The density of an equal-mass water 1 ethanol mixture is 0.913 g/cc at 20°C. If the density of water is 0.998 g/cm3 and ethanol is 0.789 g/cm3 with both at 20°C, does this equal-mass mixture possess a positive or negative excess volume at 20°C?
The Correct Answer and Explanation is :
To determine whether the equal-mass water-ethanol mixture has a positive or negative excess volume at 20°C, we first calculate the expected volume of the mixture without interactions (ideal volume) and compare it to the actual volume of the mixture. If the actual volume is less than the ideal volume, the mixture has a negative excess volume, and vice versa.
Step 1: Calculate the mass of water and ethanol in the mixture
Since it is an equal-mass mixture, let:
- Mass of water ((m_{\text{water}})) = (m_{\text{ethanol}}) = 1 g.
Step 2: Calculate the ideal volume of the mixture
The volume of each component can be calculated using the formula:
[
\text{Volume} = \frac{\text{Mass}}{\text{Density}}
]
For water:
[
V_{\text{water}} = \frac{1 \, \text{g}}{0.998 \, \text{g/cc}} = 1.002 \, \text{cc}
]
For ethanol:
[
V_{\text{ethanol}} = \frac{1 \, \text{g}}{0.789 \, \text{g/cc}} = 1.267 \, \text{cc}
]
The ideal volume of the mixture is:
[
V_{\text{ideal}} = V_{\text{water}} + V_{\text{ethanol}} = 1.002 + 1.267 = 2.269 \, \text{cc}
]
Step 3: Calculate the actual volume of the mixture
The actual volume is determined using the density of the mixture:
[
\text{Density of mixture} = \frac{\text{Total mass}}{\text{Actual volume}}
]
Rearranging for actual volume:
[
\text{Actual volume} = \frac{\text{Total mass}}{\text{Density of mixture}}
]
[
\text{Actual volume} = \frac{2 \, \text{g}}{0.913 \, \text{g/cc}} = 2.191 \, \text{cc}
]
Step 4: Calculate the excess volume
The excess volume is:
[
V_{\text{excess}} = V_{\text{actual}} – V_{\text{ideal}} = 2.191 – 2.269 = -0.078 \, \text{cc}
]
Conclusion
The mixture has a negative excess volume at 20°C, indicating that the water and ethanol molecules interact in a way that causes them to pack more efficiently than expected, reducing the total volume. This effect arises from hydrogen bonding between water and ethanol molecules, leading to structural changes in the mixture.