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Vapor pressure Osmometer

I. Summary Thermodynamic properties of a solvent like vapour pressure, boiling point, or freezing point can be altered when small amounts of a soluble substance are added to this solvent. Such colligative effects are proportional to the number of added molecules. Therefore, this method can for example be applied for the determination of the molecular weight of soluble biopolymers.

II. Introduction The chemical potential of a solvent is altered when a non-volatile substance is dis-solved in it. The property changes caused hereby are called colligative phenomena. Among them we know:

1. Lowering of the vapour pressure 2. Rising of the boiling point 3. Lowering of the freezing point 4. Changing the osmotic pressure

In dilute solutions such phenomena depend only on the number of dissolved particles but not on their chemical nature (molecule, ions), except that the substance has to be non-volatile. Therefore these phenomena can be used to find the number of particles of an unknown substance and, thus, to determine its molecular weight. For that purpose, cryoscopy, ebullioscopy and isothermal distillation were popular for a long time. A disadvantage of these methods, however, was that they were not very accurate. Cryoscopy, for example, has typical standard deviations around 10%. Using ebullioscopy the molecular weights of hydrocarbons could be determined with standard deviations around 1-2%. The applied procedure, however, is quite cumbersome. Isothermal distillation is very slow with typical durations of a few days. Ebullioscopy, on the other hand, has the disadvantage that a relatively large quantity of substance is needed. Furthermore cryoscopy and ebullioscopy are confined to a temperature region determined by the solvent. If the temperature is too high the substance decomposes or might react with the solvent. If the temperature is too low the solubility might be insufficient. Moreover, these methods cannot be used to investigate equilibrium as a function of temperature without changing the solvent. More than 40 years ago Hill proposed a different method for the determination of molecular weights. The method does not feature the disadvantages mentioned above. It is well suited for routine analysis as well as for physico-chemical investigations. Today the method is generally referred to as vapour pressure osmometry.

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III. Experimental setup and measurement principle Two thermistors are placed in a chamber saturated with solvent vapour at a well-calibrated temperature. On each thermistor one drop of solvent is placed. The system is in equilibrium, i.e. the drops and the solvent have the same temperature. The pressure in the chamber is given by the external pressure. Now one drop of the solvent is substituted with a solution of an unknown, non-volatile substance in the same solvent. The temperature of this replaced drop is initially identical with the temperature of the system. Since the vapour pressure of the solution is lower than the vapour pressure of the pure solvent, solvent molecules from the saturated vapour phase condense on this drop. The released condensation heat increases the temperature of the solution drop. Condensation continues until the vapour pressure of the solution equals that of the solvent. The change in temperature changes the resistance of the thermistor. A bridge circuit measures the resistance difference of both thermistors. As long as changes in temperature are small, the resistance difference is proportional to ΔT. In equilibrium, ΔT is proportional to the molality of the solution:

ΔT = d·cm (1) ΔT = difference of the temperatures of the two drops (solution-solvent) d = constant of evaporation cm = molality of the solution In ideal solutions, d is independent of the nature of the dissolved substance and is determined only by the respective solvent. The electrical circuit used for the measurement of the resistance change resembles that of a Wheatstone bridge circuit. The advantage of this method is its high accuracy.

IV. Theory of vapour pressure osmometry

1. Determination of molecular weights: Pure solvent is in a chamber at a constant pressure p and a constant temperature t. There is an equilibrium between the liquid (L) and the gas phase (G). Therefore the chemical potential of both phases is equal:

µL = µG (2)

with µ = µ (p,T). In a solution µ also depends on the mole fraction, i.e. µ = µ(p,T,x). In the following the solvent will be indicated with the index 1 (x1 is the mole fraction of the solvent). If a non-volatile substance (index 2) is dissolved in the solvent, the chemical potential µ of the solvent in the liquid phase, µL

1, is changed, since the mole fraction of the solvent, x1, is changed. Equation (2) therefore is not satisfied anymore. By changing its temperature (we assume p=const.), the system adjusts to a new equilibrium.

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For the total differential of the chemical potential holds:

!

dµ1L(p,T,x1

L) = (

"µ1L

"T)p,x1

L dT + ("µ1

L

"x1) p,T dx1

L + ("µ1

L

"p)T ,x1

L dp (3)

with dp = 0 when p=cost

!

dµ1G(p,T,x1

G) = (

"µ1G

"T)p,x1

G dT + ("µ1

G

"x1G) p,T dx1

G + ("µ1

G

"p)T ,x1

G dp (4)

with dp=0 at p= const and dx1

G = 0, since the substance is non-volatile.

The new equilibrium between the two phases is achieved when the two changes in chemical potential are equal, i.e. when

dμL

1 = dμ1G (5)

Putting together equations (3) and (4) in (5) reveals

!

("µ1

L

"T)p,x1

L dT + ("µ1

L

"x1L) p,T dx1

L = ("µ1

G

"T)p,x1

G dT (6)

From thermodynamics it is known:

!

("µ1L,G

"T) p,x1

dT = #S1L,G ;

!

("µ1

L

"x1) p,T dx1 = RTd lna1 (7)

and

!

(S1L" S1

G) =

#1

T (8)

where a1 and Λ1 are the activity and the molar vaporization heat of the solvent at temperature T. Inserting relations (7) and (8) into equation (6) reveals:

!

d lna1 ="1

RT2dT (9)

For small changes of temperature dT Λ1 is constant. Then equation (9) can be integrated:

!

d lna1

1

a1

" =#1R

dT

T2

T1

T2

" →

!

lna1 = "#1

R(1

T2

"1

T1

) (10)

With the already used assumptions that dT << T1,T2, or T1≈T2 equation can be rewritten:

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!

lna1 = "#1

RT12$T (11)

Also, a1= g1 x1 is valid, where g1 is the activity coefficient of the solvent. For very diluted solution (x1@1, x2<<1) g1 can be set to 1, and using ln a1 =ln g1 +ln x1 @ ln x1 = ln (1-x2) @ -x2 one obtains:

!

"T =RT1

2

#1x2 (12)

This relation shows that the change in temperature ΔT depends only on the mole fraction of the dissolved substance x2, i.e. on the number of the dissolved particles, and on the nature of the solvent. It is thus independent on the nature of the dissolved substance. Since x2 essentially depends on the molecular weight of the dissolved substance M2, equation (12) can be used for determination of the molecular weight of M2:

!

x2 =n2

n1 + n2"n2

n1

Where ni are the number of moles of the solvent (n1) and the dissolved material (n2). For strongly diluted solutions it is possible to neglect n2 in the denominator, since n1>>n2. Additional transformations reveal for x2:

!

x2

=n2

n1

=g2/M

2

g1/M

1

=g2

M2

"M1

g1

=g2

g1

" M1"1

M2

(13)

Here, g1 and g2 are the initial weights of the solvent and of the dissolved substance, respectively, and M1 is the known molecular weight of the solvent. From that it follows from equation (12):

!

"T =RT

1

2

#1

$g2

g1

$1

M2

=RT

1

2

#1

$ cm (14)

in which Λ1 = M1. λ1 is introduced. λ1 is the vaporization heat of the solvent per gram at temperature T1. When the initial weights g1 and g2 are known and the temperature difference between the solution and the solvent is measured, equation (14) can be used to determine the molecular weight M2. It is important to be aware of the approximations used when equation (14) was derived:

• A very diluted solution is considered an ideal solution • Λ1 is independent of the temperature in the interval ΔT • The activity coefficient g1 is set to 1

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2. Determination of activity coefficients:

We start with equation (11):

!

"lna1 =#1

RT12$T (11)

and we consider the change of g1 with the mole fraction of the dissolved material. For that purpose we express the term –ln a1 with the true molecular weight M2. At first, from equation (11) follows with a1 = g1x1:

!

" ln a1= " ln#

1" ln x

1=

$1

RT1

2%T (15)

ln g1 can now be written as a power series of x1 or x2, respectively:

!

ln"1 =# + $x2 + Ax22

+ ...... For increasing dilution of the solution, g1 will approach 1 with a horizontal tangent:

!

lim x2"0

ln#1 = 0 and

!

lim x2"0

# ln$1

#x2

= 0 (16)

This, however, means that a = b = 0. For this reason the first term of the expansion of lng1 is quadratic:

!

ln"1 = Ax22

+ ...... (17) For -ln x1 it thus follows:

!

"ln x1 = "ln(1" x2) = x2 +1

2x22

+ ...... (18)

Since x2<<1 we can discard terms that are of higher than quadratic order in x2 and with equations (17) and (18) we get from equation (15) :

!

"lna1 = x2 +1

2x22" Ax2

2= x2 + (

1

2" A)x2

2 (19)

The mole fraction x2 can now be expressed using the real molecular weight M2:

!

x2 =n2

n1 + n2=

g2 /M2

n1 + g2 /M2

=g2

n1M2

(1+g2

n1M2

)"1 (20)

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One assumption was that

!

g2

n1M

2

<<1. Therefore, x2 can be written as a power

series in g2/n1M2. If higher than quadratic terms are discarded again, we end up with:

!

x2 =g2

n1M2

(1"g2

n1M2

) (21)

and

!

x22

= (g2

n1M2

)2 (22)

if we insert equations (21) and (22) into equation (19) we get:

!

"lna1 =g2

n1M2

(1"g2

n1M2

) + (1

2" A)(

g2

n1M2

)2 (23)

On the other hand, equation (11) was:

!

"lna1

=#1

RT1

2$T (11)

From equations (14) we can express ΔT as:

!

"T =RT

1

2

#1

$g2

g1

$M1

M2

' (24)

Here, we have to write the apparent molecular weight since for the derivation of equation (14) the assumptions noted above had been used.

From equations (11) and (24) follows:

!

"lna1 =g2

g1.M1

M2

' (25)

M’2 can now be expressed in terms of the real molecular weight M2:

M’2 = M2 + ΔM (26) From equation (25) after writing a power series and considering only up to quadratic terms, we get for (M2’)-1:

!

1

M '2= (M2 + "M)

#1=1

M2

(1+"M

M2

) #1$1

M2

(1#"M

M2

) (27)

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with ΔM << M2 With (26) follows from (25):

!

"lna1 =g2

g1.M1

M2

(1"#M

M2

) =g2

n1.1

M2

(1"#M

M2

) (28)

Equating (23) and (28) reveals for ΔM2:

!

"M2 = (1

2+ A)

g2

n1 (29)

and:

!

"M2 = M2'#M2 = (

1

2+ A)

g2

n1 (30)

Plotting M’2 for various g2 versus g2/n1 should reveal a straight line. The slope of this line provides A and the axis intercept (g2 = 0) corresponds to the real molecular weight M2. From equation (17) the activity coefficient g1can be determined:

!

"1

= eAx

2

'2

(31) Finally, from (31) and (22) follows:

!

"1(g

2/n1) = e

A (g2 / n1M 2')2 (32)

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V. Experiment and analysis

1. Preparation of the calibration solution:

Prepare 4 solutions of different concentrations from the calibration substance received from the assistant. It is very important that the substance is well soluble in all four concentrations. Generally, the molalities of the solutions should be in the interval

0.02 Mol/Kg § cm § 0.10 Mol/Kg

The molalities of the solutions have to be known very accurately!

2. Initial operation of the vapor pressure osmometer:

First of all the cell is filled with about 20 ml of the solvent. The apparatus is then put together according to the instructions available in the lab. In the final step two syringes are filled with the solvent and four with the four different solutions of the calibration substance. It is important that the syringes are very clean. After the syringes are put into the apertures on the top of the osmometer, the apparatus is switched on. Before any measurement can be started, it is necessary to wait about 1,5-2 hours. During this time it is for example possible, to prepare additional solutions.

3. The measurement:

When the apparatus is ready, the zero point is adjusted. First, the coarsest level of sensibility (position 1024) is used. On both thermistors, one drop of pure solvent is put. After some time the temperature measurements should arrive to a constant value. With the help of the calibration button the scale is adjusted to zero. Afterwards, the sensibility of the measurement bridge is increased gradually by turning the corresponding switch twoards position 1. In the case that the scale should start to deviate at some point it has to be readjusted to zero. The calibration is redone twice by changing the solvent drops on both thermistors. The calibration should be reproducible within about ±1/2 scale units, of course without any additional manual adjustment. After changing the drops on the thermistors it takes between 3 to 10 minutes until the equilibrium is reached again and the measurement can be done. When the calibration is finished and the zero point is adjusted correctly, the measurement can be started. For this purpose the solvent drop on one thermistor is replaced with a drop of the solution of lowest concentration. In order to remove completely the pure solvent the thermistor is rinsed with 4 to 5 drops of solution. Then, also the drop of pure solvent on the second thermistor is replaced with a new

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drop of solvent. When the new equilibrium is reached the measurement is done and the result is noted. In order to check this value, the procedure is repeated and a second measurement is done. If the two values are equal, it can be proceeded to the next concentration. If the two values should differ by more than 1 scale division, a third measurement has to be done. This third value should not deviate more than 1 scale division from the second one. The average of second and third value is then the result of this measurement. Correspondingly, the measurements on the other concentrations are carried out.

4. Analysis:

a. 1. Method

For the calibration substance, the measured voltages multiplied by the measuring rangeare plotted against the corresponding molalities.

Figure 1: calibration curve

For the unknown solutions, the apparent molecular weight M’2 can now be found using this calibration curve with the equation:

!

M2'

=weight(mg)

solvent(g) "molality (33)

This method should, however, be used only for small molecular weights. In the case of larger molecular weights the second method should be used.

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b. 2. Method

In this method the measured voltages are divided by the concentration (molalities!). These values are then plotted against the molality. The obtained points are fitted to a straight line, which is extended such that it crosses the y-axes at a concentration of 0. This value reveals the calibration constant Ccal. Such a calibration line is presented in Figure 2.

Figure 2: calibration curve for the value Ccal

The values measured for the unknown substance are plotted correspondingly as shown in Figure 3. Here, the quotient g2/gl is used (g2 = initial weight of the unknown substance (g), gl = weight of solvent (kg)). The intersection of this second line with the y-axes at a concentration of 0 reveals the factor C.

Figure 3: curve for the value C

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The molecular weight is then found through the relation:

!

M =CCal

C (34)

c. 3. Method

The apparent molecular weights determined according to method 1 are plotted against the quotient g2/n1. A straight-line results that crosses the ordinate at g2/n1. = 0. This crossing point reveals the real molecular weight. Here, g2 are the initial weights of the unknown substance and n1 is the amount of moles of solvent. The slope of the line allows to determine A and finally the coefficient of activity γI for the different concentrations according to equation (30):

!

"M2 = M2'#M2 = (

1

2+ A)

g2

n1 (30)

d. Determination of the vapour pressure of a solvent

From the equation (14) follows that the measured temperature difference ΔT is proportional to the molality of the dissolved substance. The proportional constant m

!

m =RT1

2

"1 (35)

only depends on the used solvent. R is the universal gas constant, T the temperature the measurement has been done, λ1 the heat of evaporation of the solvent per g at temperature T. The constant of proportionality m corresponds to the slope of the line when method 1 is applied for analysis. When experiments are carried out with two different solvents, the following ratio can be formed:

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!

m1

m2

=RT1

2

"1."2

RT22

(36)

If T1=T2 we get:

!

"2 = "1m1

m2

(37)

If λ1 is known, λ2 can be determined. e. Temperature dependence of the evaporation heat

Equation (36) can be converted as follows:

!

"2 = "1m1

m2

T22

T12

(38)

With this equation the temperature dependence of the evaporation heat can be determined. To do so, the slopes of straight lines determined according to method 1 have to be determined at different temperatures. According to equation (38) the quotient of the slopes reveals the ratio of the evaporation heats when T1 and T2 are known.

5. Exercises:

a. Measuring the calibration curve b. Determine the molecular weight of unknown substances using different

methods. Discuss the results. c. Determine the heat of evaporation d. Calculate the activity coefficient according to equation (30).