Arun - Partial Molar Volume PDF

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Arun Ajmera CHEM 3851 sec. 002 Dr. Yumin Li November 13, 2012 Partial Molar Volume of a Salt in Aqueous Solution Introduction The purpose of this experiment was to use the accurate determination of density to calculate the partial molar volumes of the various components in a solution. Partial molar volume is the contribution that a component of a mixture makes to the overall volume of the solution [2]. The experimentally determined partial molar volumes were then compared to literature values. The literature partial molar volume of NaCl was 16.62 [2]. The mass of both water and NaCl was measured to four decimal places using an analytical balance. These experimentally determined partial molar volumes are then compared to calculated or literature values. In order to ensure accurate and precise measurements, the pycnometers were filled until the capillaries were full with solution, and any solution that dripped down the outside was dried off. Theory Before looking specifically at partial molar volume, we will consider a general property Y. For property Y at constant temperature, the pressure will be a function of the composition; if Y has two components, component 1 and 2, the partial molar quantities can be defined as equations 1a and b.

( ᵨᵨYn )

Y 1 ,m =

1 n2 , T , P

Eqn. 1a

( )

Y 2 ,m =

ᵨY ᵨ n2

Eqn. 1b

n2 , T , P

Theoretically, this relationship can be written as equation 2. In this experiment, we will specifically explore partial molar volumes, equation 3. Y =n 1 Y 1 ,m + n2 Y 2, m Eqn. 2 V =n 1 Y 1 ,m + n2 Y 2, m Eqn. 3

Partial molar volume is the contribution that a component of a mixture makes to the overall volume of the solution. It may thus be denoted as the change in volume per mole of substance x added to solution. To calculate the partial molar volumes, we first needed to calculate the mole fractions of the solute and solvent, then the total molar volume can be deduced, equations 4a and 4b. The variable n refers to the number of moles, x refers to partial molar volume, Vm refers to molar volume, component 1 refers to the water portion of the mixture, and component 2 refers to the salt. x 2=

n2 n

and x 1=

( ᵨᵨVn )

V 1 ,m =

n1 n

Eqn. 4a

1 n2

V m=

V n

Eqn. 4b

Equation 4a can then be rewritten as seen in equation 5a, and the derivative with respect to n1 can be related to the derivative with respect to x2 using the chain rule, equation 5b.

( ) ( )

( )

ᵨn V m ᵨV m ᵨV = =V m + n ᵨn1 n ᵨn1 ᵨn1 n 2

2

n2

Eqn. 5a

( ) ( )( ) ᵨV m ᵨn 1

=

n2

ᵨV m ᵨx 2

ᵨx 2

1

ᵨn1

Eqn. 5b

n2

The mole fraction of component two can be related to equation 5b to produce equation 5c. This relationship can be rearranged to be more easily visualized in equation 6a. x 2=

n2 (n1 +n2 )

( )

−n2 −n ᵨx 2 = = 22 2 ᵨn1 n ( n1+ n2 ) n 2

( )

V 1 ,m =

ᵨV ᵨn1

( )( )

=V m −n n2

V1,m = Vm – x2

( ) ∂V m ∂ x2

n2 n

2

( )

ᵨV m ᵨV m =V m −x 2 ᵨx 2 1 ᵨx 2

Eqn. 5c

1

Eqn. 6a

Equation 6b was our 3rd degree polynomial fit, where the constants a0, a1, and a3 are obtained from the curve fitting program; the derivative is seen in equation 6c. The partial molar volume of the solvent at specified concentration can be calculated by then combining equation 6a with equation 6c, as seen below in equation 7. Vm = a0 + a1x2 + a2x22 + a3x22 Eqn. 6b

( )

∂V m 2 = a1+2 a2 x 2 +3 a3 x2 Eqn. 6c ∂ x2

V 1 ,m =V m−x 2

( ) ᵨV m ᵨx 2

Eqn.7 1

The final relationship for partial molar volume of the sodium chloride solution can then be obtained by rearranging equation 7 and relationship it back to equation three to derive equations 8 and 9. Vm is again the total molar volume, x1 and x2 are the mole fractions for solvent and solute respectively, and V2,m is the partial molar volume of salt.

V =V m=x 1 V 1,m + x 2 V 2 ,m Eqn.8 n

( V m−x 1 V 1 , m )

V2,m =

x2

Eqn. 9

Experimental Chemicals and Equipment Experimental Procedure See Attached Document Data Table1: Solution Preparation Data Solution %

Mass of salt used (g)

Mass of H2O used (g)

Moles of salt used

Moles of H2O used

Total number of moles, n

Mole fraction of H2O, x1

Mole fraction of salt, x2

2%

1.5263g

75.000g

0.026mol

4.162mol

4.189mol

0.994

0.006

4%

3.0465g

75.000g

0.0521mo l

4.162mol

4.215mol

0.988

0.012

8%

6.0714g

75.000g

0.104mol

4.162mol

4.267mol

0.976

0.024

12%

9.0508g

75.000g

0.155mol

4.162mol

4.318mol

0.964

0.036

16%

12.0581g

75.000g

0.206mol

4.162mol

4.370mol

0.953

0.047

The mass of the salt and the water was obtained using an analytical balance and all figures were reported to four decimal places. The moles of salt were calculated by (grams of salt) x (1 mol/MW of salt) and the same was done for moles of water (grams of water) x (1 mol/MW of water). To find the total number of moles for each solution the moles of salt and the moles of water were added together. The mole fractions were found by either (# mol water)/ (total # mol) or (# mol salt)/ (total # mol).

Table 2: Determination of the Density of Solutions Mass of pycnomete r & soln. (g)

Mass of solution

Density of solution (g/mL)

Solution %

Pycnomete r

Pycnomete r volume (mL)

Mass of dry pycnomete r (g)

2%

A

25.00

19.1943

44.6729

25.4786 1.019

B

25.00

18.9778

44.2691

25.2913 1.012

A

25.00

19.1943

44.8214

25.6271 1.025

B

25.00

18.9778

44.4061

25.4283 1.017

A

25.00

19.1943

45.7582

26.5639 1.063

B

25.00

18.9778

45.1392

26.1614 1.046

A

25.00

19.1943

46.4254

27.2311 1.089

B

25.00

18.9778

45.7611

26.7833 1.071

A

25.00

19.1943

47.1561

27.9618 1.118

B

25.00

18.9778

46.5016

27.5238 1.101

4%

8%

12%

16%

Avg. Density (g/mL) 1.015

1.021

1.055

1.080

1.110

The volume of the pycnometer was given as a standard from the manufacture while the mass of the pycnometers was obtained by using an analytical balance. Once the pycnometers were filled and weighed again the masses of the solutions were obtained by (mass of pycnometer and solution) – (mass of pycnometer). The densities were obtained by (mass of solution)/(volume of pycnometer). Table 3: Calculated Data Solutio n

Total Molar Volume (mL/mol)

XA of Solute

Deriv. of Vm with respect to solute

Partial molar volume of solvent (mL/mol)

Partial molar volume of solute (mL/mol)

2%

17.99

0.006

17.245

1.788*10^1

35.128

4%

18.132

0.013

6.153

1.806*10^1

24.209

8%

18.017

0.026

-6.222

1.817*10^1

11.946

12%

18.018

0.04

-6.527

1.825*10^1

11.726

16%

17.954

0.055

4.298

1.775*10^1

22.049

Table 4: Error Propagation Solution

Error in salt (x10-13)

Error in H2O (x10-15)

Total Error (x10-7)

2%

1.648

0

4.06

4%

1.607

0

4.012

8%

1.531

1.003

3.925

12%

1.46

2.126

3.848

Q6%

1.392

3.598

3.779

The data shown in Tables 3 and 4 were obtained via the processes described below in the Data Work-Up. The trends and values are analyzed in the results and discussion.

Percent Error Calculation: Literature Value of Partial Molar Volume of NaCl: 16.62 mol/L Experimental Value of Partial Molar Volume of NaCl: 21.011 mol/L 21.011 −16.62 ∗100=26.42% Error 16.62 Data Work-Up Constants i 0 4 MWNaCl 58.44

Range Variable gm mol

Molecular Weight NaCl

gm

MWH2O  18.0148 mol

Molecular Weight Water Data Weights of NaCl used, in grams, for 2, 4, 8, 12, and 16% solutions from top to bottom.

 1.5263   3.0465    gmsalt  6.0714  gm  9.0508     12.0581 Weights of H2O used, in grams, for 2, 4, 8, 12, and 16% solutions from top to bottom.  75   75    gmH2O 75  gm  75     75  Calculations A. Moles, total moles, and mole fractions nsalt 

gmsalt MWNaCl

Definition for the calculation of moles of NaCl

Moles of NaCl for 2, 4, 8, 12, and 16% solutions from top to bottom.  0.026  0.052   nsalt   0.104 mol  0.155    0.206 nH2O

gmH2O MWH2O

Definition for the calculation of moles of H2O

Moles of H2O for 2, 4, 8, 12, and 16% solutions from top to bottom.  4.163   4.163    nH2O   4.163  mol  4.163     4.163  totalmolnsalt  nH2O Definition for the calculation of the total moles of each solution. Total moles of solution for 2, 4, 8, 12, and 16% solutions from top to bottom.

 4.189   4.215    totalmol   4.267  mol  4.318     4.37  nH2O X1  i

i

totalmol

i

Definition for the calculation of the mole fraction of water

Mole fractions of water for 2, 4, 8, 12, and 16% solutions from top to bottom.  0.994   0.988    X1   0.976  i  0.964     0.953  X2 i 

nsalt i totalmol

i

Definition for the calculation of the mole fraction of water

Mole fractions of NaCl for 2, 4, 8, 12, and 16% solutions from top to bottom.  0.006   0.012    X2   0.024  i  0.036     0.047  B. Pycnometer Calculations vol 25.00mL

volume of pycnometers

Pycnometer A mdryA 19.1943gm

 44.6729  44.8214   mpycsolnA  45.7582 gm  46.4254    47.1561 msolnA mpycsolnA  mdryA

Dry mass of pycnometer A

Mass of pynometer A and solution for 2, 4, 8, 12, and 16% solutions from top to bottom Definition to find the mass of solution for pynometer A.

Mass of solution in pycnometer A for 2, 4, 8, 12, and 16% solutions from top to bottom

 25.4786  25.6271   msolnA  26.5639 gm  27.2311    27.9618 dA 

msolnA vol

Definition for the density of solutions in pycnometer A

Densities for 2, 4, 8, 12, and 16% solutions from top to bottom for pycnometer A  1.019  1.025    gm dA  1.063    1.089  mL    1.118  Pycnometer B mdryB18.9778gm

Dry mass of pycnometer B

Mass of pynometer B and solution for 2, 4, 8, 12, and 16% solutions from top to bottom  44.2691  44.4061   mpycsolnB 45.1392 gm  45.7611    46.5016 msolnB mpycsolnB  mdryB Definition to find the mass of solution for pycnometer B. Mass of solution in pycnometer B for 2, 4, 8, 12, and 16% solutions from top to bottom  25.2913  25.4283   msolnB  26.1614 gm  26.7833    27.5238 dB 

msolnB vol

Definition for the density of solutions in pynometer B

Densities for 2, 4, 8, 12, and 16% solutions from top to bottom for pycnometer B  1.012   1.017    gm dB  1.046    1.071  mL    1.101 

dave 

dA  dB 2

Definition of Average Density

Average densities for 2, 4, 8, 12, and 16% solutions from top to bottom  1.015  1.021   gm dave   1.055  1.08  mL    1.11  C. Total Molar Volume mtotalgmsalt  gmH2O

Definition for the total mass of solution

Total mass of 2, 4, 8, 12, and 16% solutions from top to bottom  76.526    78.046 mtotal  81.071 gm  84.051    87.058 voltotal 

mtotal dave

 75.366  76.433   voltotal   76.881 mL  77.804    78.451 vm 

Definition of the total volume

Total volume

voltotal totalmol

   18.132   mL vm  18.017   18.018 mol    17.954

Definition of the total molar volume

17.99

Total Molar Volume

Total Molar Vo lume vs. Molar Fraction of Salt

5

Total Molar Volume (mL/mol)

1 .8 310 5

1 .8 261 0

5

1 .8 210

vm

5

1 .8 110

5

1 .81 0 5

1 .7 991 0

5

1 .7 910

0

0 .0 1

0.0 2

0 .03

3

0 .0 4

6 .3 421 0

0.0 5

0.0 6 0.0 56

X2

Molar Fraction of Salt t

th

l

f th l i

i

ti

dh

theslope slope (X2 vm) mL theslope   2.08 mol

the slope of the line

theintercept intercept(X2 vm) mL theintercept  18.075 mol

the intercept of the line

Vvm 

vm mL mol

Equation to remove units  17.99     18.132 Vvm   18.017  18.018    17.954 Units removed Polynomial Fits, Derivatives, and Partial Molar Volumes polyfit regress(X2Vvm3) Defining Polynomial Fit - equation used to make a 3rd order polynomial fit of mole fraction salt and molar volume Polynomial Fit Values

3    3     3  17.861  polyfit     31.824   3   1.303 10   4   1.428 10  2

vmfit( X2 ) polyfit  polyfit X2  polyfit X2  polyfit X2 3

4

5

3

6

Defining the function

 18.012

 18.082   vmfit( X2 )   18.069  17.985    17.962

Values of vm at X2 values

d dvmdX2(X2 )  vmfit( X2 ) dX2

Derivative of Vm with respect to salt

 i

thedvmdX2 dvmdX2 X2 i

Calculated derivative values with respect to salt

 17.245   6.153    thedvmdX2   6.222   6.527    4.298 

 

mL v1m vm  X2 dvmdX2 X2  i i i i mol

Equation to calculate the partial molar volumes of water partial molar volumes of H20 in ml/mol

1   1.788 10   1  1.806 10  mL v1m  1.817 101     mol  1.825 101     1.775 101   

v2mi 

 vmi 

X1i v1mi X2i

 Equation to calculate the partial molar volume of the salt

 35.128   24.209    mL v2m   11.946   11.726  mol    22.049 

partial molar volume of the salt

mol Averagev2m mean( v2m) mL Averagev2m 21.011

Average of partial molar volume of salt in mL/mol

l Error Propogation g 0.0001gm

Error in the weighing of the salt  gmsalt   MWNaCl  salt ( gmsaltgmH2O)   gmH2O    gmsalt   MWH2O   MWNaCl  Definition of the mole fraction of salt d dsaltdg (gmsaltgmH2O)  salt ( gmsalt gmH2O) dgmsalt errorinsalt dsaltdg gmsalt gmH2O





i

i

i

 2 (g )2

Partial derivative of molar fraction of salt with respect to water Definition for the error in our salt measurements

 1.648 10 13    13   1.607 10  errorinsalt  1.531 10 13     1.46 10 13     1.392 10 13   

Calculated Error for salt measurements

 gmH2O   MWH2O  H2O( gmsaltgmH2O)  gmH2O    gmsalt    MWH2O   MWNaCl 

Definition of the mole fraction of water

d dH2Odg ( gmsaltgmH2O)  H2O( gmsalt gmH2O) d gmH2O Partial derivative of

molar fraction of water with respect to salt

errorinH2O  dH2Odg gmsalt gmH2O i





i

 2 (g ) 2

i

Definition for the error in our water measurements

0  0    errorinH2O   1.003 10  2.126 10     3.598 10

  15   15   15  

Associated Error values for H2O 1



2

TotalErrori  errorinH2Oi  errorinsalt i

 4.06 10 7    7  4.012 10  TotalError   3.925 10 7     3.848 10 7      3.779 10 7   

Equation used to calculate total propagated error

Total propagated error

Results and Discussion The purpose of the experiment was to use the determination of density to calculate the partial molar volume NaCl in a solution. Our average experimental partial molar volume of NaCl was 21.011 mol/L, and the literature partial molar volume of NaCl at 25°C is 16.62 mol/L . The percentage of error of our experimental values compared to the literature value is 26.42%. Though the propagated error was very small, as discussed later, there is a significant difference between the literature value and experimental value of the partial molar volume of salt. This could be attributed to defective pycnometers, not completely wiping the overspill from the side of the pycnometer, or incorrectly estimating the final decimal point with taking volumetric measurements. Some noticeable trends in our data include that as the concentration of the solute increased, so did the density. As the number of moles of salt increased with increasing

concentration, the moles of solvent decreased. The mole fraction of water decreased with increasing concentration while the mole fraction of solute increased. The partial molar volume of solute increased with increasing solute concentration, while the partial molar volume of water decreased. These trends were all consistent with our knowledge of solutions, mixtures, moles, mole fraction, density, and partial molar volume. In general, the greater the concentration of solute in solution, the greater the mole fraction, number of moles, density, and partial molar volume...