This article details the derivation of equations used in the Alcohol Dilution Calculators
When alcohol (isopropanol or ethanol) and water are combined, the mass is conserved but the volume is not:
Total mass = mass of alcohol + mass of water
Total volume ≠ volume of alcohol + volume of water
Resulting volume is not equal to the sum of individual volumes. Because of the change in volume, computing for the total volume using the individual mass and density of pure alcohol and pure water will produce a degree of error in the answer (up to ~3.5%) for the final volume.
Detailed alcoholometric tables are available for ethanol which provide density values at various w/w and v/v concentrations for a wide range of temperatures. Tables for isopropanol are rare and density values are available only for a few temperatures, thus some values need to determined via interpolation and extrapolation.
To produce algorithms for the alcohol dilution calculators, these alcoholometric tabular data (and derived data such as v/v concentration in terms of w/w concentration) were entered into a polynomial regression analysis calculator to obtain polynomial equations with coefficients of determination $ R^2 $ > 0.9999, providing continuous computed values for 1. density as a function of w/w concentration at five different temperatures, and 2. w/w concentration as a function of v/v concentration for the same five temperatures.
Caveat: For the polynomial regression equations used in the calculators, alcohol concentration has been limited to the range of 30% to 100%w/w, which translates to 36% to 100%v/v. A lower bound of around 40%v/v should cover most ordinary (non-food) purposes for which isopropanol and ethanol aqueous solutions are used. Limiting the range of concentration lowers the standard deviation of the polynomial regression equations.
| Term used in this page and its meaning | Term used in the calculators page | |
|---|---|---|
| init-sol | initial alcohol-water solution (alcohol to be diluted) | concentrated alcohol |
| fin-sol | final alcohol-water solution (alcohol + added water) | diluted alcohol |
| v/v | volume/volume | v/v |
| w/w | weight/weight | w/w |
| conc. | alcohol concentration | concentration |
| ref-temp | reference temperature: the temperature at which v/v conc. is specified by alcohol manufacturer/supplier | reference temperature |
| cur-temp | current temperature: the actual, present temperature of the alcohol and water | current temperature |
| Symbol | Description | Units |
|---|---|---|
| $ C'_{V,t_{ref}} $ | init-sol v/v conc. at ref-temp | decimal |
| $ C''_{V,t_{ref}} $ | fin-sol v/v conc. at ref-temp | decimal |
| $ V'_{t_{ref}} $ | volume of init-sol at ref-temp | liters |
| $ V''_{t_{ref}} $ | volume of fin-sol at ref-temp | liters |
| $ \rho''_{t_{ref}} $ | density of fin-sol at ref-temp | kg/liter |
| $ \rho_{100\%,t_{ref}} $ | density of 100% alcohol at ref-temp | kg/liter |
| $ C_{V,t} $ | alcohol-water solution v/v conc. at cur-temp | decimal |
| $ C'_{V,t} $ | init-sol v/v conc. at cur-temp | decimal |
| $ C''_{V,t} $ | fin-sol v/v conc. at cur-temp | decimal |
| $ C_{m,t} $ | alcohol-water solution w/w conc. at cur-temp | decimal |
| $ C'_{m,t} $ | init-sol w/w conc. at cur-temp | decimal |
| $ C''_{m,t} $ | fin-sol w/w conc. at cur-temp | decimal |
| $ V'_{t} $ | volume of init-sol at cur-temp | liters |
| $ \rho'_{t} $ | density of init-sol at cur-temp | kg/liter |
| $ m'_{t} $ | mass of init-sol at cur-temp | kg |
| $ V''_{t} $ | volume of fin-sol at cur-temp | liters |
| $ \rho''_{t} $ | density of fin-sol at cur-temp | kg/liter |
| $ m''_{t} $ | mass of fin-sol at cur-temp | kg |
| $ V_{100\%,t} $ | volume of 100% alcohol at cur-temp | liters |
| $ \rho_{100\%,t} $ | density of 100% alcohol at cur-temp | kg/liter |
| $ m_{100\%,t} $ | mass of 100% alcohol at cur-temp | kg |
| $ V_{H2O,t} $ | volume of added water at cur-temp | liters |
| $ \rho_{H2O,t} $ | density of water at cur-temp | kg/liter |
| $ m_{H2O,t} $ | mass of added water at cur-temp | kg |
Note: Equations with yellow background are deemed core equations
Because of the change in volume when alcohol and water are combined, the volume of fin-sol is—for common, practical concentrations and dilutions —less than the sum of the volume of init-sol and the volume of added water: $$ V''_t \lt V'_t + V_{H2O,t} $$ Volumes must therefore be determined empirically and values computed using concentration and density data from alcoholometric tables.
Mass of fin-sol is the sum of the mass of init-sol and mass of added water: $$ m''_t = m'_t +m_{H2O,t} $$
Alcohol v/v concentration of init-sol at reference temperature is the ratio of the volume of 100% alcohol to volume of init-sol both at reference temperature: $$ C'_{V,t_{ref}} = \frac {V_{100\%,t_{ref}}} {V'_{t_{ref}}} \label{C1vtref_def} $$
Alcohol v/v concentration of fin-sol at reference temperature is the ratio of the volume of 100% alcohol to volume of fin-sol both at reference temperature: $$ C''_{V,t_{ref}} = \frac {V_{100\%,t_{ref}}} {V''_{t_{ref}}} \label{C2vtref_def} $$
Alcohol w/w concentration of init-sol is the ratio of the mass of 100% alcohol to mass of init-sol: \begin{equation} C'_{m,t} = \frac {m_{100\%,t}} {m'_t} \label{C1m_def} \end{equation}
Alcohol w/w concentration of fin-sol is the ratio of the mass of 100% alcohol to mass of fin-sol: \begin{equation} C''_{m,t} = \frac {m_{100\%,t}} {m''_t} \label{C2m_def} \end{equation}
Alcohol v/v concentration of init-sol at current temperature is the ratio of the volume of 100% alcohol to volume of init-sol at current temperature: \begin{equation} C'_{V,t} = \frac {V_{100\%,t}} {V'_{t}} \label{C1vtcur_def} \end{equation}
Alcohol v/v concentration of fin-sol at current temperature is the ratio of the volume of 100% alcohol to volume of fin-sol at current temperature : \begin{equation} C''_{V,t} = \frac {V_{100\%,t}} {V''_{t}} \label{C2vtcur_def} \end{equation}
Density $ \rho $ is defined as $ \frac{mass}{volume} $
Therefore, the masses of init-sol and fin-sol are: \begin{align} m'_t &= \rho'_t V'_t \label{m1} \\ m''_t &= \rho''_t V''_t \label{m2} \end{align}
Thus the volumes of init-sol and fin-sol are: $$ V'_t = \frac {m'_t}{\rho'_t} $$ $$ V''_t = \frac {m''_t}{\rho''_t} $$
The density of 100% alcohol is given by: \begin{equation} \rho_{100\%,t} = \frac {m_{100\%,t}} {V_{100\%,t}} \label{d100tcur} \end{equation}
The density of water is given by: \begin{equation} \rho_{H2O,t} = \frac {m_{H2O,t}} {V_{H2O,t}} \label{dH20} \end{equation}
From Eq.\ref{C1m_def} we get: $$ m_{100\%,t} = m'_t C'_{m,t} $$ and from Eq.\ref{C2m_def} we get: $$ m_{100\%,t} = m''_t C''_{m,t} $$ Therefore: \begin{equation} \bbox[#fd0,10px] { m'_t C'_{m,t} = m''_t C''_{m,t} } \label{mCm} \end{equation} In other words, the mass of 100% alcohol in both init-sol and fin-sol are equal.
Substituting using Eq.\ref{m1} and Eq.\ref{m2} we obtain: \begin{equation} \rho'_t V'_t C'_{m,t} = \rho''_t V''_t C''_{m,t} \label{dVCm} \end{equation}
From Eq.\ref{C1vtcur_def} we get: $$ V_{100\%,t} = V'_t C'_{V,t} $$ From Eq.\ref{C2vtcur_def} we get: $$ V_{100\%,t} = V''_t C''_{V,t} $$ Therefore: $$ V'_t C'_{V,t} = V''_t C''_{V,t} $$ In other words, the volume of 100% alcohol (that is, pure alcohol if extracted from the solution) in both init-sol and fin-sol are equal, provided both are at the same temperature.
In Calculator B only $ C'_{m,t} $ is available from the user. However, both $ C'_{m,t} $ and $ C''_{m,t} $ are necessary in order to calculate $ \rho'_t $ and $ \rho''_t $, respectively. All four variables are in turn necessary to calculate the various masses and volumes.
In Calculator B the givens are as follows:
From Eq.\ref{mCm} we get: $$ C''_{m,t} = \frac {m'_t C'_{m,t}} {m''_t} $$
Since $$ m''_t = m'_t + m_{H2O,t} $$ we have: \begin{equation} C''_{m,t} = \frac {m'_t C'_{m,t}} {m'_t + m_{H2O,t}} \label{B_C2m_m} \end{equation}
Since $$ m'_t = \rho'_t V'_t $$ and $$ m_{H2O,t} = \rho_{H2O,t} V_{H2O,t} $$ we have the equivalent equation: \begin{equation} C''_{m,t} = \frac {\rho'_t V'_t C'_{m,t}} {\rho'_t V'_t + \rho_{H2O,t} V_{H2O,t}} \label{B_C2m_dV} \end{equation}
Eq.\ref{B_C2m_m} and Eq.\ref{B_C2m_dV} allow the use of either mass or volume or combinations thereof of init-sol and added water
For the following equations, it is assumed that $ C'_{m,t} $ and $ C''_{m,t} $ are already known, with $ C'_{m,t} $ having been computed via a polynomial regression equation (see below) and $ C''_{m,t} $ having been computed via a regression equation or through Eq.\ref{B_C2m_m} or Eq.\ref{B_C2m_dV}. Furthermore, with $ C'_{m,t} $ and $ C''_{m,t} $ already known, $ \rho'_t $ and $ \rho''_t $ will also be known since they can be derived via regression equations for $ \rho_t = f(C_{m,t}) $
Given: volume of fin-sol
From Eq.\ref{dVCm} we get: $$ \rho'_t V'_t = \frac {\rho''_t V''_t C''_{m,t}} {C'_{m,t}} $$ $$ m'_t = \frac {\rho''_t V''_t C''_{m,t}} {C'_{m,t}} $$
Given: mass of fin-sol
From Eq.\ref{mCm} we get: $$ m'_t = \frac {m''_t C''_{m,t}} {C'_{m,t}} $$
Given: volume of init-sol
Eq.\ref{m1} gives us: $$ m'_t = \rho'_t V'_t $$
By definition the mass of additional water = difference between the masses of fin-sol and init sol: $$ m_{H2O,t} = m''_t - m'_t $$ Equivalently we have: $$ m_{H2O,t} = \rho''_t V''_t - \rho'_t V'_t $$
Given: volume of fin-sol and mass of init-sol
$$ m_{H2O,t} = \rho''_t V''_t - m'_t $$
Given: mass of init-sol
From Eq.\ref{dVCm} we get: $$ \rho''_t V''_t = \frac {\rho'_t V'_t C'_{m,t}} {C''_{m,t}} $$
Substituting: \begin{gather*} m_{H2O,t} = \frac {\rho'_t V'_t C'_{m,t}} {C''_{m,t}} - \rho'_t V'_t \\ m_{H2O,t} = \rho'_t V'_t \left ( \frac {C'_{m,t}}{C''_{m,t}} - 1 \right ) \\ m_{H2O,t} = m'_t \left ( \frac {C'_{m,t}}{C''_{m,t}} - 1 \right ) \end{gather*}
From Eq.\ref{dH20} we have: $$ V_{H2O,t} = \frac {m_{H2O,t}} {\rho_{H2O,t}} $$
Volume loss is defined and calculated as follows, where $ V'_t + V_{H2O,t} $ is the sum of the volumes of init-sol and added water and $ V''_t $ is the actual volume of fin-sol: $$ = \frac {(V'_t + V_{H2O,t}) - V''_t} {V'_t + V_{H2O,t}} $$ $$ = 1- \frac {V''_t} {V'_t + V_{H2O,t}} $$ Multiplying by 100 gives volume loss as a percentage.
If the result is negative then volume has expanded: $ V''_t > V'_t + V_{H2O,t} $ This occurs when $ V_{H2O,t} \gg V'_t $
From Eq.\ref{d100tcur} we get: $$ V_{100\%,t} = \frac {m_{100\%,t}} {\rho_{100\%,t}} $$
From Eq.\ref{C1m_def} we get: $$ m_{100\%,t} = {m'_t} C'_{m,t} $$
From Eq.\ref{C2m_def} we get: $$ m_{100\%,t} = {m''_t} C''_{m,t} $$
Therefore, we obtain: $$ V_{100\%,t} = \frac {{m'_t} C'_{m,t}} {\rho_{100\%,t}} = \frac {{m''_t} C''_{m,t}} {\rho_{100\%,t}} $$
Eq.\ref{C1vtcur_def} and Eq.\ref{C2vtcur_def} give us: \begin{gather*} C'_{V,t} = \frac {V_{100\%,t}} {V'_t} \\ C''_{V,t} = \frac {V_{100\%,t}} {V''_t} \end{gather*}
Substituting we get: $$ C'_{V,t} = \frac { \left ( \frac {{m'_t} C'_{m,t}} {\rho_{100\%,t}} \right) } {V'_t} = \frac { {m'_t} C'_{m,t}} {\rho_{100\%,t} V'_t} $$ $$ C''_{V,t} = \frac { \left ( \frac {{m''_t} C''_{m,t}} {\rho_{100\%,t}} \right ) } {V''_t} = \frac {{m''_t} C''_{m,t}} {\rho_{100\%,t} V''_t} $$
From Eq.\ref{m1} and Eq.\ref{m2} we get: $$ \rho'_t = \frac {m'_t}{V'_t} $$ $$ \rho''_t = \frac {m''_t}{V''_t} $$
Substituting we get: \begin{equation} C'_{V,t} = \frac { \rho'_t C'_{m,t}} {\rho_{100\%,t}} \label{C1vtcur} \end{equation} \begin{equation} C''_{V,t} = \frac {\rho''_t C''_{m,t}} {\rho_{100\%,t}} \label{C2vtcur} \end{equation}
We can generalize the above equations as: \begin{equation} \bbox[#fd0,10px] { C_{V,t} = \frac { \rho_t C_{m,t}} {\rho_{100\%,t}} } \label{Cvt} \end{equation}
Eq.\ref{Cvt} is used to convert from w/w to v/v concentration and vice versa using density values from alcoholometric tables (from which polynomial regression equations for $ C_{m,t} = f(C_{V,t}) $ are obtained).
In Calculator A the user provides both init-sol and fin-sol v/v conc. at reference temperature, thus both w/w conc. can be obtained directly through the use of polynomial regression equations derived from alcoholometric tables. $ C'_m $ and $ C''_m $ are assumed to be constant over the range of temperatures from 15 to 35°C. Thus the temperature subscript can be omitted: \begin{gather*} C'_{m} = f(C'_{V,t_{ref}}) \\ C''_{m} = f(C''_{V,t_{ref}}) \end{gather*}
In calculator B the user provides the init-sol v/v conc. at reference temperature and thus $ C'_m $ will be constant over the entire temperature range. However, $ C''_m $ has to be computed from $ C'_m $ and user-provided alcohol and water weight and volume. If one of the values entered by the user is in terms of volume $ C''_m $ will vary depending on the current temperature. Therefore for Calculator B fin-sol w/w conc. may be temperature-dependent: \begin{gather*} C'_{m} = f(C'_{V,t_{ref}}) \\ C''_{m,t} = f(C''_{V,t}) \end{gather*}
To avoid possible confusion init-sol and fin-sol w/w conc. are both designated with a temperature subscript: \begin{gather*} C'_{m,t} \\ C''_{m,t} \end{gather*}
Once w/w conc. is known, density can be computed as a function of $ C_{m,t} $ using polynomial regressions equations derived from alcoholometric tables: \begin{gather*} \rho'_{t} = f(C'_{m,t}) \\ \rho''_{t} = f(C''_{m,t}) \end{gather*}
In Calculators A and B init-sol and fin-sol v/v conc. at current temperature are displayed and are given by Eq.\ref{C1vtcur} and Eq.\ref{C2vtcur}, respectively.
In Calculator B, the user provides only the concentration for init-sol. Therefore, besides init-sol and fin-sol v/v conc. at current temperature, the fin-sol v/v conc. at reference temperature is also shown. The equation is as follows:
We have been assuming that w/w conc. of the alcohol solution remains constant throughout the temperature range of 15 to 35°C. Therefore, it follows that $ C_{m,t_{ref}} = C_{m,t} $. In other words, if we raise or lower the temperature of the alcohol solution so that it reaches ref-temp, its w/w conc. will remain the same (because the mass of alcohol and water do not change). Since we know $ C''_{m,t} $, and using Eq.\ref{Cvt}, fin-sol v/v conc. at reference temperature is: $$ C''_{V,t_{ref}} = \frac {\rho''_{t_{ref}} C''_{m,t}} {\rho_{100\%,t_{ref}}} $$ where fin-sol density at reference temperature is obtained through the use of polynomial regression equations derived from alcoholometric tables, again where $ C_{m,t_{ref}} = C_{m,t} $: $$ \rho''_{t_{ref}} = f(C''_{m,t_{ref}}) $$
To create a table of corresponding w/w - v/v conc. at particular temperatures, Eq.\ref{Cvt} was applied to density values (from alcoholometric tables) in terms of w/w conc.
Detailed alcoholometric data are available for ethanol and therefore tables for density as a function of w/w conc. was used to obtain corresponding v/v conc. values for temperatures of 15, 20, 25, 30, and 35°C.
Only one table for isopropanol could be found which lists densities at various w/w conc. but only for temperatures of 0, 15, 20, and 30°C. Density values were therefore interpolated for 25°C and extrapolated for 35°C using data for 20 and 30°C. Eq.\ref{Cvt} was then used to obtain the corresponding v/v conc. for temperatures of 15, 20, 25, 30, and 35°C.
The calculated values above were then fed to a polynomial regression calculator to obtain equations for w/w conc. as a function of v/v conc. for various temperatures.
Using the same tables as above, equations for density as a function of w/w concentration were obtained using the same regression calculator.
Using data from an alcoholometric table, density values for 25 and 35°C were obtained through interpolation and extrapolation of data for 20 and 30°C.
$ C_{V,t} $ values were then computed using Eq.\ref{Cvt} for temperatures of 15, 20, 25, 30, and 35°C. $ C_{m,t} $ range was from 30%w/w to 100%w/w in increments of 1%.
The following equations for $ C_{m,t} $ were then obtained using polynomial regression analysis using the values for $ C_{V,t} $ and $ C_{m,t} $ obtained above. Coefficient of determination $ R^2 > 0.999999 $
$ C_{m,15°C} = 0.18611877237610175 - 1.1654496420442664C_{V,15°C} + 8.453804361680312C^2_{V,15°C} - 18.837457942418727C^3_{V,15°C} + 23.81193022954337C^4_{V,15°C} - 15.755209244329187C^5_{V,15°C} + 4.30632611030398C^6_{V,15°C} $
$ C_{m,20°C} = -0.20690345525202378 + 3.0050151936546583C_{V,20°C} - 9.498803283454661C^2_{V,20°C} + 21.263047408577382C^3_{V,20°C} - 25.309092418431906C^4_{V,20°C} + 15.610183102917432C^5_{V,20°C} - 3.8633707325934425C^6_{V,20°C} $
$ C_{m,25°C} = 0.014540379974508398 + 0.6719431851731379C_{V,25°C} + 0.43919513783361563C^2_{V,25°C} - 0.6916968817401634C^3_{V,25°C} + 1.2565801579169842C^4_{V,25°C} - 1.1193577377099946C^5_{V,25°C} + 0.42865327131693126C^6_{V,25°C} $
$ C_{m,30°C} = 0.2479276131479107 - 1.7788644006822556C_{V,30°C} + 10.848061992142636C^2_{V,30°C} - 23.62525994197691C^3_{V,30°C} + 28.93870122421419C^4_{V,30°C} - 18.512130742272536C^5_{V,30°C} + 4.881165018878908C^6_{V,30°C} $
$ C_{m,35°C} = 0.49333593354528527 - 4.3477970320004085C_{V,35°C} + 21.727826678026474C^2_{V,35°C} - 47.534463875967035C^3_{V,35°C} + 57.729860605180455C^4_{V,35°C} - 36.561417900025404C^5_{V,35°C} + 9.491958302169577C^6_{V,35°C} $
Using data from alcoholometric tables, $ C_{V,t} $ values were first computed using Eq.\ref{Cvt} for temperatures of 15, 20, 25, 30, and 35°C. $ C_{m,t} $ range was from 30%w/w to 100%w/w in increments of 1%.
The following equations for $ C_{m,t} $ were then obtained using polynomial regression analysis using the values for $ C_{m,t} $ and $ C_{V,t} $ obtained above. Coefficient of determination $ R^2 > 0.999999 $.
$ C_{m,15°C} = 0.3465566932367186 - 2.7867516916298882C_{V,15°C} + 15.253566677638556C^2_{V,15°C} - 33.80023530105008C^3_{V,15°C} + 41.92884684315532C^4_{V,15°C} - 27.232809990231424C^5_{V,15°C} + 7.290508530857499C^6_{V,15°C} $
$ C_{m,20°C} = 0.354163239042076 - 2.869214203422433C_{V,20°C} + 15.582350444879886C^2_{V,20°C} - 34.47412571991754C^3_{V,20°C} + 42.679592105999724C^4_{V,20°C} - 27.663879080836843C^5_{V,20°C} + 7.390807390189705C^6_{V,20°C} $
$ C_{m,25°C} = 0.3670026743242395 - 3.0022390677766446C_{V,25°C} + 16.11016345509771C^2_{V,25°C} - 35.554771301523786C^3_{V,25°C} + 43.8849017132032C^4_{V,25°C} - 28.357922542432014C^5_{V,25°C} + 7.552556607231511C^6_{V,25°C} $
$ C_{m,30°C} = 0.37362525016741066 - 3.0692406088807953C_{V,30°C} + 16.356012177893817C^2_{V,30°C} - 36.01234868129796C^3_{V,30°C} + 44.33664445765785C^4_{V,30°C} - 28.577396976614754C^5_{V,30°C} + 7.592374182581728C^6_{V,30°C} $
$ C_{m,35°C} = 0.3873381051766422 - 3.2048244882817447C_{V,35°C} + 16.87337220868026C^2_{V,35°C} - 37.03151779779187C^3_{V,35°C} + 45.42691263265177C^4_{V,35°C} - 29.1745385281625C^5_{V,35°C} + 7.722857018182701C^6_{V,35°C} $
In the alcohol calculators the density values of isopropanol aqueous solutions are computed as a function of $ C_{m,t} $
Using data from an alcoholometric table, density values for 25 and 35°C were derived through interpolation and extrapolation, respectively, of data for 20 and 30°C.
The following equations for density were then obtained using polynomial regression analysis for $ C_{m,t} $ from 30%w/w to 100%w/w in increments of 1% for temperatures of 15, 20, 25, 30, and 35°C. Coefficients of determination $ R^2 > 0.9999 $
$ \rho_{15°C} = 0.9936843779301617 - 0.013592091801382409C_{m,15°C} - 0.5344205148363754C^2_{m,15°C} + 0.5704719057064189C^3_{m,15°C} - 0.22709596286929676C^4_{m,15°C} $
$ \rho_{20°C} = 0.9858784444373024 + 0.018461500230157968C_{m,20°C} - 0.5875733839794985C^2_{m,20°C} + 0.5727953713341042C^3_{m,20°C} - 0.20468649016693682C^4_{m,20°C} $
$ \rho_{25°C} = 0.9894312233029401 - 0.031027161995539194C_{m,25°C} - 0.4781726682425268C^2_{m,25°C} + 0.47684905890001916C^3_{m,25°C} - 0.17599507079846727^4C_{m,25°C} $
$ \rho_{30°C} = 0.9929840021685773 - 0.0805158242212347C_{m,30°C} - 0.36877195250555916C^2_{m,30°C} + 0.3809027464659384C^3_{m,30°C} - 0.14730365142999938C^4_{m,30°C} $
$ \rho_{35°C} = 0.9965367810342147 - 0.1300044864469307C_{m,35°C} - 0.2593712367685903C^2_{m,35°C} + 0.28495643403185644C^3_{m,35°C} - 0.11861223206153099C^4_{m,35°C} $
In the alcohol calculators the density values of isopropanol aqueous solutions are computed as a function of $ C_{m,t} $
Using data from alcoholometric tables, the following equations for density were obtained using polynomial regression analysis for $ C_{m,t} $ from 30%w/w to 100%w/w in increments of 1% for temperatures of 15, 20, 25, 30, and 35°C. Coefficient of determination $ R^2 > 0.99999 $.
$ \rho_{15°C} = 0.9664881317059026 + 0.14986649233581079C_{m,15°C} - 0.8220218752496306C^2_{m,15°C} + 0.8098296128082298C^3_{m,15°C} - 0.31051862078965836C^4_{m,15°C} $
$ \rho_{20°C} = 0.9690755073150866 + 0.11575810563718208C_{m,20°C} - 0.7542800757698563C^2_{m,20°C} + 0.7476802900377703C^3_{m,20°C} - 0.28884956335859774C^4_{m,20°C} $
$ \rho_{25°C} = 0.9712308871587023 + 0.0835229240561932C_{m,25°C} - 0.6898473031170309C^2_{m,25°C} + 0.6878753155009547C^3_{m,25°C} - 0.2676668418810413C^4_{m,25°C} $
$ \rho_{30°C} = 0.9726387036273973 + 0.05541514312706141C_{m,30°C} - 0.6342950513951708C^2_{m,30°C} + 0.6361735375820552C^3_{m,30°C} - 0.24909454346326781C^4_{m,30°C} $
$ \rho_{35°C} = 0.9731998196119197 + 0.03199135531545214C_{m,35°C} - 0.5886979727179072C^2_{m,35°C} + 0.5934477175931433C^3_{m,35°C} - 0.23339509763450106C^4_{m,35°C} $
Sources: isopropanol table and ethanol table
| Isopropanol Concentration w/w |
Density (kg/liter or grams/mL) NOTE: Isopropanol density values for 25°C and 35°C are interpolated and extrapolated, respectively, from the density values for 20°C and 30°C. |
Computed v/v Concentration Using Eq.\ref{Cvt} | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 15°C | 20°C | 25°C | 30°C | 35°C | 15°C | 20°C | 25°C | 30°C | 35°C | ||
| 0.30 | 0.95493 | 0.952 | 0.9483 | 0.9446 | 0.9409 | 0.36305 | 0.36364 | 0.36417 | 0.36471 | 0.36526 | |
| 0.31 | 0.953 | 0.95 | 0.9463 | 0.9426 | 0.9389 | 0.37439 | 0.37497 | 0.37552 | 0.37607 | 0.37663 | |
| 0.32 | 0.951 | 0.9481 | 0.9443 | 0.9405 | 0.9367 | 0.38565 | 0.38629 | 0.38681 | 0.38734 | 0.38787 | |
| 0.33 | 0.9489 | 0.946 | 0.94215 | 0.9383 | 0.93445 | 0.39683 | 0.39748 | 0.39799 | 0.39851 | 0.39903 | |
| 0.34 | 0.9468 | 0.944 | 0.94005 | 0.9361 | 0.93215 | 0.40795 | 0.40866 | 0.40914 | 0.40962 | 0.41011 | |
| 0.35 | 0.9446 | 0.9419 | 0.93785 | 0.9338 | 0.92975 | 0.41897 | 0.41974 | 0.42018 | 0.42063 | 0.42108 | |
| 0.36 | 0.9424 | 0.9399 | 0.9357 | 0.9315 | 0.9273 | 0.42994 | 0.43082 | 0.43120 | 0.43158 | 0.43197 | |
| 0.37 | 0.9401 | 0.9377 | 0.93345 | 0.9292 | 0.92495 | 0.44080 | 0.44175 | 0.44211 | 0.44248 | 0.44285 | |
| 0.38 | 0.9379 | 0.9355 | 0.9312 | 0.9269 | 0.9226 | 0.45166 | 0.45262 | 0.45296 | 0.45331 | 0.45366 | |
| 0.39 | 0.9356 | 0.9333 | 0.92895 | 0.9246 | 0.92025 | 0.46241 | 0.46344 | 0.46376 | 0.46408 | 0.46441 | |
| 0.40 | 0.9333 | 0.931 | 0.9267 | 0.9224 | 0.9181 | 0.47310 | 0.47415 | 0.47450 | 0.47485 | 0.47521 | |
| 0.41 | 0.9311 | 0.9287 | 0.9244 | 0.9201 | 0.9158 | 0.48378 | 0.48481 | 0.48516 | 0.48551 | 0.48587 | |
| 0.42 | 0.9288 | 0.9264 | 0.92205 | 0.9177 | 0.91335 | 0.49436 | 0.49540 | 0.49573 | 0.49605 | 0.49639 | |
| 0.43 | 0.9266 | 0.9239 | 0.91965 | 0.9154 | 0.91115 | 0.50493 | 0.50583 | 0.50621 | 0.50659 | 0.50698 | |
| 0.44 | 0.9243 | 0.9215 | 0.91725 | 0.913 | 0.90875 | 0.51539 | 0.51625 | 0.51663 | 0.51701 | 0.51740 | |
| 0.45 | 0.922 | 0.9191 | 0.91485 | 0.9106 | 0.90635 | 0.52579 | 0.52660 | 0.52699 | 0.52737 | 0.52777 | |
| 0.46 | 0.9197 | 0.9165 | 0.91235 | 0.9082 | 0.90405 | 0.53613 | 0.53678 | 0.53723 | 0.53767 | 0.53813 | |
| 0.47 | 0.9174 | 0.9141 | 0.91 | 0.9059 | 0.9018 | 0.54642 | 0.54702 | 0.54749 | 0.54797 | 0.54845 | |
| 0.48 | 0.915 | 0.9117 | 0.90765 | 0.9036 | 0.89955 | 0.55658 | 0.55719 | 0.55770 | 0.55821 | 0.55873 | |
| 0.49 | 0.9127 | 0.9093 | 0.9053 | 0.9013 | 0.8973 | 0.56675 | 0.56730 | 0.56784 | 0.56839 | 0.56894 | |
| 0.50 | 0.9104 | 0.9069 | 0.90295 | 0.899 | 0.89505 | 0.57686 | 0.57735 | 0.57792 | 0.57851 | 0.57910 | |
| 0.51 | 0.9081 | 0.9044 | 0.9005 | 0.8966 | 0.8927 | 0.58691 | 0.58727 | 0.58788 | 0.58850 | 0.58913 | |
| 0.52 | 0.9058 | 0.902 | 0.89815 | 0.8943 | 0.89045 | 0.59690 | 0.59720 | 0.59785 | 0.59850 | 0.59916 | |
| 0.53 | 0.9035 | 0.8996 | 0.89575 | 0.8919 | 0.88805 | 0.60684 | 0.60706 | 0.60772 | 0.60837 | 0.60904 | |
| 0.54 | 0.9011 | 0.8971 | 0.8933 | 0.8895 | 0.8857 | 0.61664 | 0.61680 | 0.61749 | 0.61819 | 0.61889 | |
| 0.55 | 0.8988 | 0.8946 | 0.89085 | 0.8871 | 0.88335 | 0.62646 | 0.62647 | 0.62720 | 0.62793 | 0.62868 | |
| 0.56 | 0.8964 | 0.8921 | 0.8884 | 0.8847 | 0.881 | 0.63615 | 0.63608 | 0.63685 | 0.63762 | 0.63841 | |
| 0.57 | 0.894 | 0.8896 | 0.88595 | 0.8823 | 0.87865 | 0.64577 | 0.64562 | 0.64643 | 0.64725 | 0.64807 | |
| 0.58 | 0.8917 | 0.8874 | 0.8837 | 0.88 | 0.8763 | 0.65541 | 0.65532 | 0.65610 | 0.65689 | 0.65768 | |
| 0.59 | 0.8893 | 0.885 | 0.88135 | 0.8777 | 0.87405 | 0.66492 | 0.66482 | 0.66564 | 0.66646 | 0.66730 | |
| 0.60 | 0.8869 | 0.8825 | 0.87885 | 0.8752 | 0.87155 | 0.67436 | 0.67418 | 0.67500 | 0.67583 | 0.67667 | |
| 0.61 | 0.8845 | 0.88 | 0.8764 | 0.8728 | 0.8692 | 0.68375 | 0.68347 | 0.68434 | 0.68521 | 0.68609 | |
| 0.62 | 0.8821 | 0.8776 | 0.874 | 0.8704 | 0.8668 | 0.69307 | 0.69278 | 0.69365 | 0.69453 | 0.69541 | |
| 0.63 | 0.8798 | 0.8751 | 0.87155 | 0.868 | 0.86445 | 0.70241 | 0.70195 | 0.70286 | 0.70378 | 0.70471 | |
| 0.64 | 0.8775 | 0.8727 | 0.86915 | 0.8656 | 0.86205 | 0.71170 | 0.71114 | 0.71205 | 0.71298 | 0.71391 | |
| 0.65 | 0.8752 | 0.8702 | 0.86665 | 0.8631 | 0.85955 | 0.72092 | 0.72018 | 0.72110 | 0.72203 | 0.72297 | |
| 0.66 | 0.8728 | 0.8679 | 0.8643 | 0.8607 | 0.8571 | 0.73001 | 0.72933 | 0.73021 | 0.73110 | 0.73200 | |
| 0.67 | 0.8705 | 0.8656 | 0.86195 | 0.8583 | 0.85465 | 0.73911 | 0.73842 | 0.73926 | 0.74010 | 0.74096 | |
| 0.68 | 0.8682 | 0.8632 | 0.85955 | 0.8559 | 0.85225 | 0.74816 | 0.74736 | 0.74820 | 0.74905 | 0.74991 | |
| 0.69 | 0.8658 | 0.8609 | 0.8572 | 0.8535 | 0.8498 | 0.75707 | 0.75633 | 0.75713 | 0.75793 | 0.75875 | |
| 0.70 | 0.8634 | 0.8584 | 0.85475 | 0.8511 | 0.84745 | 0.76591 | 0.76506 | 0.76591 | 0.76676 | 0.76762 | |
| 0.71 | 0.8611 | 0.856 | 0.85235 | 0.8487 | 0.84505 | 0.77478 | 0.77382 | 0.77467 | 0.77552 | 0.77638 | |
| 0.72 | 0.8588 | 0.8537 | 0.85005 | 0.8464 | 0.84275 | 0.78360 | 0.78261 | 0.78346 | 0.78431 | 0.78517 | |
| 0.73 | 0.8564 | 0.8513 | 0.84765 | 0.844 | 0.84035 | 0.79226 | 0.79125 | 0.79209 | 0.79295 | 0.79381 | |
| 0.74 | 0.8541 | 0.8489 | 0.84525 | 0.8416 | 0.83795 | 0.80096 | 0.79983 | 0.80067 | 0.80152 | 0.80238 | |
| 0.75 | 0.8517 | 0.8464 | 0.8428 | 0.8392 | 0.8356 | 0.80950 | 0.80825 | 0.80914 | 0.81004 | 0.81095 | |
| 0.76 | 0.8493 | 0.8439 | 0.84035 | 0.8368 | 0.83325 | 0.81798 | 0.81661 | 0.81754 | 0.81849 | 0.81945 | |
| 0.77 | 0.847 | 0.8415 | 0.83795 | 0.8344 | 0.83085 | 0.82650 | 0.82500 | 0.82594 | 0.82688 | 0.82784 | |
| 0.78 | 0.8446 | 0.8391 | 0.8356 | 0.8321 | 0.8286 | 0.83486 | 0.83333 | 0.83432 | 0.83531 | 0.83632 | |
| 0.79 | 0.8422 | 0.8366 | 0.83315 | 0.8297 | 0.82625 | 0.84316 | 0.84150 | 0.84254 | 0.84358 | 0.84464 | |
| 0.80 | 0.83979 | 0.8342 | 0.83075 | 0.8273 | 0.82385 | 0.85139 | 0.84971 | 0.85074 | 0.85179 | 0.85285 | |
| 0.81 | 0.8374 | 0.8317 | 0.82825 | 0.8248 | 0.82135 | 0.85958 | 0.85775 | 0.85878 | 0.85983 | 0.86089 | |
| 0.82 | 0.835 | 0.8292 | 0.8258 | 0.8224 | 0.819 | 0.86770 | 0.86573 | 0.86682 | 0.86791 | 0.86902 | |
| 0.83 | 0.8326 | 0.8268 | 0.8234 | 0.82 | 0.8166 | 0.87575 | 0.87375 | 0.87484 | 0.87593 | 0.87704 | |
| 0.84 | 0.8302 | 0.8243 | 0.8209 | 0.8175 | 0.8141 | 0.88375 | 0.88160 | 0.88269 | 0.88378 | 0.88489 | |
| 0.85 | 0.8278 | 0.8219 | 0.8185 | 0.8151 | 0.8117 | 0.89169 | 0.88950 | 0.89058 | 0.89168 | 0.89279 | |
| 0.86 | 0.8254 | 0.8194 | 0.81605 | 0.8127 | 0.80935 | 0.89956 | 0.89723 | 0.89837 | 0.89951 | 0.90067 | |
| 0.87 | 0.8229 | 0.8169 | 0.81355 | 0.8102 | 0.80685 | 0.90727 | 0.90489 | 0.90603 | 0.90717 | 0.90833 | |
| 0.88 | 0.8205 | 0.8145 | 0.81115 | 0.8078 | 0.80445 | 0.91502 | 0.91261 | 0.91374 | 0.91488 | 0.91604 | |
| 0.89 | 0.818 | 0.812 | 0.80865 | 0.8053 | 0.80195 | 0.92260 | 0.92014 | 0.92127 | 0.92242 | 0.92357 | |
| 0.90 | 0.8155 | 0.8096 | 0.80625 | 0.8029 | 0.79955 | 0.93011 | 0.92773 | 0.92886 | 0.93000 | 0.93115 | |
| 0.91 | 0.813 | 0.8072 | 0.8038 | 0.8004 | 0.797 | 0.93756 | 0.93526 | 0.93633 | 0.93741 | 0.93850 | |
| 0.92 | 0.8104 | 0.8047 | 0.8013 | 0.7979 | 0.7945 | 0.94483 | 0.94261 | 0.94367 | 0.94475 | 0.94583 | |
| 0.93 | 0.8079 | 0.8023 | 0.79885 | 0.7954 | 0.79195 | 0.95216 | 0.95001 | 0.95101 | 0.95202 | 0.95305 | |
| 0.94 | 0.8052 | 0.7998 | 0.79635 | 0.7929 | 0.78945 | 0.95918 | 0.95723 | 0.95823 | 0.95924 | 0.96025 | |
| 0.95 | 0.8026 | 0.7973 | 0.79385 | 0.7904 | 0.78695 | 0.96625 | 0.96439 | 0.96538 | 0.96638 | 0.96739 | |
| 0.96 | 0.7999 | 0.7949 | 0.79135 | 0.7878 | 0.78425 | 0.97314 | 0.97161 | 0.97247 | 0.97334 | 0.97422 | |
| 0.97 | 0.7972 | 0.7925 | 0.78885 | 0.7852 | 0.78155 | 0.97996 | 0.97877 | 0.97950 | 0.98024 | 0.98098 | |
| 0.98 | 0.7945 | 0.7901 | 0.78635 | 0.7826 | 0.77885 | 0.98671 | 0.98586 | 0.98646 | 0.98706 | 0.98767 | |
| 0.99 | 0.7918 | 0.7877 | 0.7838 | 0.7799 | 0.776 | 0.99339 | 0.99290 | 0.99329 | 0.99369 | 0.99410 | |
| 1.00 | 0.7891 | 0.7854 | 0.7812 | 0.777 | 0.7728 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | |
| Ethanol Concentration w/w |
Density (kg/liter or grams/mL) | Computed v/v Concentration Using Eq.\ref{Cvt} | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| 15°C | 20°C | 25°C | 30°C | 35°C | 15°C | 20°C | 25°C | 30°C | 35°C | ||
| 0.30 | 0.95682 | 0.95378 | 0.95063 | 0.94737 | 0.94399 | 0.36174 | 0.36254 | 0.36332 | 0.36407 | 0.36480 | |
| 0.31 | 0.95521 | 0.95209 | 0.94887 | 0.94553 | 0.94209 | 0.37317 | 0.37396 | 0.37474 | 0.37547 | 0.37620 | |
| 0.32 | 0.95355 | 0.95036 | 0.94706 | 0.94367 | 0.94017 | 0.38454 | 0.38533 | 0.38609 | 0.38682 | 0.38754 | |
| 0.33 | 0.95185 | 0.94858 | 0.94522 | 0.94177 | 0.93821 | 0.39585 | 0.39662 | 0.39738 | 0.39811 | 0.39882 | |
| 0.34 | 0.95010 | 0.94677 | 0.94335 | 0.93983 | 0.93623 | 0.40710 | 0.40786 | 0.40861 | 0.40933 | 0.41004 | |
| 0.35 | 0.94831 | 0.94492 | 0.94144 | 0.93787 | 0.93422 | 0.41828 | 0.41904 | 0.41978 | 0.42049 | 0.42119 | |
| 0.36 | 0.94648 | 0.94303 | 0.93949 | 0.93588 | 0.93218 | 0.42940 | 0.43015 | 0.43088 | 0.43158 | 0.43228 | |
| 0.37 | 0.94462 | 0.94111 | 0.93752 | 0.93386 | 0.93012 | 0.44046 | 0.44120 | 0.44192 | 0.44262 | 0.44331 | |
| 0.38 | 0.94271 | 0.93915 | 0.93551 | 0.93181 | 0.92803 | 0.45145 | 0.45218 | 0.45289 | 0.45358 | 0.45427 | |
| 0.39 | 0.94077 | 0.93716 | 0.93348 | 0.92973 | 0.92592 | 0.46238 | 0.46309 | 0.46380 | 0.46448 | 0.46516 | |
| 0.40 | 0.93880 | 0.93515 | 0.93142 | 0.92764 | 0.92379 | 0.47324 | 0.47395 | 0.47464 | 0.47532 | 0.47599 | |
| 0.41 | 0.93679 | 0.93310 | 0.92934 | 0.92552 | 0.92164 | 0.48403 | 0.48473 | 0.48542 | 0.48609 | 0.48675 | |
| 0.42 | 0.93476 | 0.93103 | 0.92724 | 0.92338 | 0.91947 | 0.49476 | 0.49545 | 0.49613 | 0.49679 | 0.49745 | |
| 0.43 | 0.93270 | 0.92894 | 0.92511 | 0.92122 | 0.91729 | 0.50543 | 0.50611 | 0.50678 | 0.50743 | 0.50809 | |
| 0.44 | 0.93062 | 0.92682 | 0.92296 | 0.91905 | 0.91508 | 0.51603 | 0.51670 | 0.51736 | 0.51801 | 0.51865 | |
| 0.45 | 0.92851 | 0.92469 | 0.92080 | 0.91686 | 0.91287 | 0.52656 | 0.52723 | 0.52788 | 0.52852 | 0.52916 | |
| 0.46 | 0.92638 | 0.92253 | 0.91862 | 0.91465 | 0.91064 | 0.53703 | 0.53769 | 0.53833 | 0.53896 | 0.53960 | |
| 0.47 | 0.92424 | 0.92037 | 0.91643 | 0.91244 | 0.90839 | 0.54743 | 0.54809 | 0.54873 | 0.54935 | 0.54996 | |
| 0.48 | 0.92208 | 0.91818 | 0.91422 | 0.91021 | 0.90614 | 0.55777 | 0.55842 | 0.55905 | 0.55966 | 0.56028 | |
| 0.49 | 0.91990 | 0.91598 | 0.91200 | 0.90796 | 0.90388 | 0.56805 | 0.56869 | 0.56931 | 0.56991 | 0.57052 | |
| 0.50 | 0.91771 | 0.91377 | 0.90977 | 0.90571 | 0.90160 | 0.57826 | 0.57889 | 0.57951 | 0.58010 | 0.58070 | |
| 0.51 | 0.91551 | 0.91155 | 0.90753 | 0.90345 | 0.89932 | 0.58841 | 0.58904 | 0.58964 | 0.59023 | 0.59081 | |
| 0.52 | 0.91329 | 0.90931 | 0.90527 | 0.90118 | 0.89703 | 0.59849 | 0.59911 | 0.59971 | 0.60029 | 0.60086 | |
| 0.53 | 0.91106 | 0.90707 | 0.90301 | 0.89890 | 0.89473 | 0.60851 | 0.60913 | 0.60971 | 0.61028 | 0.61085 | |
| 0.54 | 0.90882 | 0.90481 | 0.90074 | 0.89661 | 0.89242 | 0.61847 | 0.61907 | 0.61966 | 0.62021 | 0.62077 | |
| 0.55 | 0.90658 | 0.90255 | 0.89846 | 0.89431 | 0.89011 | 0.62837 | 0.62896 | 0.62953 | 0.63008 | 0.63063 | |
| 0.56 | 0.90432 | 0.90028 | 0.89617 | 0.89201 | 0.88779 | 0.63820 | 0.63879 | 0.63935 | 0.63988 | 0.64042 | |
| 0.57 | 0.90205 | 0.89799 | 0.89388 | 0.88970 | 0.88546 | 0.64797 | 0.64854 | 0.64910 | 0.64962 | 0.65014 | |
| 0.58 | 0.89977 | 0.89570 | 0.89157 | 0.88738 | 0.88313 | 0.65767 | 0.65824 | 0.65878 | 0.65930 | 0.65981 | |
| 0.59 | 0.89749 | 0.89340 | 0.88926 | 0.88506 | 0.88079 | 0.66731 | 0.66787 | 0.66840 | 0.66891 | 0.66941 | |
| 0.60 | 0.89520 | 0.89110 | 0.88694 | 0.88272 | 0.87845 | 0.67689 | 0.67744 | 0.67796 | 0.67845 | 0.67894 | |
| 0.61 | 0.89289 | 0.88878 | 0.88461 | 0.88038 | 0.87610 | 0.68640 | 0.68693 | 0.68745 | 0.68793 | 0.68841 | |
| 0.62 | 0.89058 | 0.88646 | 0.88228 | 0.87804 | 0.87374 | 0.69584 | 0.69637 | 0.69688 | 0.69735 | 0.69781 | |
| 0.63 | 0.88827 | 0.88413 | 0.87994 | 0.87568 | 0.87137 | 0.70523 | 0.70574 | 0.70624 | 0.70669 | 0.70714 | |
| 0.64 | 0.88594 | 0.88179 | 0.87759 | 0.87332 | 0.86900 | 0.71455 | 0.71505 | 0.71553 | 0.71597 | 0.71641 | |
| 0.65 | 0.88361 | 0.87945 | 0.87523 | 0.87096 | 0.86663 | 0.72380 | 0.72429 | 0.72476 | 0.72520 | 0.72562 | |
| 0.66 | 0.88127 | 0.87709 | 0.87287 | 0.86858 | 0.86424 | 0.73299 | 0.73346 | 0.73392 | 0.73434 | 0.73476 | |
| 0.67 | 0.87892 | 0.87473 | 0.87050 | 0.86620 | 0.86186 | 0.74212 | 0.74257 | 0.74302 | 0.74342 | 0.74383 | |
| 0.68 | 0.87656 | 0.87237 | 0.86812 | 0.86382 | 0.85946 | 0.75117 | 0.75162 | 0.75205 | 0.75245 | 0.75283 | |
| 0.69 | 0.87420 | 0.86999 | 0.86573 | 0.86142 | 0.85706 | 0.76016 | 0.76060 | 0.76101 | 0.76139 | 0.76177 | |
| 0.70 | 0.87183 | 0.86761 | 0.86334 | 0.85902 | 0.85465 | 0.76909 | 0.76951 | 0.76991 | 0.77027 | 0.77064 | |
| 0.71 | 0.86945 | 0.86522 | 0.86094 | 0.85662 | 0.85224 | 0.77795 | 0.77835 | 0.77873 | 0.77909 | 0.77944 | |
| 0.72 | 0.86706 | 0.86283 | 0.85854 | 0.85420 | 0.84982 | 0.78674 | 0.78713 | 0.78750 | 0.78784 | 0.78818 | |
| 0.73 | 0.86467 | 0.86043 | 0.85613 | 0.85178 | 0.84739 | 0.79546 | 0.79585 | 0.79620 | 0.79651 | 0.79684 | |
| 0.74 | 0.86227 | 0.85802 | 0.85371 | 0.84936 | 0.84495 | 0.80412 | 0.80449 | 0.80482 | 0.80513 | 0.80543 | |
| 0.75 | 0.85986 | 0.85560 | 0.85129 | 0.84693 | 0.84251 | 0.81271 | 0.81306 | 0.81339 | 0.81368 | 0.81396 | |
| 0.76 | 0.85745 | 0.85317 | 0.84885 | 0.84449 | 0.84007 | 0.82124 | 0.82156 | 0.82187 | 0.82215 | 0.82242 | |
| 0.77 | 0.85502 | 0.85074 | 0.84641 | 0.84204 | 0.83761 | 0.82969 | 0.83000 | 0.83029 | 0.83055 | 0.83080 | |
| 0.78 | 0.85258 | 0.84830 | 0.84397 | 0.83958 | 0.83515 | 0.83806 | 0.83837 | 0.83865 | 0.83888 | 0.83912 | |
| 0.79 | 0.85014 | 0.84585 | 0.84151 | 0.83712 | 0.83268 | 0.84638 | 0.84666 | 0.84692 | 0.84715 | 0.84736 | |
| 0.80 | 0.84768 | 0.84339 | 0.83904 | 0.83465 | 0.83020 | 0.85461 | 0.85489 | 0.85513 | 0.85534 | 0.85553 | |
| 0.81 | 0.84521 | 0.84091 | 0.83656 | 0.83216 | 0.82771 | 0.86277 | 0.86303 | 0.86326 | 0.86345 | 0.86363 | |
| 0.82 | 0.84273 | 0.83843 | 0.83407 | 0.82967 | 0.82521 | 0.87086 | 0.87111 | 0.87131 | 0.87149 | 0.87165 | |
| 0.83 | 0.84024 | 0.83593 | 0.83157 | 0.82716 | 0.82270 | 0.87888 | 0.87910 | 0.87930 | 0.87945 | 0.87960 | |
| 0.84 | 0.83772 | 0.83341 | 0.82905 | 0.82464 | 0.82018 | 0.88680 | 0.88701 | 0.88719 | 0.88733 | 0.88747 | |
| 0.85 | 0.83519 | 0.83088 | 0.82651 | 0.82210 | 0.81764 | 0.89465 | 0.89485 | 0.89500 | 0.89513 | 0.89525 | |
| 0.86 | 0.83264 | 0.82832 | 0.82396 | 0.81955 | 0.81509 | 0.90241 | 0.90258 | 0.90274 | 0.90285 | 0.90296 | |
| 0.87 | 0.83007 | 0.82575 | 0.82139 | 0.81698 | 0.81252 | 0.91008 | 0.91025 | 0.91039 | 0.91049 | 0.91058 | |
| 0.88 | 0.82747 | 0.82315 | 0.81879 | 0.81439 | 0.80993 | 0.91766 | 0.91781 | 0.91794 | 0.91803 | 0.91811 | |
| 0.89 | 0.82485 | 0.82053 | 0.81617 | 0.81177 | 0.80733 | 0.92515 | 0.92528 | 0.92540 | 0.92548 | 0.92556 | |
| 0.90 | 0.82220 | 0.81788 | 0.81353 | 0.80913 | 0.80470 | 0.93254 | 0.93266 | 0.93277 | 0.93283 | 0.93291 | |
| 0.91 | 0.81952 | 0.81521 | 0.81085 | 0.80647 | 0.80204 | 0.93983 | 0.93994 | 0.94003 | 0.94010 | 0.94016 | |
| 0.92 | 0.81681 | 0.81249 | 0.80815 | 0.80377 | 0.79936 | 0.94701 | 0.94710 | 0.94719 | 0.94725 | 0.94732 | |
| 0.93 | 0.81407 | 0.80975 | 0.80541 | 0.80104 | 0.79665 | 0.95410 | 0.95417 | 0.95424 | 0.95429 | 0.95437 | |
| 0.94 | 0.81128 | 0.80697 | 0.80263 | 0.79827 | 0.79390 | 0.96105 | 0.96112 | 0.96117 | 0.96122 | 0.96130 | |
| 0.95 | 0.80845 | 0.80414 | 0.79981 | 0.79547 | 0.79110 | 0.96789 | 0.96793 | 0.96798 | 0.96803 | 0.96810 | |
| 0.96 | 0.80558 | 0.80127 | 0.79695 | 0.79261 | 0.78827 | 0.97460 | 0.97463 | 0.97468 | 0.97471 | 0.97479 | |
| 0.97 | 0.80266 | 0.79836 | 0.79404 | 0.78971 | 0.78538 | 0.98119 | 0.98121 | 0.98123 | 0.98126 | 0.98133 | |
| 0.98 | 0.79968 | 0.79538 | 0.79107 | 0.78676 | 0.78243 | 0.98762 | 0.98762 | 0.98764 | 0.98767 | 0.98773 | |
| 0.99 | 0.79663 | 0.79235 | 0.78805 | 0.78374 | 0.77941 | 0.99389 | 0.99390 | 0.99391 | 0.99392 | 0.99395 | |
| 1.00 | 0.79351 | 0.78924 | 0.78495 | 0.78065 | 0.77631 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | |