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Page 1

Partial Molal Volume

Purpose: Determine the partial molal volumes of sodium chloride and water in a series of

solutions from 0.3 to 3.0 m.

Reading Assignment: Section 7.1, P. W. Atkins, Physical Chemistry, 7th Ed.

Introduction

Partial molal quantities tell us how the properties of solutions change with concentration. We

need to know partial molal quantities for all the extensive properties of a solution, including V,

G, H, S, and A. For example, the partial molal volume is important in oceanography and aquatic

environmental science, which is why we measure the partial molal volume of NaCl solutions in

this lab. Another example is that partial molal volume is needed in biochemistry for careful

calculations of the molecular weights of proteins and nucleic acids using ultracentrifugation. The

partial molal Gibbs Free energy, which is called the chemical potential, is central to the study of

solutions.

Take an example of a two-component solution with n1 moles of component 1 and n2 moles of

component 2. The change in Gibbs Free Energy with concentration at constant temperature and

pressure is

dG = (∂G∂n1)

n2

dn1 + (

∂G

∂n2

) n1dn2 1

The partial molal Gibbs Free Energy with respect to changes in the number of moles of

component 1 is

(∂G∂n1)

n2

= µ1 2

The partial molal Gibbs Free Energy is the Gibbs Free Energy per mole of compound in the

solution. The partial molal Gibbs Free Energy with respect to changes in the number of moles of

component 2 is

(∂G∂n2)

n1

= µ2 3

Substituting these definitions into Equation 1 gives:

dG = µ1 dn1 + µ2 dn2 4

which determines the change in Gibbs Free Energy for changes in concentration. The important

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Partial Molal Volume 6

φV =

V1kg - 1000/d1

m (1 kg) 21

where d1 is the density of the pure solvent.

where w is the weight in grams of the solute in volume V of solution, for any volume of solution.

The Solvent. In order to calculate to calculate the partial molal volume of the solvent, we simply

use the total volume and the partial molal volume of the solute and Equation 8. For V the volume

of the solution containing 1kg of solvent, and n2 = m(1kg), then Equation 8 becomes:

V1kg = (1000/M1) V1 + m(1kg) V2 22

Measuring the Density. The density will be measured using a density meter. The meter has a

glass U-tube that is filled with the sample. The U-tube is made to vibrate and the vibration

frequency is measured. The vibration frequency is a sensitive measure of the density of the

solution. The more dense the solution, the smaller the vibration frequency. The meter is computer

controlled, and the computer converts the vibration frequency into the measured density using a

polynomial calibration equation. Density is a strong function of temperature. The meter also

determines the temperature and applies a correction for the density calculation. The meter is

calibrated using reagent grade water.

To cause the U-tube to vibrate, a small magnet is glued onto the end of the U-tube. The magnet

is placed in a coil that has an oscillating current in it, Figure 3. The frequency of the oscillating

current is varied until the natural vibration frequency, that is the resonance frequency, of the U-

tube is found. In operation, the density meter is very similar to the speaker on a stereo.

Marginal

Oscillator

ν

sample in

tube

Figure 3. The operation of the density meter.

Experimental

Apparatus

In addition to sodium chloride, the experiment requires a Mettler density meter, five 50-ml

stoppered flasks, and a constant temperature bath set to 25.0°C.

Page 7

Partial Molal Volume 7

Procedure

Prepare about 25 ml of five solutions of sodium chloride in distilled water ranging in molal

concentration from about 0.3 to 3.0. Weigh the solute with an analytical balance and solvent

with a milligram balance into dry stoppered flasks. You should calibrate the density meter while

you are making the solutions, to save time. Degass the sample for about a minute to avoid bubble

formation in the density meter. Suspend the flasks in the 25°C bath to reach constant

temperature.

The densities of the solutions are measured with a Mettler density meter that has been

calibrated by determining the density of water (see tables in the CRC or page 6 of the meter

instructions) at the temperature of the measurement using the following instructions. First, rinse

the meter with a few portions of reagent grade water using the following steps. Fill the meter by

dipping the inlet tube into the liquid and pulling the liquid into the oscillator U-tube using the

syringe. Carefully examine the U-tube to ensure that there are no bubbles. Select "DENS" as the

measured value by pressing the "MEAS" key and then pressing the ∆ or ∇ . Monitor the density

reading for 30 sec. If the reading doesn't stabilize there is an air bubble in the measurement U-

tube: empty and refill if necessary. Press the "ENTER" and "CALIB" key simultaneously. The

"CALIB" symbol will be flashing and "AUTO" is displayed. The automatic calibration is at an

end when "MEAS" is again displayed, which can take a minimum of one minute and a maximum

of 15 minutes.

Check the calibration by emptying the measuring tube and refilling with reagent grade water. If

the difference between the measured and tabulated values is greater than 0.001 g/cm3, check to

see that the measuring U-tube is clean and repeat the measurement. If the difference is less than

0.001 g/cm3, repeat the determination two more times and use the average difference between the

measured and tabulated values to correct all subsequent readings.

Exercise extreme care in filling, rinsing, cleaning, and handling the meter.

Measure the density of the sodium chloride solutions, taking the same care used in calibrating

the meter. Keep the sample flask immersed in the thermostat bath while filling the meter.

Results and Calculations

Equations 16-22 are all based on using the volume of the solution that contains 1 kg of solvent.

Use Equation 16 to calculate the volume of each solution for 1000g of solvent. Fit the V using a

cubic polynomial in m, Eqn. 17; include pure water in your fit. From the fit coefficients and

Equation 18, calculate the partial molal volume of the solute at each molality. Then use Equation

22 to calculate the partial molal volume of the solvent for each solution. With Equation 21

calculate the apparent molal volumes of the solute for each solution6,7. Plot the partial molal

volumes versus m. Also, separately, plot the apparent molal volume versus m.

For the error analysis we need to get estimates of the uncertainty in the fit coefficients. Use the

Nonlinear Least Squares Curve Fitting- 4 Parameters applet5 that calculates the uncertainties in

the fit coefficients. Alternatively, you can also use a curve fitting spreadsheet that calculates the

uncertainties. The cubest.xls spreadsheet has a link from the lab manual page. Based on the fit

coefficient uncertainties, use significant figure rules or propagation of errors to calculate the

uncertainty in a typical partial molal volume for one solution for the solute and the solvent.

Partial Molal Volume

Purpose: Determine the partial molal volumes of sodium chloride and water in a series of

solutions from 0.3 to 3.0 m.

Reading Assignment: Section 7.1, P. W. Atkins, Physical Chemistry, 7th Ed.

Introduction

Partial molal quantities tell us how the properties of solutions change with concentration. We

need to know partial molal quantities for all the extensive properties of a solution, including V,

G, H, S, and A. For example, the partial molal volume is important in oceanography and aquatic

environmental science, which is why we measure the partial molal volume of NaCl solutions in

this lab. Another example is that partial molal volume is needed in biochemistry for careful

calculations of the molecular weights of proteins and nucleic acids using ultracentrifugation. The

partial molal Gibbs Free energy, which is called the chemical potential, is central to the study of

solutions.

Take an example of a two-component solution with n1 moles of component 1 and n2 moles of

component 2. The change in Gibbs Free Energy with concentration at constant temperature and

pressure is

dG = (∂G∂n1)

n2

dn1 + (

∂G

∂n2

) n1dn2 1

The partial molal Gibbs Free Energy with respect to changes in the number of moles of

component 1 is

(∂G∂n1)

n2

= µ1 2

The partial molal Gibbs Free Energy is the Gibbs Free Energy per mole of compound in the

solution. The partial molal Gibbs Free Energy with respect to changes in the number of moles of

component 2 is

(∂G∂n2)

n1

= µ2 3

Substituting these definitions into Equation 1 gives:

dG = µ1 dn1 + µ2 dn2 4

which determines the change in Gibbs Free Energy for changes in concentration. The important

Page 6

Partial Molal Volume 6

φV =

V1kg - 1000/d1

m (1 kg) 21

where d1 is the density of the pure solvent.

where w is the weight in grams of the solute in volume V of solution, for any volume of solution.

The Solvent. In order to calculate to calculate the partial molal volume of the solvent, we simply

use the total volume and the partial molal volume of the solute and Equation 8. For V the volume

of the solution containing 1kg of solvent, and n2 = m(1kg), then Equation 8 becomes:

V1kg = (1000/M1) V1 + m(1kg) V2 22

Measuring the Density. The density will be measured using a density meter. The meter has a

glass U-tube that is filled with the sample. The U-tube is made to vibrate and the vibration

frequency is measured. The vibration frequency is a sensitive measure of the density of the

solution. The more dense the solution, the smaller the vibration frequency. The meter is computer

controlled, and the computer converts the vibration frequency into the measured density using a

polynomial calibration equation. Density is a strong function of temperature. The meter also

determines the temperature and applies a correction for the density calculation. The meter is

calibrated using reagent grade water.

To cause the U-tube to vibrate, a small magnet is glued onto the end of the U-tube. The magnet

is placed in a coil that has an oscillating current in it, Figure 3. The frequency of the oscillating

current is varied until the natural vibration frequency, that is the resonance frequency, of the U-

tube is found. In operation, the density meter is very similar to the speaker on a stereo.

Marginal

Oscillator

ν

sample in

tube

Figure 3. The operation of the density meter.

Experimental

Apparatus

In addition to sodium chloride, the experiment requires a Mettler density meter, five 50-ml

stoppered flasks, and a constant temperature bath set to 25.0°C.

Page 7

Partial Molal Volume 7

Procedure

Prepare about 25 ml of five solutions of sodium chloride in distilled water ranging in molal

concentration from about 0.3 to 3.0. Weigh the solute with an analytical balance and solvent

with a milligram balance into dry stoppered flasks. You should calibrate the density meter while

you are making the solutions, to save time. Degass the sample for about a minute to avoid bubble

formation in the density meter. Suspend the flasks in the 25°C bath to reach constant

temperature.

The densities of the solutions are measured with a Mettler density meter that has been

calibrated by determining the density of water (see tables in the CRC or page 6 of the meter

instructions) at the temperature of the measurement using the following instructions. First, rinse

the meter with a few portions of reagent grade water using the following steps. Fill the meter by

dipping the inlet tube into the liquid and pulling the liquid into the oscillator U-tube using the

syringe. Carefully examine the U-tube to ensure that there are no bubbles. Select "DENS" as the

measured value by pressing the "MEAS" key and then pressing the ∆ or ∇ . Monitor the density

reading for 30 sec. If the reading doesn't stabilize there is an air bubble in the measurement U-

tube: empty and refill if necessary. Press the "ENTER" and "CALIB" key simultaneously. The

"CALIB" symbol will be flashing and "AUTO" is displayed. The automatic calibration is at an

end when "MEAS" is again displayed, which can take a minimum of one minute and a maximum

of 15 minutes.

Check the calibration by emptying the measuring tube and refilling with reagent grade water. If

the difference between the measured and tabulated values is greater than 0.001 g/cm3, check to

see that the measuring U-tube is clean and repeat the measurement. If the difference is less than

0.001 g/cm3, repeat the determination two more times and use the average difference between the

measured and tabulated values to correct all subsequent readings.

Exercise extreme care in filling, rinsing, cleaning, and handling the meter.

Measure the density of the sodium chloride solutions, taking the same care used in calibrating

the meter. Keep the sample flask immersed in the thermostat bath while filling the meter.

Results and Calculations

Equations 16-22 are all based on using the volume of the solution that contains 1 kg of solvent.

Use Equation 16 to calculate the volume of each solution for 1000g of solvent. Fit the V using a

cubic polynomial in m, Eqn. 17; include pure water in your fit. From the fit coefficients and

Equation 18, calculate the partial molal volume of the solute at each molality. Then use Equation

22 to calculate the partial molal volume of the solvent for each solution. With Equation 21

calculate the apparent molal volumes of the solute for each solution6,7. Plot the partial molal

volumes versus m. Also, separately, plot the apparent molal volume versus m.

For the error analysis we need to get estimates of the uncertainty in the fit coefficients. Use the

Nonlinear Least Squares Curve Fitting- 4 Parameters applet5 that calculates the uncertainties in

the fit coefficients. Alternatively, you can also use a curve fitting spreadsheet that calculates the

uncertainties. The cubest.xls spreadsheet has a link from the lab manual page. Based on the fit

coefficient uncertainties, use significant figure rules or propagation of errors to calculate the

uncertainty in a typical partial molal volume for one solution for the solute and the solvent.