Study Guide in CHE 111: CHEMISTRY FOR ENGINEERS UNITS AND DIMENSIONS

UNITS and DIMENSIONS

At an early stage the student of engineering will discover that the data which he/she uses are expressed in a
great variety of different units, so that he/she must convert his/her quantities into a common system before
proceeding with his/her calculations.
Most of the physical properties determined in the laboratory will have been expressed in the c.g.s. system,
whereas the dimensions of the full-scale plant, its throughput, design, and operating characteristics will have appeared
either in some form of general engineering units or in special units which have their own origin in the history of the
particular industry. This inconsistency is quite unavoidable and is a reflection of the fact that chemical engineering has
in many cases developed as a synthesis of scientific knowledge and practical experience. Familiarity with the various
systems of units and an ability to convert from one another is therefore essential.

Dimensions- these are the basic concepts of measurements such as length, time, mass, temperature and so on.
Units- are a means of expressing the dimensions such as feet or centimeters for length, or hours or seconds in time.

Fundamental Dimensions:
Dimension Symbol
Mass m
Length l
Time t
Temperature T
Force F

Systems of Units

1. The English System (FPS)- the first system of measurement to be developed It is a system that uses biological
standards
2. The Metric System (CGS)- the system of units used for scientific measurements. Metric units are based on decimal
system, related to powers of 10. The first measurement system to use earth as a standard
3. The Systeme International d'Unites (SI) - the established (standard) system of measurement agreed upon in 1960
A measurement system that covers the entire-field-of science and engineering, including electromagnetic and
illumination. It is based on physical rather than biological standards.

A system of units has the following components:

1. Base units for mass, lengths, time, electric current and light intensity.
2. Multiple units, which are define as multiples or fractions of base units such as minutes, hours and
milliseconds, is all of which are defined in terms of the base unit of second. Multiple units are defined
for convenience rather than necessity.
3. Derived units, obtained in two ways:
a) By multiplying or dividing base or multiple units (cm2, ft/min, kg۰m/s2). Derived units of this type
are referred to as compound units
b) As defined as equivalents of compound units (I lb f= 32.174 1bm۰ft/s2)

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International System of Units: The seven (7) base units

Dimensions Unit Definition of unit Symbol

Mass Measure of the kilogram The kilogram is equal to the mass of the
amount of matter international prototype of the kilogram. The kg
contained in an primary standard is a cylinder of Platinum-
object, heaviness lridium alloy. (This cylinder is kept at a Bureau
or lightness of weights and measures at Sevres, France)
Length Measure of meter The meter is the length of the path traveled m
distance by light in a vacuum during a meter time
interval of (1/299,792,458) a second
Time Measure of second The second is the duration 9,192,631,770. S
duration of periods of the radiation corresponding to the
period transition between the two hyperfine levels
of the ground state of the cesium-133 atom.
Or of simply, the duration required for
9,192,631,770 cycles of the Cesium resonator.
Temperature Measure of the Kelvin Kelvin, unit of thermodynamic temperature, K
hotness or is the fraction (1/273.16) of the
coldness of an thermodynamic temperature of the triple
object point of water.
Amount of Dimensionless mole The mole is the amount of substance of a mol
Substance expression of the system which contains as many elementary
number of entities as there are atoms in 0.012 kilogram
particles in a of carbon-12. When the mole is used, the
sample elementary entities must be specified and
may be atoms, ions, electrons, other
molecules, particles, or specified group of
particles.
Electric Measure the flow ampere The ampere is that constant current which if A
Current of electricity maintained in a 2 straight parallel conductors
through a of infinite length of negligible circular cross-
conductor section, and placed 1 meter apart in a
vacuum, would produce between these
conductors a force equal to 2 x 10 Newton
per meter of length
Luminous An expression of candela The candela is the luminous intensity given cd
Intensity the amount of direction of a source that emits
light power monochromatic radiation of frequency 540 x
emanating from a 1012 hertz and that has a radiant intensity in
point of source that direction of (1/683) watt per steradian. It
within a solid is the luminous intensity, in the perpendicular
angle of one direction of a surface of 1/600,000 m of a
solution. black body at a temperature of freezing
platinum under a pressure of 101,325 N/m.

Physical Quantities
Any physical quantity consists of two parts: (a) a unit- tells what the quantity is and gives the standard by
which it is measured (b) a number - tells how many units are needed to make up the quantity
By attaching units to all numbers that are not fundamentally dimensionless, you get the following very
practical benefits:
1. Diminished possibility of inadvertent inversion of any portion of the calculation.
2. Reduced intermediate calculations and time in problem solving.

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Study Guide in CHE 111: CHEMISTRY FOR ENGINEERS UNITS AND DIMENSIONS

3. A logical approach to the problem rather than remembering a formula and plugging into it.
4. Easy interpretation of the physical meaning of the numbers you use.

Physical Symbol Absolute Systems

Quantity Internationally adopted units for ordinary and scientific use
SI cgs fps
Mass m kilogram, kg gram, g pounds,𝑙𝑏𝑚
Length l meter, m centimeter, cm foot, ft
Time t seconds, s seconds, s seconds, s
Temperature T Kelvin, K Degree Centigrade Degree Rankine
Force F Newton (kg.m/s2) Dynes (g.cm/s2) poundal
3 3
Volume V cubic meter, m cubic centimeter, cm cubic foot, ft3
Pressure p 𝑁𝑒𝑤𝑡𝑜𝑛 𝐷𝑦𝑛𝑒𝑠 𝑝𝑜𝑢𝑛𝑑𝑎𝑙
, pascal
𝑚2 𝑐𝑚2
𝑓𝑡 2
2
Work/Energy W/E 𝑘𝑔.𝑚
, Joules erg ft.lbf
𝑠2
Heat H Joule, J Calorie BTU
Power P 𝐽 𝑒𝑟𝑔 𝑓𝑡. 𝑙𝑏𝑓
, watts
𝑠
𝑠 𝑠
Density ρ 𝑘𝑔 𝑔 𝑙𝑏𝑚
𝑚3 𝑐𝑚3 𝑓𝑡 3
Velocity u 𝑚 𝑐𝑚 𝑓𝑡
𝑠 𝑠 𝑠
Acceleration a 𝑚 𝑐𝑚 𝑓𝑡
𝑠2 𝑠2 𝑠2

SI PREFIXES
Factor Prefix Symbol Factor Prefix Symbol Factor Prefix Symbol
1024 Yotta Y 103 Kilo k 10-9 nano n
1021 Zeta Z 102 Hecta h 10-12 pico p
1018 Exa E 101 Deca da 10-15 fempto f
1015 Peta P 10-1 deci d 10-18 atto a
1012 Tera T 10-2 centi c 10-21 zepto z
109 Giga G 10-3 milli m 10-24 yocto y
106 Mega M 10-6 micro μ

Note: The distinction between the uppercase ad lowercase letters should be followed, even if the symbol appears in
the applications where the other lettering is in uppercase.

Conversion of Units and Conversion Factors

RULES IN HANDILING UNITS
1. Treat the units as you would with the algebraic symbols.
2. Add, subtract or equate numerical quantities only if the units of the quantities are the same.
3. In multiplication and division, you can multiply or divide units, but you cannot cancel them out unless they
are the same.
4. When a compound unit is formed by multiplication of two or more units, its symbols consists of the symbols
for separate units joined by a centered dot (). Hyphens should not be used in symbols for compound units.
5. Positive and negative exponents may be used with the symbols for units.
6. If a compound is formed by division of one unit by another, its symbols consist of the symbols for the
separate unit either separated by solidus (0) or multiplied by using negative powers.

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Conversion of Units

➢ A measured quantity can be expressed in terms of any units having the appropriate dimension. The
equivalence between two expressions of the same quantity may be defined in terms of a ratio.

1𝑚 100𝑐𝑚
m= 100 cm can be expressed as or
100𝑐𝑚 1𝑚

The above ratios are known as conversions factors.

➢ To convert a quantity expressed in terms of one unit to its equivalent in terms of another unit multiply the
given quantity by the conversion factor, new unit/old unit.
➢ If you are given a quantity giving a compound unit and you wish to convert it to its equivalent in terms of
another set of units, set up dimensional equation. Write the given quantity and its units on the left, write
the units of conversion factors that cancel the old units and replace them with the desired ones. Fill the
values of the conversion factors and carry out the indicated arithmetic to find the desire value.
➢ Every valid equation must be dimensionally homogeneous, that is, all additive terms on both sides of the
equation must have the same dimensions (dimensional homogeneity)

Common Conversion Factors:

Physical Conversion Factor Physical Conversion Factor

Quantity Quantity
1kg 2.204 lbm 1N 𝑘𝑔. 𝑚
1
2.2 lbm 𝑠2
1000 g 1x105 dyne
Force
1g 1000 mg 1 lbf 4.4482 N
𝑙𝑏𝑚 . 𝑓𝑡
Mass 1 short ton 2000 lbm 32.174
𝑠2
1 long ton 2240 lbm 1 atm 760 mmHg
1 metric ton 1000 kg 29.92 inHg
2200 lbm 14.7 psi
1 slug 32.174 lbm 𝑁
1m 3.28 ft 101,325 2 𝑜𝑟 𝑃𝑎
Pressure 𝑚
100 cm 33.899 ft H2O
1000 mm 1.01325 bar
1 ft 30.48 cm 760 torr
12 inch 1 bar 100 kPa
Length 1 in 2.54 cm 1x105 Pa
1 yard 3 ft 1 Joules 1 N.m
1 mile 5280 ft 1x107 erg
1.609 km 1 BTU 252 cal
1 fathom 6 ft Work/Energy 1054.5 Joules
1 Angstrom 1 x 10-10 m 778 ft.lbf
1 liter 1000 cm3 1 erg 10.409 L.atm
1 m3 1000 liter 1 cal 4.184 Joules
1 Poise 𝑔
1 ft3 7.481 gal 1
Volume 𝑐𝑚. 𝑠
1 gal 3.7854 liter 100 cP
4 quarts 0.1 Pa.s
1 quart 2 pints Viscosity 𝑘𝑔
1 cP 1𝑥10−3
1 hectare 1000 m 2 𝑚. 𝑠
Area 𝑙𝑏𝑚
1 acre 43, 560 ft2 6.72𝑥10−4
𝑓𝑡. 𝑠

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Additional Important Quantities:

Quantity Value
Gravitational acceleration, g 𝑚
9.8
𝑠2
𝑓𝑡
32.174 2
𝑠
Universal gas constant, R 𝐿. 𝑎𝑡𝑚
0.08205
𝑚𝑜𝑙. 𝐾
𝑚3 . 𝑎𝑡𝑚
0.08205
𝑘𝑚𝑜𝑙. 𝐾
𝐵𝑇𝑈
1.987
𝑙𝑏𝑚𝑜𝑙. ⁰𝑅
𝑓𝑡 3 . 𝑎𝑡𝑚
0.7302
𝑙𝑏𝑚𝑜𝑙. ⁰𝑅
𝑓𝑡 3 . 𝑝𝑠𝑖
10.73
𝑙𝑏𝑚𝑜𝑙. ⁰𝑅
𝑚3 . 𝑃𝑎
8.314
𝑚𝑜𝑙. 𝐾
Dimensional Constant, gc 𝑘𝑔. 𝑚
1
𝑁. 𝑠 2
𝑓𝑡. 𝑙𝑏𝑚
32.174
𝑙𝑏𝑓 . 𝑠 2

PROCESS VARIABLES

Variables or parameters are necessary to describe systems and processes. These variables are
properties that can be measured and recorded to define the conditions of a system at any time.

The most common process variables that an engineer in concerned with and measures are:
(a)Mass (d)Composition (g) Pressure
(b)Volume (e)Concentration (h) Flow rate
(c)Density (f) Temperature

MASS AND VOLUME

A. Density (ρ) - it is defined as the mass per unit volume of a substance

- densities of a substance can be used as a conversion factor to relate the mass and the
volume of a quantity of the substance
- densities of pure solids and liquids are essentially independent of pressure and vary
relatively and slightly with temperature
- densities of mixtures vary with composition
𝑚𝑎𝑠𝑠 𝑘𝑔 𝑙𝑏𝑚 𝑔
ρ= 𝑣𝑜𝑙𝑢𝑚𝑒 UNITS: 𝑚3 ; ;
𝑓𝑡 3 𝑐𝑚3

B. Specific Gravity (SG or Sp.Gr.) - the ratio of two densities, that of the substance of interest to
the density of a reference substance

Specific gravity= density of the substance

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density of the reference substance

-The reference used for solids and liquids is water at 4°C while the reference substance used
for gases is usually air at 60°F, or other specified gas.

ρwater at 4C lg/cm3= 1000 kg/m3 = 62.4 Ibm/ft3

Example: SG = 1.2 20°/4° signifies that the specific gravity of the substance at 20°C with
reference to water at 4°C is 1.2.

Note: If the specific gravity of a substance is given, multiply it with the density of the
reference substance in any units to get the density of the substance in the same units. In case the
temperature for which the specific gravity is state are unknown, assume ambient temperature and
4°C respectively

Special Specific Gravity Units:

1. Degrees Baumè (°Bè) - it is a hydrometer scale used to indicate the density of liquids.
-an expression of the specific gravity of a liquid at 60°F in relation to water at 60°F
(a) For liquids heavier than water (SG>I)

°Bè = 145 - 145

SG

(b) For liquids less dense than water (SG<1)

°Bè = 140 - 130

SG

2. Degrees Twadell (°Tw) - an arbitrary hydrometer scale usually used for liquids heavier than
water, mostly used in England. (Example: in the leather industry, to check tanning solutions)
𝜌
°Tw = 200200 (𝜌𝑙𝑖𝑞𝑢𝑖𝑑 − 1)
𝑤𝑎𝑡𝑒𝑟

3. Brix scale (°Brix) - a hydrometer scale calibrated so that the readings at a specified temperature
(in the US, usually at 20°c) is equal to the percentage by weight of sugar in a sugar solution (i.e. the
number of grams of sugar in 100 grams of liquid)
- it describes the sugar content of grape juice from which wine is made (maybe found in wine
labels)
-introduced in 1879 by Adolf F. Brix (Austrian)

400
°𝐵𝑟𝑖𝑥 = − 400
𝑆𝐺

4. Degrees API (°API) - American Petroleum institute (used in the Petroleum industry)

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141.5
°𝐴𝑃𝐼 = − 131.5
𝑆𝐺

C. Specific Volume - the inverse of the density, that is, the volume per unit mass or unit amount of
material.
1
Specific volume = 𝜌

FLOWRATES

A. Mass flowrate, ṁ -is the mass of the material flowing per unit time

𝑚𝑎𝑠𝑠 𝑚𝑎𝑠𝑠 𝑣𝑜𝑙𝑢𝑚𝑒

ṁ= 𝑡𝑖𝑚𝑒
= 𝑣𝑜𝑙𝑢𝑚𝑒
× 𝑡𝑖𝑚𝑒

ṁ = 𝜌ὐ = 𝜌𝑢𝐴 where A – area perpendicular to the direction

of the flow

B. Volumetric flowrate, ὐ - is the volume of a material flowing per unit time

ὐ = 𝑎𝑟𝑒𝑎 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 = 𝐴𝑢

ṁ
ὐ=
𝜌

C. Flow velocity, u- refers to the distance traversed by a material per unit time

𝑣𝑜𝑙𝑢𝑚𝑒 1
𝑢= ×
𝑡𝑖𝑚𝑒 𝑎𝑟𝑒𝑎

ὐ ṁ
𝑢= =
𝐴 𝜌𝐴

D. Mass velocity, G - is the mass flowrate divided by the cross-sectional area of flow

𝑚𝑎𝑠𝑠 1
𝐺= ×
𝑡𝑖𝑚𝑒 𝑎𝑟𝑒𝑎

ṁ 𝜌𝑢𝐴
𝐺= = = 𝜌𝑢
𝐴 𝐴

CONCENTRATION

-This pertains to the amount of dissolved substance (solute) present in a specified amount
of solvent or solution. The concentration is a ratio of two quantities being either of the

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following:

amount of solute or amount of solute

amount of solvent amount of solution

-Knowledge of concentration of solution is very important in quantitative study

-Concentration is expressed in several ways, among them and the most common are
Normality , Molarity (M), and Molality (m)

Ways of Expressing Concentration:

A. In terms of PHYSICAL UNITS:

(a) Weight-weight ratio / Percent by weight
(b) Volume-volume ratio / Percent by volume
(c) Weight-Volume ratio
(d) Parts per million (ppm)- approximately equivalent to mg/kg, μg/g, mg/L,
μg/ml

ppm - this expression is used in reporting trace or small quantities which is very
common in wastewater treatment

mass of solute
ppm solute = 𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 × 106

(e) Parts per billion (ppb)- approximately equivalent to μg/kg, ng/g, ug/L, ng/ml

mass of solute
ppb solute = 𝑡𝑜𝑡𝑎𝑙 𝑚𝑎𝑠𝑠 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 × 109

B. In terms of CHEMICAL UNITS:

(a) NORMALITY, N- the normality of af solution refers to the number of gram
equivalent of solute per liter of solution

𝑚𝑎𝑠𝑠𝑠𝑜𝑙𝑢𝑡𝑒
𝑛𝑜. 𝑜𝑓 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡𝑠 𝑀𝑊𝑠𝑜𝑙𝑢𝑡𝑒 𝑥 𝑓𝑎𝑐𝑡𝑜𝑟 𝑛𝑠𝑜𝑙𝑢𝑡𝑒𝑠 𝑥 𝑓
𝑁= = =
𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝐿) 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 (𝐿) 𝑉𝑙 𝑜𝑓 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛

(b) MOLARITY, M- the Molarity of a solution refers to the number of gram-moles of

solute per liter of solution

moles of solute
M=
moles Liter of solution liter

(c) MOLALITY, m- this is the number of gram-moles of solute present per kilogram
of solvent

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moles of solute moles

𝑚=
kilogram of solvent kg

PRESSURE

➢ One of the most important parameter in process variables

➢ Force per unit area
➢ Force is perpendicular to the area

𝐹 𝑚𝑎 𝑚𝑔
𝑃= = =
𝐴 𝑔𝑐 𝐴 𝑔𝑐 𝐴

𝑚𝑎𝑠𝑠
Since: 𝜌 = 𝑉𝑜𝑙𝑢𝑚𝑒 then 𝑚 = 𝜌𝑉 = ρAh

V=Axh

𝑚𝑔 ρAh g ρhg
Substitute: 𝑃 = 𝑔 = =
𝑐𝐴 𝑔𝑐 𝐴 𝑔𝑐

Pressure is expressed in: Pa or N/m; lbf/ft3; dyne/cm2; psi or Ibf,/in2; atm; bar

Devices used to Measure Pressure:

1. Manometer device for pressure difference
➢ Open-end manometer
➢ Sealed-end manometer
➢ Differential manometer

2. Barometer - device for measuring atmospheric pressure

* Pressure is constant but varies from height to height.

Ways of Expressing Pressure:
1. Atmospheric or Barometric Pressure varies in height
2. Standard Atmosphere= 1 atm (sea level)
3. Gauge Pressure, Pg
4. Absolute Pressure
Pabsolute = Pg + Patm
Pabsolute = Patm - Pvacuum
5. Vacuum or Draft Pressure= negative Pressure
6. Head of a fluid – in terms of height

TEMPERATURE

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Measure of hotness or coldness of-a substance

Measure of the average kinetic energy of a molecule.
thermometer
Discovered by Galileo Galilei in 1592
Then by Gabriel Fahrenheit Degrees(⁰F)
Followed by Andres Celsius in Degrees Celsius (⁰C)
William Thompson who introduced the Kelvin Scale(absolute scale)

Basic Temperature Formula:

TF =1.8 Tc + 32

𝑇𝐹 − 32
Tc =
1.8

𝑇𝐾 = 𝑇𝐶 + 273.15

𝑇𝑅 = 𝑇𝐹 + 460

GAS LAWS

A. Pressure- Volume Relationship

BOYLE'S LAW by Robert Boyle
"The volume of a mixed quantity of gas maintained at constant temperature is inversely
proportional to the pressure".

V1 P1 = V2 P2

B. TEMPERATURE-VOLUME RELATIONSHIP
CHARLES' LAW - by Jacques Charles
"The volume of a fixed amount of gas maintained at constant pressure is directly
proportional to its absolute temperature".

𝑉1 𝑉2
=
𝑇1 𝑇2

C.TEMPERATURE- PRESSURE RELATIONSHIP

AMONTON'S Law by Guillaume Amonton
"At constant volume, the pressure of a gas sample is directly proportional to the absolute
temperature".

𝑃1 𝑃2
=
𝑇1 𝑇2

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D.QUANTITY - VOLUME RELATIONSHIP

AVOGADRO'S LAW
“The volume of gas maintained at a temperature and pressure and directly proportional
to the number of moles of the gas."
𝑉1 𝑉2
=
𝑛1 𝑛2

E. COMBINED GAS LAW

𝑃1 𝑉1 𝑃2 𝑉2
=
𝑇1 𝑇2

F. IDEAL GAS LAW

PV = nRT

Exercises:

1) If glycerine has a specific gravity of 1.261 at 20°C, what is its density in g/cm3; in lb/ft3 ; in
kg/m3?
2) The density of benzene at 60°F is 0.879 g/cm3. What is its specific gravity (SG)?
3) A student needs 15.0 g of ethanol for an experiment. If the density of the alcohol is 0.789 g/ml,
how many milliliters of alcohol are needed?
4) A certain material whose density is 0.7652 g/cm3 is flowing through a 4-cm ID tube at a rate
of 2300kg/hr. Determine (a) the flow velocity, cm/s (b) the mass velocity, kg/cm2.s (c) the
volumetric flow rate, cm3 /s

D=4cm
ṁx= 2300 kg/hr
ρx= 0.7652 g/cm3

5) For the given system, calculate the pressure at the interface and at the bottom of the tank.
Assume: SGoil= 0.82

Patm

Oil 2ft
P1
Water
6ft

P2

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Required: a. pressure at the interphase (P1) b. pressure at the base (P2)

6) The system in the figure below is at 20 °C. If the atmospheric pressure is 101.33 kPa and the
absolute pressure at the bottom of the tank is 237 kPa, what is the specific gravity of fluid x?
(SGoil=0.89 SGHg=13.6)

Oil 1m

2m
water

3m
Fluid X

Hg 0.5 m

Pb= 237 kPa

7) A sample of Freon gas used in air conditioner has a volume of 325L and a pressure of 96.3
KPa at 20°C. What will be the pressure of the gas when its volume is 975L at 20°C.
8) A sample of CO occupies 300 ml at 10 °C and 750 torr. What volume will the gas have at 30 °C
and 750 torr.
9) An inflated balloon has a volume of 6.0L at sea level. It is allowed to ascend in altitude until
the pressure in 0.45 atm. During ascent, the temperature of the gas falls from 22 °C to -21°C.
Calculate the volume of the balloon at its final altitude.
10) The gas pressure in an aerosol can is 1.5 atm at 25°C. What would be the pressure be if the
can was heated to 450 °C?
11) A 0.50 mol sample of oxygen gas is confined at 0°C and 1.0 atm in a cylinder with a movable
piston. The piston compresses the gas so that the final volume is half the initial volume, and the
final pressure is 2.2 atm. Calculate the temperature of the gas at the final state.

PANGASINAN STATE UNIVERSITY 12