chemical problem solving using dimensional analysis

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons

Margin Size

• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

1.6: Dimensional Analysis

• Last updated
• Save as PDF
• Page ID 21696

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

Learning Objectives

• To be introduced to the dimensional analysis and how it can be used to aid basic chemistry problem solving.
• To use dimensional analysis to identify whether an equation is set up correctly in a numerical calculation
• To use dimensional analysis to facilitate the conversion of units.

Dimensional analysis is amongst the most valuable tools physical scientists use. Simply put, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors. This is valuable because certain measurements are more accurate or easier to find than others.

A Macroscopic Example: Party Planning

If you have every planned a party, you have used dimensional analysis. The amount of beer and munchies you will need depends on the number of people you expect. For example, if you are planning a Friday night party and expect 30 people you might estimate you need to go out and buy 120 bottles of sodas and 10 large pizza's. How did you arrive at these numbers? The following indicates the type of dimensional analysis solution to party problem:

\[(30 \; \cancel{humans}) \times \left( \dfrac{\text{4 sodas}}{1 \; \cancel{human}} \right) = 120 \; \text{sodas} \label{Eq1} \]

\[(30 \; \cancel{humans}) \times \left( \dfrac{\text{0.333 pizzas}}{1 \; \cancel{human}} \right) = 10 \; \text{pizzas} \label{Eq2} \]

Notice that the units that canceled out are lined out and only the desired units are left (discussed more below). Finally, in going to buy the soda, you perform another dimensional analysis: should you buy the sodas in six-packs or in cases?

\[(120\; { sodas}) \times \left( \dfrac{\text{1 six pack}}{6\; {sodas}} \right) = 20 \; \text{six packs} \label{Eq3} \]

\[(120\; {sodas}) \times \left( \dfrac{\text{1 case }}{24\; {sodas}} \right) = 5 \; \text{cases} \label{Eq4} \]

Realizing that carrying around 20 six packs is a real headache, you get 5 cases of soda instead.

In this party problem, we have used dimensional analysis in two different ways:

• In the first application (Equations \(\ref{Eq1}\) and Equation \(\ref{Eq2}\)), dimensional analysis was used to calculate how much soda is needed need. This is based on knowing: (1) how much soda we need for one person and (2) how many people we expect; likewise for the pizza.
• In the second application (Equations \(\ref{Eq3}\) and \(\ref{Eq4}\)), dimensional analysis was used to convert units (i.e. from individual sodas to the equivalent amount of six packs or cases)

Using Dimensional Analysis to Convert Units

Consider the conversion in Equation \(\ref{Eq3}\):

\[(120\; {sodas}) \times \left( \dfrac{\text{1 six pack}}{6\; {sodas}} \right) = 20 \; \text{six packs} \label{Eq3a} \]

If we ignore the numbers for a moment, and just look at the units (i.e. dimensions ), we have:

\[\text{soda} \times \left(\dfrac{\text{six pack}}{\text{sodas}}\right) \nonumber \]

We can treat the dimensions in a similar fashion as other numerical analyses (i.e. any number divided by itself is 1). Therefore:

\[\text{soda} \times \left(\dfrac{\text{six pack}}{\text{sodas}}\right) = \cancel{\text{soda}} \times \left(\dfrac{\text{six pack}}{\cancel{\text{sodas}}}\right) \nonumber \]

So, the dimensions of the numerical answer will be "six packs".

How can we use dimensional analysis to be sure we have set up our equation correctly? Consider the following alternative way to set up the above unit conversion analysis:

\[ 120 \cancel{\text{soda}} \times \left(\dfrac{\text{6 sodas}}{\cancel{\text{six pack}}}\right) = 720 \; \dfrac{\text{sodas}^2}{\text{1 six pack}} \nonumber \]

• While it is correct that there are 6 sodas in one six pack, the above equation yields a value of 720 with units of sodas 2 /six pack .
• These rather bizarre units indicate that the equation has been setup incorrectly (and as a consequence you will have a ton of extra soda at the party).

Using Dimensional Analysis in Calculations

In the above case it was relatively straightforward keeping track of units during the calculation. What if the calculation involves powers, etc? For example, the equation relating kinetic energy to mass and velocity is:

\[E_{kinetics} = \dfrac{1}{2} \text{mass} \times \text{velocity}^2 \label{KE} \]

An example of units of mass is kilograms (kg) and velocity might be in meters/second (m/s). What are the dimensions of \(E_{kinetic}\)?

\[(kg) \times \left( \dfrac{m}{s} \right)^2 = \dfrac{kg \; m^2}{s^2} \nonumber \]

The \(\frac{1}{2}\) factor in Equation \ref{KE} is neglected since pure numbers have no units. Since the velocity is squared in Equation \ref{KE}, the dimensions associated with the numerical value of the velocity are also squared. We can double check this by knowing the the Joule (\(J\)) is a measure of energy, and as a composite unit can be decomposed thusly:

\[1\; J = kg \dfrac{m^2}{s^2} \nonumber \]

Units of Pressure

Pressure ( P ) is a measure of the Force ( F ) per unit area ( A ):

\[ P =\dfrac{F}{A} \nonumber \]

Force, in turn, is a measure of the acceleration (\(a\)) on a mass (\(m\)):

\[ F= m \times a \nonumber \]

Thus, pressure (\(P\)) can be written as:

\[ P= \dfrac{m \times a}{A} \nonumber \]

What are the units of pressure from this relationship? ( Note: acceleration is the change in velocity per unit time )

\[ P =\dfrac{kg \times \frac{\cancel{m}}{s^2}}{m^{\cancel{2}}} \nonumber \]

We can simplify this description of the units of Pressure by dividing numerator and denominator by \(m\):

\[ P =\dfrac{\frac{kg}{s^2}}{m}=\dfrac{kg}{m\; s^2} \nonumber \]

In fact, these are the units of a the composite Pascal ( Pa ) unit and is the SI measure of pressure.

Performing Dimensional Analysis

The use of units in a calculation to ensure that we obtain the final proper units is called dimensional analysis . For example, if we observe experimentally that an object’s potential energy is related to its mass, its height from the ground, and to a gravitational force, then when multiplied, the units of mass, height, and the force of gravity must give us units corresponding to those of energy.

Energy is typically measured in joules, calories, or electron volts (eV), defined by the following expressions:

• 1 J = 1 (kg·m 2 )/s 2 = 1 coulomb·volt
• 1 cal = 4.184 J
• 1 eV = 1.602 × 10 −19 J

Performing dimensional analysis begins with finding the appropriate conversion factors . Then, you simply multiply the values together such that the units cancel by having equal units in the numerator and the denominator. To understand this process, let us walk through a few examples.

Example \(\PageIndex{1}\)

Imagine that a chemist wants to measure out 0.214 mL of benzene, but lacks the equipment to accurately measure such a small volume. The chemist, however, is equipped with an analytical balance capable of measuring to \(\pm 0.0001 \;g\). Looking in a reference table, the chemist learns the density of benzene (\(\rho=0.8765 \;g/mL\)). How many grams of benzene should the chemist use?

\[0.214 \; \cancel{mL} \left( \dfrac{0.8765\; g}{1\;\cancel{mL}}\right)= 0.187571\; g \nonumber \]

Notice that the mL are being divided by mL, an equivalent unit. We can cancel these our, which results with the 0.187571 g. However, this is not our final answer, since this result has too many significant figures and must be rounded down to three significant digits. This is because 0.214 mL has three significant digits and the conversion factor had four significant digits. Since 5 is greater than or equal to 5, we must round the preceding 7 up to 8.

Hence, the chemist should weigh out 0.188 g of benzene to have 0.214 mL of benzene.

Example \(\PageIndex{2}\)

To illustrate the use of dimensional analysis to solve energy problems, let us calculate the kinetic energy in joules of a 320 g object traveling at 123 cm/s.

To obtain an answer in joules, we must convert grams to kilograms and centimeters to meters. Using Equation \ref{KE}, the calculation may be set up as follows:

\[ \begin{align*} KE &=\dfrac{1}{2}mv^2=\dfrac{1}{2}(g) \left(\dfrac{kg}{g}\right) \left[\left(\dfrac{cm}{s}\right)\left(\dfrac{m}{cm}\right) \right]^2 \\[4pt] &= (\cancel{g})\left(\dfrac{kg}{\cancel{g}}\right) \left(\dfrac{\cancel{m^2}}{s^2}\right) \left(\dfrac{m^2}{\cancel{cm^2}}\right) = \dfrac{kg⋅m^2}{s^2} \\[4pt] &=\dfrac{1}{2}320\; \cancel{g} \left( \dfrac{1\; kg}{1000\;\cancel{g}}\right) \left[\left(\dfrac{123\;\cancel{cm}}{1 \;s}\right) \left(\dfrac{1 \;m}{100\; \cancel{cm}}\right) \right]^2=\dfrac{0.320\; kg}{2}\left[\dfrac{123 m}{s(100)}\right]^2 \\[4pt] &=\dfrac{1}{2} 0.320\; kg \left[ \dfrac{(123)^2 m^2}{s^2(100)^2} \right]= 0.242 \dfrac{kg⋅m^2}{s^2} = 0.242\; J \end{align*} \nonumber \]

Alternatively, the conversions may be carried out in a stepwise manner:

Step 1: convert \(g\) to \(kg\)

\[320\; \cancel{g} \left( \dfrac{1\; kg}{1000\;\cancel{g}}\right) = 0.320 \; kg \nonumber \]

Step 2: convert \(cm\) to \(m\)

\[123\;\cancel{cm} \left(\dfrac{1 \;m}{100\; \cancel{cm}}\right) = 1.23\ m \nonumber \]

Now the natural units for calculating joules is used to get final results

\[ \begin{align*} KE &=\dfrac{1}{2} 0.320\; kg \left(1.23 \;ms\right)^2 \\[4pt] &=\dfrac{1}{2} 0.320\; kg \left(1.513 \dfrac{m^2}{s^2}\right)= 0.242\; \dfrac{kg⋅m^2}{s^2}= 0.242\; J \end{align*} \nonumber \]

Of course, steps 1 and 2 can be done in the opposite order with no effect on the final results. However, this second method involves an additional step.

Example \(\PageIndex{3}\)

Now suppose you wish to report the number of kilocalories of energy contained in a 7.00 oz piece of chocolate in units of kilojoules per gram.

To obtain an answer in kilojoules, we must convert 7.00 oz to grams and kilocalories to kilojoules. Food reported to contain a value in Calories actually contains that same value in kilocalories. If the chocolate wrapper lists the caloric content as 120 Calories, the chocolate contains 120 kcal of energy. If we choose to use multiple steps to obtain our answer, we can begin with the conversion of kilocalories to kilojoules:

\[120 \cancel{kcal} \left(\dfrac{1000 \;\cancel{cal}}{\cancel{kcal}}\right)\left(\dfrac{4.184 \;\cancel{J}}{1 \cancel{cal}}\right)\left(\dfrac{1 \;kJ}{1000 \cancel{J}}\right)= 502\; kJ \nonumber \]

We next convert the 7.00 oz of chocolate to grams:

\[7.00\;\cancel{oz} \left(\dfrac{28.35\; g}{1\; \cancel{oz}}\right)= 199\; g \nonumber \]

The number of kilojoules per gram is therefore

\[\dfrac{ 502 \;kJ}{199\; g}= 2.52\; kJ/g \nonumber \]

Alternatively, we could solve the problem in one step with all the conversions included:

\[\left(\dfrac{120\; \cancel{kcal}}{7.00\; \cancel{oz}}\right)\left(\dfrac{1000 \;\cancel{cal}}{1 \;\cancel{kcal}}\right)\left(\dfrac{4.184 \;\cancel{J}}{1 \; \cancel{cal}}\right)\left(\dfrac{1 \;kJ}{1000 \;\cancel{J}}\right)\left(\dfrac{1 \;\cancel{oz}}{28.35\; g}\right)= 2.53 \; kJ/g \nonumber \]

The discrepancy between the two answers is attributable to rounding to the correct number of significant figures for each step when carrying out the calculation in a stepwise manner. Recall that all digits in the calculator should be carried forward when carrying out a calculation using multiple steps. In this problem, we first converted kilocalories to kilojoules and then converted ounces to grams.

Converting Between Units: Converting Between Units, YouTube(opens in new window) [youtu.be]

Dimensional analysis is used in numerical calculations, and in converting units. It can help us identify whether an equation is set up correctly (i.e. the resulting units should be as expected). Units are treated similarly to the associated numerical values, i.e., if a variable in an equation is supposed to be squared, then the associated dimensions are squared, etc.

Contributors and Attributions

• Mark Tye (Diablo Valley College)

Mike Blaber ( Florida State University )

6 Dimensional Analysis

LumenLearning

Converting from One Unit to Another

Converting units using dimensional analysis makes working with large and small measurements more convenient.

LEARNING OBJECTIVES

Describe the purpose of unit analysis.

KEY TAKEAWAYS

• Dimensional analysis is the process of converting between units.
• The International System of Units (SI) specifies a set of seven base units from which all other units of measurement are formed. Derived units are based on those seven base units.
• Unit analysis is a form of proportional reasoning where a given measurement can be multiplied by a known proportion or ratio to give a result having a different unit or dimension.
• Dimensional analysis involves using conversion factors, which are ratios of related physical quantities expressed in the desired units.
• dimensional analysis : A method of converting from one unit to another. It is also sometimes called unit conversion.

Base and Derived Units

For most quantities, a unit is absolutely necessary to communicate values of that physical quantity. Imagine you need to buy some rope to tie something onto the roof of a car. How would you tell the salesperson how much rope you need without using some unit of measurement?

However, not all quantities require a unit of their own. Using physical laws, units of quantities can be expressed as combinations of units of other quantities. Therefore, only a small set of units is required. These units are called base units , and other units are derived units . Derived units are a matter of convenience, as they can be expressed in terms of basic units.

Different systems of units are based on different choices of base units. The most widely used system of units is the International System of Units, or SI. There are seven SI base units, and all other SI units can be derived from these base units.

The seven base SI units are: [Physical Quantity: unit symbol (unit name)]

• Length: m (meter)
• Mass: kg (kilogram)
• Time: s (second)
• Electric Current: A (Ampere)
• Thermodynamic Temperature: K (degrees Kelvin)
• Amount of Substance: mol (mole)
• Luminous Intensity: cd (candela)

The base units of SI are actually not the smallest set possible; smaller sets have been defined. For example, there are unit sets in which the electric and magnetic field have the same unit. This is based on physical laws that show that electric and magnetic fields are actually different manifestations of the same phenomenon.

Derived units are based on units from the SI system of units. For example, volume is a derived unit because volume is based on length. To calculate the volume of something, you multiply the width x length x height, all in meters. Therefore, the derived unit for volume is m 3 . Here is a list of some commonly derived units:

• Volume: m 3
• Velocity: m/s
• Acceleration: m/s 2
• Density: g/mL or g/cm 3
• Force: [latex]kg \cdot m/s^2[/latex], or the Newton (N)
• Energy: [latex]N \cdot m[/latex], or the Joule (J)

Dimensional Analysis

Sometimes, it is necessary to deal with measurements that are very small (as in the size of an atom) or very large (as in numbers of atoms). In these cases, it is often necessary to convert between units of metric measurement. For example, a mass measured in grams may be more convenient to work with if it was expressed in mg (10 –3 g). Converting between metric units is called unit analysis or dimensional analysis.

Unit analysis is a form of proportional reasoning where a given measurement can be multiplied by a known proportion or ratio to give a result having a different unit or dimension. Algebraically, we know that any number multiplied by one will be unchanged. If, however, the number has units, and we multiply it by a ratio containing units, the units in the number will multiply and divide by the units of the ratio, giving the original number (remember you are multiplying by one) but with different units.

This method can be generalized as multiply or divide a given number by a known ratio to find your answer. The given number is a numerical quantity (with its units). The ratios used are based upon the units and are set up so that the units in the denominator of the ratio match the numerator units of the given and the units in the numerator of the ratio match those in either the next ratio or the final answer. When these are multiplied, the given number will now have the correct units for your answer.

“Converting Units with Conversion Factors” – YouTube: How to convert units using conversion factors and canceling units.

If you had a sample of a substance with a mass of 0.0034 grams and you wanted to express that mass in mg, you could use the following dimensional analysis. The given quantity is the mass of 0.0034 grams. The quantity that you want to find is the mass in mg, and we know that 1 mg = 10 -3 g. Expressing this as a proportion or ratio, there is one mg per 10 -3 grams, or 1000 mg/1 g.

Therefore, 0.0034g x (1000 mg/1 g) = 3.4 mg

Strategy for General Problem Solving

To convert a measured quantity to a different unit of measure without changing the relative amount, use a conversion factor.

Apply knowledge of dimensional analysis to convert between units in chemistry problems.

• Chemistry, along with other sciences and engineering, makes use of many different units.
• In mathematics and chemistry, a conversion factor is used to convert a measured quantity to a different unit of measure without changing the relative amount.
• Units behave just like numbers in products and quotients—they can be multiplied and divided.
• conversion factor : A conversion factor changes one unit to a new unit.

Chemistry, along with other sciences and engineering, makes use of many different units. Some of the common ones include mass (ton, pounds, ounces, grains, grams), length (yard, feet, inches, meters), and energy (Joule, erg, kcal, eV). Since there are so many different units that can be used, it is necessary to be able to convert between the various units. To do this, one uses a conversion factor.

In mathematics, specifically algebra, a conversion factor is used to convert a measured quantity to a different unit of measure without changing the relative amount. To accomplish this, a ratio (fraction) is established that equals one (1). In the ratio, the conversion factor is a multiplier that, when applied to the original unit, converts the original unit into a new unit, by multiplication with the ratio.

When doing dimensional analysis problems, follow this list of steps:

• Identify the given (see previous concept for additional information).
• Identify conversion factors that will help you get from your original units to your desired unit.
• Set up your equation so that your undesired units cancel out to give you your desired units. A unit will cancel out if it appears in both the numerator and the denominator during the equation.
• Multiply through to get your final answer. Don’t forget the units and sig figs (significant figures)!

EXAMPLE PROBLEM 1

Here is an example problem: How many hours are in 3 days?

• Identify the given: 3 days
• Identify conversion factors that will help you get from your original units to your desired unit: [latex]\frac{24 \text{ hours}}{1 \text{ day}}[/latex]
• Set up your equation so that your undesired units cancel out to give you your desired units: [latex]3 \text{ days} \cdot \frac{24 \text{ hours}}{1 \text{ day}}[/latex]
• Multiply through to get your final answer: 72 hours

Flipping the Conversion Factor

Don’t forget that if need be, you can flip a conversion factor. After all, if a = b, then a/b = 1 and b/a = 1. For example, days are converted to hours by multiplying the days by the conversion factor of 24. The conversion can be reversed by dividing the hours by 24 to get days. The reciprocal 1/24 could be considered the reverse conversion factor for an hours-to-days conversion. The term “conversion factor” is the multiplier, not divisor, which yields the result.

Consider the relationship between feet and inches.

1 foot = 12 inches

1 foot/12 inches = 1 = 12 inches/1 foot.

Both fractions are equal to 1. If the units are ignored, the quotients do not numerically equal 1, but 1/12 or 12. However, with the inclusions of the units, both the numerators and denominators describe the exact same length, so the quotients are equal to 1. Since the two quotients are equal to 1, multiplying or dividing by the quotients is the same as multiplying or dividing by 1. It does not change the equation, only the relative numerical values within the various units.

EXAMPLE PROBLEM 2

You can also use these quotients to convert from inches to feet or from feet to inches. For example, how many inches are in 5 feet?

• The given is 5 feet.
• The conversion factor is [latex]\frac{12 \text{ inches}}{1 \text{ foot}}[/latex]
• Set up the equation: [latex]5 \text{ feet} \cdot \frac{12 \text{ inches}}{1 \text{ foot}}[/latex]
• Multiply through: 60 inches

Another example is: how many feet are in 30 inches?

[latex]30 \text{ inches} \cdot \frac{1 \text{ foot}}{12 \text{ inches}} = 2.5 \text{ feet}[/latex]

If there is confusion regarding which quotient to use in the conversion, just make sure the units cancel out correctly. In the first equation, the unit (feet) is in both the numerator and denominator of the expression, so they cancel. The units behave just like numbers in products and quotients—they can be multiplied and divided.

LICENSES AND ATTRIBUTIONS

Cc licensed content, shared previously.

• Curation and Revision. Provided by : Boundless.com. License : CC BY-SA: Attribution-ShareAlike

CC LICENSED CONTENT, SPECIFIC ATTRIBUTION

• Units of measurement.  Provided by : Wikipedia.  Located at :  http://en.wikipedia.org/wiki/Units_of_measurement%23Base_and_derived_units .  License :  CC BY-SA: Attribution-ShareAlike
• Mathematics for Chemistry /”Units and dimensions.” Provided by : Wikibooks.  Located at :  http://en.wikibooks.org/wiki/Mathematics_for_Chemistry/Units_and_dimensions .  License :  CC BY-SA: Attribution-ShareAlike
• Introductory Chemistry Online /”Measurements and Atomic Structure.” Provided by : Wikibooks.  Located at :  http://en.wikibooks.org/wiki/Introductory_Chemistry_Online/Measurements_and_Atomic_Structure%23.C2.A0.C2.A01.5.09Unit_Conversion_within_the_Metric_System .  License :  CC BY-SA: Attribution-ShareAlike
• strobilus.  Provided by : Wiktionary.  Located at :  http://en.wiktionary.org/wiki/strobilus .  License :  CC BY-SA: Attribution-ShareAlike
• International System of Units.  Provided by : Wikipedia.  Located at :  http://en.wikipedia.org/wiki/International%20System%20of%20Units .  License :  CC BY-SA: Attribution-ShareAlike
• dimensional analysis.  Provided by : Wiktionary.  Located at :  http://en.wiktionary.org/wiki/dimensional_analysis .  License :  CC BY-SA: Attribution-ShareAlike
• “Converting Units with Conversion Factors” – YouTube. Located at :  http://www.youtube.com/watch?v=7N0lRJLwpPI .  License :  Public Domain: No Known Copyright .  License Terms : Standard YouTube license
• Steve Davis, Dimensional Analysis. September 17, 2013.  Provided by : OpenStax CNX.  Located at :  http://cnx.org/content/m42293/latest/ .  License :  CC BY: Attribution
• Conversion factor.  Provided by : Wikipedia.  Located at :  http://en.wikipedia.org/wiki/Conversion_factor .  License :  CC BY-SA: Attribution-ShareAlike
• conversion factor.  Provided by : Wikipedia.  Located at :  http://en.wikipedia.org/wiki/conversion%20factor .  License :  CC BY-SA: Attribution-ShareAlike

This chapter is an adaptation of the chapter “ Dimensional Analysis ” in Boundless Chemistry by LumenLearning and is licensed under a CC BY-SA 4.0 license.

Dimensional Analysis Copyright © by LumenLearning is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

Internet Archive Audio

• This Just In
• Grateful Dead
• Old Time Radio
• 78 RPMs and Cylinder Recordings
• Audio Books & Poetry
• Computers, Technology and Science
• Music, Arts & Culture
• News & Public Affairs
• Spirituality & Religion
• Radio News Archive

• Flickr Commons
• Occupy Wall Street Flickr
• NASA Images
• Solar System Collection
• Ames Research Center

• All Software
• Old School Emulation
• MS-DOS Games
• Historical Software
• Classic PC Games
• Software Library
• Kodi Archive and Support File
• Vintage Software
• CD-ROM Software
• CD-ROM Software Library
• Software Sites
• Tucows Software Library
• Shareware CD-ROMs
• Software Capsules Compilation
• CD-ROM Images
• ZX Spectrum
• DOOM Level CD

• Smithsonian Libraries
• FEDLINK (US)
• Lincoln Collection
• American Libraries
• Canadian Libraries
• Universal Library
• Project Gutenberg
• Children's Library
• Biodiversity Heritage Library
• Books by Language
• Additional Collections

• Prelinger Archives
• Democracy Now!
• Occupy Wall Street
• TV NSA Clip Library
• Animation & Cartoons
• Arts & Music
• Computers & Technology
• Cultural & Academic Films
• Ephemeral Films
• Sports Videos
• Videogame Videos
• Youth Media

Search the history of over 866 billion web pages on the Internet.

Mobile Apps

• Wayback Machine (iOS)
• Wayback Machine (Android)

Browser Extensions

Archive-it subscription.

• Explore the Collections
• Build Collections

Save Page Now

Capture a web page as it appears now for use as a trusted citation in the future.

Please enter a valid web address

• Donate Donate icon An illustration of a heart shape

Chemical problem-solving by dimensional analysis; a self-instructional program

Bookreader item preview, share or embed this item, flag this item for.

• Graphic Violence
• Explicit Sexual Content
• Hate Speech
• Misinformation/Disinformation
• Marketing/Phishing/Advertising
• Misleading/Inaccurate/Missing Metadata

cut off text due to tight binding

plus-circle Add Review comment Reviews

22 Previews

DOWNLOAD OPTIONS

No suitable files to display here.

PDF access not available for this item.

IN COLLECTIONS

Uploaded by station11.cebu on January 14, 2023

SIMILAR ITEMS (based on metadata)

One-Dimensional Model of Vertical Transport of Chemical Components in the Mars Atmosphere up to the Lower Thermosphere

• Published: 20 May 2024
• Volume 58 , pages 187–195, ( 2024 )

Cite this article

• Y. Kylivnyk 1 , 2 ,
• A. S. Petrosyan 1 , 2 ,
• A. A. Fedorova 1 &
• O. I. Korablev 1 , 2  

Explore all metrics

This study is devoted to the analysis of transport of chemical components of the Mars atmosphere. We investigate the turbulent diffusion of chemical components of the Mars atmosphere. To solve this problem in the approximation of diffusion of the minor component, we composed the continuity equation and the corresponding difference scheme. We formulated the boundary conditions in accordance with the available experimental and theoretical data and obtained the required temperature and pressure profiles. For simulation, we chose two models of turbulent diffusion, which were used in subsequent calculations. The simulation was performed using the modified Newton method. The models showed significant differences in the distribution of minor components of the atmosphere, in particular, hydrogen-containing molecules, which indicates the importance of choosing a model for describing turbulent diffusion when constructing a one-dimensional photochemical model of the atmosphere.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price includes VAT (Russian Federation)

Instant access to the full article PDF.

Rent this article via DeepDyve

Institutional subscriptions

Bougher, S., Jakosky, B., Halekas, J., Grebowsky, J., Luhmann, J., Mahaffy, P., Connerney, J., Eparvier, F., Ergun, R., Larson, D., and 74 co-authors, Early MAVEN Deep Dip campaign reveals thermosphere and ionosphere variability, Science, 2015, vol. 350, no. 6261, p. aad0459.

Article   Google Scholar  

Bougher, S., Roeten, K., Olsen, K., Mahaffy, P., Benna, M., Elrod, M., Jain, S., Schneider, N.M., Deighan, J., Thiemann, E., and 3 co-authors, The structure and variability of mars dayside thermosphere from MAVEN NGIMS and IUVS measurements: Seasonal and solar activity trends in scale heights and temperatures, J. Geophys. Res.: Space Phys., 2017, vol. 122, no. 1, pp. 1296–1313.

Article   ADS   Google Scholar  

Chaffin, M.S., Deighan, J., Schneider, N.M., and Stewart, A.I.F., Elevated atmospheric escape of atomic hydrogen from mars induced by high-altitude water, Nat. Geosci., 2017, vol. 10, no. 3, pp. 174–178.

Courtney, W.G., Kinetics of condensation of water vapor, J. Chem. Phys., 1962, vol. 36, no. 8, pp. 2018–2025.

Fedorova, A.A., Montmessin, F., Rodin, A.V., Korablev, O.I., Maattanen, A., Maltagliati, L., and Bertaux, J.L., Evidence for a bimodal size distribution for the suspended aerosol particles on Mars, Icarus, 2014, vol. 231, pp. 239–260.

Hirschfelder, J.O., Curtiss, C.F., and Bird, R.B., The Molecular Theory of Gases and Liquids, Hoboken, NJ: John Wiley and Sons, 1964.

Google Scholar  

Hunten, D.M., Aeronomy of the lower atmosphere of Mars, Rev. Geophys., 1974, vol. 12, no. 3, pp. 529–535.

Izakov, M.N., Structure and dynamics of the upper atmospheres of Venus and Mars, Sov. Phys. Usp., 1976, vol. 19, no. 6, pp. 503–529.

Hairer, E. and Wanner, G., Solving ordinary differential equations (announcement), Matem. Model., 1999, vol. 11, no. 8, p. 127.

Krasnopol’skii, V.A., Fotokhimiya atmosfer Marsa i Venery (Photochemistry of the Atmospheres of Mars and Venus), Moscow: Nauka, 1982.

Kulikov, Yu.N., Modeling the chemical composition of the atmosphere of Mars. Preliminary results of comparison of the altitude profile of atomic oxygen with measurement data from the SPICAM spectrometer, Tr. Kol’skogo Nauchn. Tsentra Ross. Akad. Nauk. Geliogeofiz., 2018, vol. 4, no. 5 (9), pp. 202–216.

Kulikov, Iu.N. and Rykhletskii, M.V., Modeling of the vertical distribution of water in the atmosphere of Mars, Sol. Syst. Res., 1984, vol. 17, no. 3, pp. 112–118.

ADS   Google Scholar  

Lindzen, R.S., Turbulence and stress owing to gravity wave and tidal breakdown, J. Geophys. Res.: Oceans, 1981, vol. 86, no. C10, pp. 9707–9714.

Marov, M.Ya. and Kolesnichenko, A.V., Vvedenie v planetarnuyu aeronomiyu (Introduction to Planetary Aeronomy), Moscow: Nauka, 1987.

Mikrin, E.A., Mikhailov, M.V., Rozhkov, S.N., Semenov, A.S., Krasnopol’skii I.A., Pochukaev, V.N., Markov, Yu.G., and Perepelkin, V.V., High-precision forecast of spacecraft orbits, analysis of the influence of various disturbing factors on the movement of low-orbit and high-orbit spacecraft, in XXI Sankt-Peterburgskaya mezhdunarodnaya konferentsiya po integrirovannym navigatsionnym sistemam (XXI St. Petersburg International Conference on Integrated Navigation Systems), 2014, pp. 77–88.

Shaposhnikov, D.S., Rodin, A.V., Medvedev, A.S., Fedorova, A.A., Kuroda, T., and Hartogh, P., Modeling the hydrological cycle in the atmosphere of Mars: Influence of a bimodal size distribution of aerosol nucleation particles, J. Geophys. Res.: Planets, 2018, vol. 123, no. 2, pp. 508–526.

Shimazaki, T. and Shimizu, M., The seasonal variation of ozone density in the Martian atmosphere, J. Geophys. Res.: Space Phys., 1979, vol. 84, no. A4, pp. 1269–1276.

Uddin, A.F., Numata, K., Shimasaki, J., Shigeishi, M., and Ohtsu, M., Mechanisms of crack propagation due to corrosion of reinforcement in concrete by AE-SiGMA and BEM, Construct. Build. Mater., 2004, vol. 18, no. 3, pp. 181–188.

Download references

ACKNOWLEDGMENTS

The authors are grateful to D.A. Klimachkov for fruitful discussions and his help in investigations.

This study was supported by State assignment “Planet” from the Ministry of Science and Higher Education of the Russian Federation.

Author information

Authors and affiliations.

Space Research Institute, Russian Academy of Sciences, 117997, Moscow, Russia

Y. Kylivnyk, A. S. Petrosyan, A. A. Fedorova & O. I. Korablev

Moscow Institute of Physics and Technology (National Research University), 141701, Dolgoprudny, Moscow oblast, Russia

Y. Kylivnyk, A. S. Petrosyan & O. I. Korablev

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Y. Kylivnyk .

Ethics declarations

The authors of this work declare that they have no conflict of interests.

Additional information

Translated by N. Wadhwa

Publisher’s Note.

Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Kylivnyk, Y., Petrosyan, A.S., Fedorova, A.A. et al. One-Dimensional Model of Vertical Transport of Chemical Components in the Mars Atmosphere up to the Lower Thermosphere. Sol Syst Res 58 , 187–195 (2024). https://doi.org/10.1134/S0038094623700041

Download citation

Received : 24 October 2023

Revised : 09 November 2023

Accepted : 17 November 2023

Published : 20 May 2024

Issue Date : April 2024

DOI : https://doi.org/10.1134/S0038094623700041

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

• Mars atmosphere
• gravitational diffusion
• Lindzen model
• turbulent diffusion
• Find a journal
• Publish with us
• Track your research

Download the free Kindle app and start reading Kindle books instantly on your smartphone, tablet, or computer - no Kindle device required .

Read instantly on your browser with Kindle for Web.

Using your mobile phone camera - scan the code below and download the Kindle app.

Follow the author

Image Unavailable

• To view this video download Flash Player

Chemical problem-solving by dimensional analysis;: A self-instructional program Paperback – January 1, 1974

There is a newer edition of this item:.

• Print length 367 pages
• Language English
• Publisher Houghton Mifflin
• Publication date January 1, 1974
• ISBN-10 0395169704
• ISBN-13 978-0395169704
• See all details

Product details

• Publisher ‏ : ‎ Houghton Mifflin (January 1, 1974)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 367 pages
• ISBN-10 ‏ : ‎ 0395169704
• ISBN-13 ‏ : ‎ 978-0395169704
• Item Weight ‏ : ‎ 1.75 pounds
• Best Sellers Rank: #9,950,772 in Books ( See Top 100 in Books )

About the author

Arnold b. loebel.

Discover more of the author’s books, see similar authors, read author blogs and more

Customer reviews

Customer Reviews, including Product Star Ratings help customers to learn more about the product and decide whether it is the right product for them.

To calculate the overall star rating and percentage breakdown by star, we don’t use a simple average. Instead, our system considers things like how recent a review is and if the reviewer bought the item on Amazon. It also analyzed reviews to verify trustworthiness.

• Sort reviews by Top reviews Most recent Top reviews

Top reviews from the United States

There was a problem filtering reviews right now. please try again later..

• Amazon Newsletter
• About Amazon
• Accessibility
• Sustainability
• Press Center
• Investor Relations
• Amazon Devices
• Amazon Science
• Sell on Amazon
• Sell apps on Amazon
• Supply to Amazon
• Protect & Build Your Brand
• Become an Affiliate
• Become a Delivery Driver
• Start a Package Delivery Business
• Advertise Your Products
• Self-Publish with Us
• Become an Amazon Hub Partner
• › See More Ways to Make Money
• Amazon Visa
• Amazon Store Card
• Amazon Secured Card
• Amazon Business Card
• Shop with Points
• Credit Card Marketplace
• Reload Your Balance
• Amazon Currency Converter
• Your Account
• Your Orders
• Shipping Rates & Policies
• Amazon Prime
• Returns & Replacements
• Manage Your Content and Devices
• Recalls and Product Safety Alerts
• Conditions of Use
• Privacy Notice
• Consumer Health Data Privacy Disclosure
• Your Ads Privacy Choices