Question

dimensional analysis (unit factor method)
using this method, it is possible to solve many problems by using the relationship of one unit to another. for example, 12 inches = one foot. since these two numbers represent the same value, the fractions 12 in./1 ft. and 1 ft./12 in. are both equal to one. when you multiply another number by the number one, you do not change its value. however, you may change its unit.
example 1: convert 2 miles to inches.
2 miles×5280 ft./1 mile×12 inches/1 ft.=126720 in.
example 2: how many seconds are in 4 days?
4 days×24 hr./1 day×60 min./1 hr.×60 sec./1 min.=345600 sec.
solve each problem. round irrational numbers to the thousandths place.

1. 3 hr. = ______________ sec.
2. 0.035 mg = ______________ cg
3. 5.5 kg = ______________ lb.
4. 2.5 yd. = ______________ in.
5. 1.3 yr. = ______________ hr.
6. 3 moles = ______________ molecules (1 mole = 6.02×10²³ molecules)
7. 2.5×10²⁴ molecules = ______________ moles
8. 5 moles = ______________ liters (1 mole = 22.4 liters)
9. 100. liters = ______________ moles
10. 50. liters = ______________ molecules
11. 5.0×10²⁴ molecules = ______________ liters
12. 7.5×10³ ml = ______________ liters

Explanation:

Step1: Convert hours to seconds

We know that 1 hour = 3600 seconds. So for 3 hours, we multiply 3 by 3600.
$3\times3600 = 10800$

Step2: Convert milligrams to centigrams

Since 1 centigram = 10 milligrams, to convert 0.035 mg to cg, we divide 0.035 by 10.
$0.035\div10=0.0035$

Step3: Convert kilograms to pounds

1 kg is approximately 2.205 lb. For 5.5 kg, we multiply 5.5 by 2.205.
$5.5\times2.205 = 12.1275\approx12.128$

Step4: Convert yards to inches

1 yard = 36 inches. For 2.5 yards, we multiply 2.5 by 36.
$2.5\times36=90$

Step5: Convert years to hours

1 year has 365 days (ignoring leap - years for simplicity), 1 day has 24 hours. For 1.3 years, we calculate $1.3\times365\times24$.
$1.3\times365\times24=11388$

Step6: Convert moles to molecules

1 mole = $6.02\times10^{23}$ molecules. For 3 moles, we multiply 3 by $6.02\times10^{23}$.
$3\times6.02\times10^{23}=1.806\times 10^{24}$

Step7: Convert molecules to moles

If 1 mole = $6.02\times10^{23}$ molecules, then for $2.5\times10^{24}$ molecules, we divide by $6.02\times10^{23}$.
$\frac{2.5\times10^{24}}{6.02\times10^{23}}\approx4.153$

Step8: Convert moles to liters

1 mole = 22.4 liters. For 5 moles, we multiply 5 by 22.4.
$5\times22.4 = 112$

Step9: Convert liters to moles

If 1 mole = 22.4 liters, then for 100 liters, we divide 100 by 22.4.
$\frac{100}{22.4}\approx4.464$

Step10: Convert liters to molecules

First, find moles from liters. If 1 mole = 22.4 liters, for 50 liters, the number of moles is $\frac{50}{22.4}$. Then convert moles to molecules.
Number of moles $n=\frac{50}{22.4}\approx2.232$
Number of molecules $=2.232\times6.02\times10^{23}\approx1.344\times10^{24}$

Step11: Convert molecules to liters

First, find moles from molecules. If 1 mole = $6.02\times10^{23}$ molecules, for $5.0\times10^{24}$ molecules, the number of moles is $\frac{5.0\times10^{24}}{6.02\times10^{23}}\approx8.306$. Then convert moles to liters.
Number of liters $=8.306\times22.4\approx186.054$

Step12: Convert milliliters to liters

Since 1 liter = 1000 mL, for $7.5\times10^{3}$ mL, we divide by 1000.
$\frac{7.5\times10^{3}}{1000}=7.5$

Answer:

1. 10800
2. 0.0035
3. 12.128
4. 90
5. 11388
6. $1.806\times 10^{24}$
7. 4.153
8. 112
9. 4.464
10. $1.344\times10^{24}$
11. 186.054
12. 7.5