-8}cm (by definition)10

^{-8}cm x (1 m / 100 cm) = 10^{-10}m1 Å = 10

^{-10}m

2) Convert 2.1 x 10^{-10} m to nm:

2.1 x 10^{-10}m times (10^{9}nm / 1 m) = 0.21 nm

**Problem #4:** Convert 0.0030 mm to nm

**Solution:**

3.0 x 10^{-3}mm x (10^{6}nm / 1 mm) = 3.0 x 10^{3}nm

**Problem #5:** Convert 6.25 x 10¯^{4} s to ms

The answer is 0.625 ms

**Problem #6:** Suppose a football, diameter 22 cm, is scaled up so that it becomes as big as the earth, diameter 13000 km. Calculate whether an atom of diameter 0.32 nm will become as big as:

a) a pin head, diameter 1 mm

b) a coin, diameter 1.9 cm

c) a football, diameter 22 cm

d) a weather balloon, diameter 1.8 m

**Solution:**

1) Convert all measurements to the same value. I used meters:

football: 0.22 m

earth: 13000000 m

atom: 3.2 x 10¯^{10}m

2) Solve via a ratio and proportion technique:

football diameter : earth diameter = atom diameter : an unknown value0.22 m ÷ 13000000 m = 3.2 x 10¯

^{10}÷ xSolving for x gives 0.0189 m

3) Answer b (1.9 cm) is the correct answer, since 0.0189 m = 1.9 cm.

**Problem #7:** Convert 10.6 kg/m^{3} to mg/μL

**Solution:**

1) Convert kg to mg:

10.6 kg/m^{3}times (10^{6}mg/kg) = 1.06 x 10^{7}mg/m^{3}

2) Convert m^{3} to L:

I have made a video on this topic: How to Convert between Cubic Meters and Liters.1 m

^{3}= 1000 dm^{3}= 1000 L1.06 x 10

^{7}mg/m^{3}times (1 m^{3}/ 1000 L) = 1.06 x 10^{4}mg/L

3) Convert L to μL:

1.06 x 10^{4}mg/L x (1 L / 10^{6}μL) = 1.06 x 10^{-2}mg/μL = 0.0106 mg/μL

Suppose the problem had been to convert kg/m^{3} to mg/mL? Then, the last part of the solution would look like this:

1.06 x 10^{4}mg/L x (1 L / 10^{3}mL) = 1.06 x 10^{1}mg/mL = 10.6 mg/mL10.6 kg/m

^{3}= 10.6 mg/mL

**Problem #8:** Light travels at a speed of 3.00 x 10^{10} cm/s. What is the speed of light in km/h?

**Solution:**

1) Convert cm/s to km/s:

answer = 3.00 x 10^{5}km/s

2) Convert seconds to hours:

answer = 1.08 x 10^{9}km/hr

**Problem #9:** (a) Convert 38.0 centiliters to kiloliters; (b) How many Mg are there in 100,000,000 mg? (c) Convert 0.050 kg to mg.

**Solution:**

(a) 38.0 cL x (1 kL / 10^{5}cL) = 3.80 x 10¯^{4}kL

(b) 1 x 10^{8}mg x (1 Mg / 10^{9}mg) = 0.1 Mg

0.050 kg x (10^{6}mg / 1 kg) = 5.0 x 10^{4}mg

Note that, in each case, a '1' is associated with the __larger__ of the two units in the conversion factor. Those units are, in turn, kL, Mg, and kg.

**Problem #10:** If an oil tanker carrying 2.0 x 10^{5} metric tons of crude oil of density 0.70 g/cm^{3} spills its entire load at sea, what area of ocean in square kilometers will be polluted if the oil spreads out to a thickness of 0.10 mm? (A metric ton = 10^{3} kg.)

**Solution:**

1) Determine kg of oil:

2.0 x 10^{5}times 1000 = 2.0 x 10^{8}kg of oil

2) Change kg to g:

2.0 x 10^{8}kg times 1000 = 2.0 x 10^{11}g

3) Determine volume of oil:

2.0 x 10^{11}g divided by 0.70 g/cm^{3}= 2.857 x 10^{11}cm^{3}

4) Change 0.10 mm to cm:

0.10 mm times (1 cm / 10 mm) = 0.010 cm

5) Determine area of oil:

2.857 x 10^{11}cm^{3}divided by 0.010 cm = 2.857 x 10^{13}cm^{2}

6) Change cm^{2} to km^{2}:

2.857 x 10^{13}cm^{2}times (1 km^{2}/ 10^{10}cm^{2}) = 2857 km^{2}

7) Comment: 1 km^{2} is 1 km on each side. 1 km = 10^{5} cm, so the 1 km^{2} is 10^{5} cm on each side, so:

10^{5}cm times 10^{5}= 10^{10}cm^{2}(which equals 1 km^{2})

**Problem #11:** A water bed has the following dimensions: 2.00 m x 1.40 m x 15.0 cm. Calculate the mass of water in (kg, please) in the bed ? Assume the density of the water to be 1.00 g/mL.

**Solution:**

1) Change water bed dimensions to cm:

200 cm x 140 cm x 15 cm = 420000 cm^{3}

2) Change volume of water to mass of water:

1cm^{3}= 1 mL, so 420000 cm^{3}= 420000 mL420000 mL x 1.00 g/mL = 420000 g

3) Change g to kg:

420. kg

**Problem #12:** (a) Convert 722 μg to mg. (b) How many pg in a ng? (c) How many mm in a km? (d) How many cm in a km? (e) How many mg in 722 micrograms?

**Solutions:**

(a) 722 μg times (1 mg / 1000 μg) = 0.722 mg

(b) 1 ng times (1000 pg / 1 ng) = 1000 pg (nano- means 10^{-9}and pico- means 10^{-12})

(c) 1 km times (10^{6}mm / 1 km) = 1,000,000 mm

(d) 1 km times (10^{5}cm / 1 km) = 100,000 cm

(e) 722 μg times (1 mg / 1000 μg) = 0.722 mg

Note the reversal of the order of the units in parts (a) and (e) of this problem. They are, of course, asking for the exact same thing.

**Problem #13:** Convert 3000 mL to daL.

**Solution:**
1) milli- means 10^{-3} and deca- means 10^{1}, therefore:

3000 mL times (1 daL / 10^{4}mL) = 0.3 daL