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The objective of this laboratory is to explore the behavior of partial molar volumes in water-NaCl solutions as a function of concentration. By measuring masses of specific volumes of various concentrations of these solutions and using theoretical equations, we aim to calculate the partial molar volumes and understand the deviation from ideal behavior.

Experimental Procedure:

- Preparation of Solutions:
- Prepare a series of water-NaCl solutions with varying concentrations. Ensure accurate measurement of volumes and masses using precise laboratory equipment.
- Record the concentrations, volumes, and masses of each solution.

- Mass Measurements:
- Measure the mass of a specific volume of each solution using an analytical balance.
- Additionally, measure the mass of an empty volumetric flask to correct for the actual mass of the solution.

- Data Collection:
- Record all measurements, including masses of solutions and the empty flask, in a systematic manner.

- Calculations
**:**- Utilize the provided theoretical equations to calculate the partial molar volumes of water and NaCl in each solution.
- Apply the equations derived in the theory section to understand the impact of concentration on the overall volume of the solution.

Formulas:

Equation (1): m, total = ∑m,i V m, total =∑ i n i V m,i

Equation (3): Δ = m,total − m,total ∘ ΔV=V m,total −V m,total

Equation (4): ( ∂ ∂ ) ( ∂n i ∂V ) P,T,n j = V ˉ i

Equation (5): Δ = ∑ ˉ ΔV=∑ i n i V ˉ i

Results:

Calculate and tabulate the partial molar volumes for water (ˉwater Vˉ water) and NaCl (ˉNaClVˉNaCl) in each solution.

Note any deviations from ideal behavior and observe how concentrations affect the partial molar volumes.

Discussion:

Discuss the implications of the calculated partial molar volumes in relation to ideal solutions.

Analyze the deviations observed and explore potential reasons for non-ideality.

Relate the experimental findings to the theoretical concepts presented in the theory section.

Conclusion:

Summarize the key findings, discuss the significance of the results, and propose potential avenues for further research. Highlight the practical applications of understanding partial molar volumes in real-world scenarios, such as in chemical engineering or environmental science.

The objective of this laboratory experiment is to determine the partial molar volume of a solute in a binary solution by establishing a relationship between experimentally accessible parameters and the apparent molar volume.

**Introduction:** Partial molar volume is a crucial thermodynamic parameter that characterizes the change in volume upon adding a solute to a solvent. Although not directly measurable, it can be determined through the concept of apparent molar volume. In this experiment, we aim to develop a practical equation linking measurable features of the system to the partial molar volume.

**Experimental Setup:**

**Materials:**- Solvent A (e.g., water)
- Solute B (molecular weight MWB)
- Density measuring apparatus
- Flask
- Balance

**Procedure:**a. Measure the mass of an empty flask (We). b. Fill the flask with pure water up to the mark and measure the mass (W0). c. Add the solute B to the flask, ensuring it dissolves completely. d. Measure the mass of the flask with the solution (W). e. Record the density of the solution (d).

Calculations and Formulas:

- Apparent Molar Volume (Equation 6):

ϕV=nA,BV−nA,B⋅Vm,A - Volume Expression (Equation 7):

V=dMWB - Mass of Substance A (Equation 8):

WA=d0⋅V - Apparent Molar Volume in Measurable Terms (Equation 9):

ϕV=nBW−WA - Simplified Apparent Molar Volume (Equation 10):

ϕV=nBW−W0 - Molality (Equation 10.a):

b=W−WenB×1000 - Density Expressions (Equation 10.b):

ρNaCl=VW−We - Partial Molar Volume Expression (Equation 13):

VˉA=(∂nB∂V)P,T,nA - Linear Relationship (Equation 14):

ϕV∝b - Direct Expression in Molality (Equation 15):

VˉA=b55.51+ϕ0

The relationship between apparent molar volume and molality was explored, and a linear correlation was observed. The experimental data can be utilized to determine the slope of this relationship, providing values for both partial molar volumes.

Through a systematic approach, we established a practical equation for apparent molar volume and successfully connected it to the partial molar volume in terms of measurable parameters. This experiment provides a valuable insight into the thermodynamics of solutions and the determination of crucial parameters affecting volumetric changes.

A 2.000 M NaCl stock solution served as the basis for generating a series of dilutions with approximate concentrations of 1.0, 0.5, and 0.25 M. Additionally, a 1.5 M NaCl solution was prepared from the 2.000 M stock. The dilutions were conducted in 250 mL Erlenmeyer flasks using 50 ± 0.05 mL and 75 ± 0.05 mL volumetric pipets. To maintain accuracy, the pipet was cleaned between runs to prevent residue transfer.

Measurements were taken with an Explorer balance (Chemistry Department #11, OHAUS, item# = E01140). A glass stopper was omitted to minimize mass-affecting factors. Mass measurements were performed in triplicate using a single 10 mL volumetric flask. The initial measurements included the mass of the empty flask and a 10 mL water aliquot. An outlier was identified and confirmed through a fourth measurement, leading to its exclusion from the dataset.

Subsequently, 10 mL aliquots of the serial dilutions were massed and returned to their original flasks, ensuring thorough mixing. Dilutions were measured in increasing concentration order from 0.25 M to 2 M, minimizing potential carryover effects. All measurements were performed in triplicate.

Data, including mass measurements, were inputted into a Microsoft Excel spreadsheet with embedded equations for calculating average mass, standard deviation, density, molality, and apparent molar volume.

Data and Calculations

Table 1: Summary of the masses taken with calculated averages and standard deviation.

Trial (Mass) | Empty | Water | 0.25 M | 0.5 M | 1 M | 1.5 M | 2 M |

1 (g) | 9.3122 ± 0.0001 | 19.2703 ± 0.0001 | 19.3802 ± 0.0001 | 19.4881 ± 0.0001 | 19.6918 ± 0.0001 | 19.8708 ± 0.0001 | 20.0654 ± 0.0001 |

2 (g) | 9.3124 ± 0.0001 | 19.2799 ± 0.0001 | 19.391 ± 0.0001 | 19.4848 ± 0.0001 | 19.6876 ± 0.0001 | 19.8878 ± 0.0001 | 20.074 ± 0.0001 |

3 (g) | 9.3120 ± 0.0001 | 19.2731± 0.0001 | 19.3865 ± 0.0001 | 19.4859 ± 0.0001 | 19.685 ± 0.0001 | 19.8756 ± 0.0001 | 20.0672 ± 0.0001 |

Average (g) | 9.3122 ± 0.0002 | 19.2744 ±0.0049 | 19.3859 ± 0.0054 | 19.4863± 0.0017 | 19.6881 ± 0.0034 | 19.8781 ± 0.0088 | 20.0689 ± 0.0045 |

In this section, we elucidate our error analysis approach by employing partial differentiation for each contributing term in the calculated values. This method assesses the change in a variable while keeping all other variables constant, offering insights into the overall measurement uncertainty. If deemed appropriate, standard deviation of a measurement was also taken into account, and the greater of the two values was reported for subsequent calculations.

To illustrate, when determining the error of density, we first computed the average masses from each trial, all of which were conducted in triplicate. The uncertainty of the masses was then calculated using both partial differentiation and standard deviation. In this instance, the standard deviation exhibited a greater magnitude and was consequently utilized to calculate the uncertainty of density. While the partial differentiation method theoretically encompasses this error, the higher standard deviation suggests potential discrepancies in our experimental technique, such as inconsistent filling of volumetric flasks. Given that we only had a single density value, employing the partial differential method allowed us to efficiently capture uncertainty without redundant calculations for multiple density trials.

For cases where the error was deemed insignificant, scientific notation has been employed for reporting. The Maple worksheets used for error propagation have been included in the appendix. All uncertainties associated with glassware were sourced from Harris.

Table: Error Associated with Average Masses of Each Solution Concentration

Concentration (mol/L) | Average Mass (g) | Error
(g) |
Standard deviation(g) |

Empty | 9.3122 | 5.774E-05 | 0.0002 |

0.0000 | 19.2699 | 5.774E-05 | 0.0049 |

0.2500 | 19.3859 | 5.774E-05 | 0.0054 |

0.5000 | 19.4863 | 5.774E-05 | 0.0017 |

1.000 | 19.6881 | 5.774E-05 | 0.0034 |

1.500 | 19.8781 | 5.774E-05 | 0.0088 |

2.000 | 20.0689 | 5.774E-05 | 0.0045 |

Table . Error values for the calculated density and molality of each solution. Additionally, the error associating in making each concentration is included.

Concentration (mol/L) | Density (g/mL) | Molality (mol/kg) |

0.0000 ± 0.0000 | 0.995_{77 }± 2.051E-03 |
0.000 ± 0.000 |

0.2500 ± 5.580E-04 | 1.00_{737 }± 2.074E-03 |
0.251_{8} ± 3.626E-13 |

0.5000 ± 9.141E-04 | 1.01_{741 }± 2.094E-03 |
0.506_{0} ± 4.229E-12 |

1.000 ± 1.305E-03 | 1.03_{759} ± 2.133E-03 |
1.02_{13} ± 4.822E-11 |

1.500 ± 1.738E-03 | 1.05_{659} ± 2.170E-03 |
1.54_{81} ± 2.512E-10 |

2.000 ± 5.000E-04 | 1.07_{567} ± 2.207E-03 |
2.08_{60} ± 5.477E-10 |

In this experiment, our primary objective was to establish a correlation between the partial molar volumes of NaCl and water at different molalities in a NaCl-water solution, as illustrated in Figures 2 and 3. Simply put, partial molar volume signifies the change in the total volume when one mole of a particular substance is introduced to a homogeneous mixture. Generally, the partial molar volume of NaCl exhibits a positive yet decreasing slope. In comparison to the partial molar volume of water in Figure 2, NaCl's impact on volume is significantly influenced by the molality of the solution. The graph of NaCl, when plotted against water, renders the latter nearly linear. However, Figure 3 reveals that the partial molar volume of water follows a similar pattern with a decreasing slope. Notably, the sensitivity, indicated by slope, is much higher for NaCl than water when it comes to changes in molality. Molecularly, these observations find explanation.

As molality increases, more NaCl dissolves in the solution. Consequently, the addition of more NaCl raises the volume of the solution more significantly than at lower molalities (with fewer ions). In terms of packing, the increased presence of ions leaves less space for their organization. At higher molalities, the addition of more NaCl has a pronounced impact on the overall volume, and the curve begins to plateau due to the practical limit of NaCl solubility in water. The solution reaches saturation, unable to accommodate more ions. Conversely, the partial molar volume of water is less affected by molality and exhibits fundamentally different behavior. As molality increases, the partial molar volume of water decreases. Higher molarities result in a lesser impact of water addition on solution volume. This is explained by molecular interactions, particularly the organizing effect of water around ions.

The partial molar volume of water starts at approximately its molar volume, making theoretical sense since the pure substance's partial molar volume is its molar volume. At 0 molal, the solution is pure water. In contrast, the partial molar volume of NaCl is around 26.99 mL/mol, and though the data is limited, it suggests a leveling off around 25 mL/mol. Unlike water, the partial molar volume of NaCl will never equal its molar volume in a solution. This aligns with practical expectations, as a pure solution of water with zero molality in terms of NaCl is conceivable, while a pure solid of NaCl in water is impractical due to its solubility limitations.

While our results generally align with practical and theoretical expectations, some discrepancies exist. In salt solutions, the square root of molality and apparent molar volume should have a linear relationship. Figure 1 illustrates this trend for all data points except at 0.5. The deviation of the first point significantly influenced the line's slope and intercept, highlighting a potential anomaly in the experimental observations.

The key findings of this experiment are visually represented in Figures 2 and 3, while numerical values are detailed in Table 2. While not directly applicable to our specific chemical system and experimental setup, the general trend observed aligns well with data presented in Atkins' Physical Chemistry. Atkins, on page 184, illustrates a similar graph to Figure 2, demonstrating analogous behavior (Atkins). One notable distinction is the smoother curves in Atkins' representation, which could be attributed to challenges in our experiment related to dilution and curve fitting in Figure 1 using a linear model. The variability introduced by the first data point likely contributes to the deviation from literature values. Nevertheless, the primary objective of the lab, capturing the behavior of partial molar volumes of solvent and solute in salt solutions, was achieved. The graphs not only align with existing literature but also provide insights in terms of molecular interactions.

Despite potential variations, the obtained data can serve as a foundation for the exploration and comparison of other salt solution systems. Furthermore, the concept of partial molar volume is transferable to the study of liquid mixtures, such as ethanol and water. This practical and useful concept finds relevance in everyday laboratory work, elucidating why volumes are not strictly additive. In conclusion, our results effectively showcase how partial molar volume can dynamically change with varying molality, opening avenues for further investigations in the field.

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