Solution Manual For
A Mathematics Boot Camp for Science and Engineering Students, 1E Ying Ma
Chapters 1-10
Topic 1. Physica
l Meaning and Units
1.1. In a multi-component mixture with components, the concentration of component is represented
by either its mass fraction, , or molar fraction, . Let the molar mass of each component be .
a. Find an expression to calculate if and for each component in the mixture are known.
b. Find an expression to calculate if and for each component in the mixture are known.
Solution
a. Consider one mole of the mixture. Because the molar fraction of component is , there are
moles of component and its mass is
The mass fraction of component is then the mass of component divided by the total mass, or:
∑
b. Consider one gram of the mixture. Because the mass fraction of component is , the mass of
component is then , which corresponds to moles of component .
The molar fraction of component is then:
∑
Reflection
This problem gives the formula to convert between mass and molar fractions. In our derivation, we
assumed a fixed amount of the mixture (one mole or one gram), which simplified the reasoning. We could
,have assumed a different amount, and we would arrive at the same result. Note that is typically given
in , while both and are dimensionless.
1.2. The experimentally measured bonding energy of a metallic material is . What is the
bonding energy in the unit of ?
Solution
We can use the factor-label method for the conversion.
1.3. In SI units, the gas constant . Using the factor-label method, express R in the unit
of .
Solution
Using the factor-label method, we have:
Reflection
,While less common, is a valid unit for energy. In fact, gives energy. Thus, a
unit of volume times a unit of pressure results in a unit of energy. Similarly, is also a unit of
energy and . Note that is the abbreviation for cubic centimeter.
1.4. The units of force, energy, pressure, and viscosity1 are dyne (dyn), erg (erg), barye (Ba), and poise
(P), respectively, in the CGS system. Note that symbols for these units are given in the parenthesis.
Show that:
a.
b.
c.
d.
Analysis
In the CGS system, the base unit for length is and the base unit for mass is . To convert the SI unit to
CGS, we need to write the SI unit in base units including and , and then convert them to and ,
respectively.
Solution
a. Because , we have:
b. Because , thus:
c. Because , thus:
1
Viscosity is a measure of a fluid’s resistance to deformation. The SI unit of viscosity is .
, d. Because , thus:
1.5. Answer the following questions.
a. Convert 1 to
b. Express elementary charge, in and .
c. Compare units and . Which one is a bigger unit? If the same charge is expressed in
and , which one has a bigger numerical value (refer to b)?
Solution:
a. is the unit of charge in e.m.u., where the Coulomb’s law is given by:
Consider two point charges of 1 that are 1 apart, then:
. /
( )
So,
Now, we need to find the amount of charge in e.s.u. that gives rise to such a force. In e.s.u.,
So,
√ √ ( )
Thus:
Here c is the speed of light measured in CGS.
A Mathematics Boot Camp for Science and Engineering Students, 1E Ying Ma
Chapters 1-10
Topic 1. Physica
l Meaning and Units
1.1. In a multi-component mixture with components, the concentration of component is represented
by either its mass fraction, , or molar fraction, . Let the molar mass of each component be .
a. Find an expression to calculate if and for each component in the mixture are known.
b. Find an expression to calculate if and for each component in the mixture are known.
Solution
a. Consider one mole of the mixture. Because the molar fraction of component is , there are
moles of component and its mass is
The mass fraction of component is then the mass of component divided by the total mass, or:
∑
b. Consider one gram of the mixture. Because the mass fraction of component is , the mass of
component is then , which corresponds to moles of component .
The molar fraction of component is then:
∑
Reflection
This problem gives the formula to convert between mass and molar fractions. In our derivation, we
assumed a fixed amount of the mixture (one mole or one gram), which simplified the reasoning. We could
,have assumed a different amount, and we would arrive at the same result. Note that is typically given
in , while both and are dimensionless.
1.2. The experimentally measured bonding energy of a metallic material is . What is the
bonding energy in the unit of ?
Solution
We can use the factor-label method for the conversion.
1.3. In SI units, the gas constant . Using the factor-label method, express R in the unit
of .
Solution
Using the factor-label method, we have:
Reflection
,While less common, is a valid unit for energy. In fact, gives energy. Thus, a
unit of volume times a unit of pressure results in a unit of energy. Similarly, is also a unit of
energy and . Note that is the abbreviation for cubic centimeter.
1.4. The units of force, energy, pressure, and viscosity1 are dyne (dyn), erg (erg), barye (Ba), and poise
(P), respectively, in the CGS system. Note that symbols for these units are given in the parenthesis.
Show that:
a.
b.
c.
d.
Analysis
In the CGS system, the base unit for length is and the base unit for mass is . To convert the SI unit to
CGS, we need to write the SI unit in base units including and , and then convert them to and ,
respectively.
Solution
a. Because , we have:
b. Because , thus:
c. Because , thus:
1
Viscosity is a measure of a fluid’s resistance to deformation. The SI unit of viscosity is .
, d. Because , thus:
1.5. Answer the following questions.
a. Convert 1 to
b. Express elementary charge, in and .
c. Compare units and . Which one is a bigger unit? If the same charge is expressed in
and , which one has a bigger numerical value (refer to b)?
Solution:
a. is the unit of charge in e.m.u., where the Coulomb’s law is given by:
Consider two point charges of 1 that are 1 apart, then:
. /
( )
So,
Now, we need to find the amount of charge in e.s.u. that gives rise to such a force. In e.s.u.,
So,
√ √ ( )
Thus:
Here c is the speed of light measured in CGS.
