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Chapter 8: Problem 51

Estimate the heat of vaporization of diethyl ether at its normal boiling point using Trouton's rule and Chen's rule and compare the results with a tabulated value of this quantity. Calculate the percentage error that results from using each estimation. Then estimate \(\Delta \hat{H}_{\mathrm{v}}\) at \(100^{\circ} \mathrm{C}\) using Watson's correlation.

Short Answer

Expert verified
The step-by-step guide calculates the heat of vaporization using Trouton's rule, Chen's rule, and Watson's correlation. The percentage error of the results for Trouton and Chen's rules relative to the known tabulated value is calculated as well.

Step by step solution

01

Estimation Using Trouton's Rule

Trouton's rule states that the entropy change during vaporization is approximately constant for many substances under their normal boiling points. The heat of vaporization (\(\Delta H_{vap}\)) can be estimated using the following formula: \[ \Delta H_{vap}= T_{B} \cdot 88\ J/K \cdot mol\] where \( T_B \) is the normal boiling point in Kelvin.
02

Estimation Using Chen's Rule

Chen鈥檚 rule estimates the heat of vaporization (\(\Delta H_{vap}\)) using: \[ \Delta H_{vap}= (18.5 - 0.0156 \cdot T_b) \cdot T_b \cdot R \] where \(T_B\) is the normal boiling point in Kelvin and \(R\) is the gas constant in \(J/K \cdot mol\).
03

Calculate the Percentage Error

The percentage error for each estimate can be calculated using the formula: \[ \% Error = \left|\frac{(Experimental Value - Tabulated Value)}{Tabulated Value}\right| \cdot 100\% \] where the experimental value is the heat of vaporization obtained from Trouton's or Chen's rule, and the tabulated value is the actual known value.
04

Estimation using Watson's Correlation

Lastly, Watson's correlation helps to estimate how the heat of vaporization changes with temperature. The formula to be used is: \[ \Delta \hat{H}_{2} = \Delta \hat{H}_{1} \cdot \left(\frac{T_2}{T_1}\right)^{0.38} \] Where \(\Delta \hat{H}_{1}\) is the molar heat of vaporization at temperature \(T_1\) (which is the normal boiling point), and \(\Delta \hat{H}_{2}\) is the molar heat of vaporization at \(T_2\) (which is 100掳C).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trouton's Rule
Trouton's Rule serves as a handy guideline for predicting the heat of vaporization of a liquid at its normal boiling point. It is premised on the principle that the entropy of vaporization is fairly constant for various substances, typically around 88 J/K鈥ol. This reflects that when a substance moves from liquid to vapor at its boiling point, the disorder or randomness (entropy) increase is about the same across different substances.

To apply this rule, the calculation is straightforward. Imagine a substance with a known boiling point in Kelvin, you would simply multiply this temperature by the entropy value provided by Trouton's Rule. However, keep in mind that Trouton's Rule is a rough estimate. It works better for nonpolar substances with relatively low molecular weights and typical boiling points. For polar substances, hydrogen-bonded liquids, or those with high molecular weights, deviations from this rule are expected.
Chen's Rule
Going a step beyond Trouton's Rule, Chen鈥檚 Rule adds a temperature-dependent term to estimate the heat of vaporization. It is particularly useful because it acknowledges that entropy change can vary slightly with the boiling point of a substance. The rule stipulates that you take into account the boiling point of the substance and plug it into a formula that includes an empirical constant and the gas constant.

By using Chen's Rule, one can gain a more refined estimate that considers the specific boiling point of the substance in question. While still being an estimate and allowing for some degree of error, Chen's Rule is a nod towards the complexity of molecular interactions during the phase transition, hence providing a somewhat more tailored value for a substance's heat of vaporization.
Watson's Correlation
When the temperature of a substance changes, so does its heat of vaporization. Watson's Correlation accounts for temperature variations that move away from the normal boiling point. It provides a means to estimate the change in the heat of vaporization with temperature, factoring in the intermolecular forces that become weaker with rising temperature.

To use Watson's Correlation, one requires the heat of vaporization at a known temperature, and from there, they can estimate the heat of vaporization at a new temperature. This is highly relevant in industrial applications where processes operate at various temperatures, not just at the normal boiling points of substances. Watson's Correlation is particularly invaluable when scaling the operation of equipment or processes in chemical engineering.
Percentage Error Calculation
In any scientific or engineering task, gauging the accuracy of an estimate or a measurement is vital. This is where the significance of percentage error calculation shines through. It expresses the deviation of an experimental value from a true or accepted value as a percentage. This metric provides a relative sense of accuracy and is crucial for comparing the reliability of different methods.

Calculating percentage error involves taking the absolute difference between the estimated and actual values, dividing by the actual value, and then multiplying by 100 to get a percentage. It not only allows students to see how close their estimates are to the real values, but also helps in identifying which empirical rules or correlations are more reliable under specific conditions, guiding future predictions and experiments.

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Most popular questions from this chapter

The heat capacity at constant pressure of hydrogen cyanide is given by the expression $$ C_{p}\left[J /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]=35.3+0.0291 T\left(^{\circ} \mathrm{C}\right) $$ (a) Write an expression for the heat capacity at constant volume for HCN, assuming ideal-gas behavior. (b) Calculate \(\Delta \hat{H}(\mathrm{J} / \mathrm{mol})\) for the constant- pressure process $$ \mathrm{HCN}\left(\mathrm{v}, 25^{\circ} \mathrm{C}, 0.80 \mathrm{atm}\right) \rightarrow \mathrm{HCN}\left(\mathrm{v}, 200^{\circ} \mathrm{C}, 0.80 \mathrm{atm}\right) $$(c) Calculate \(\Delta \hat{U}(\mathrm{J} / \mathrm{mol})\) for the constant- volume process $$\mathrm{HCN}\left(\mathrm{v}, 25^{\circ} \mathrm{C}, 50 \mathrm{m}^{3} / \mathrm{kmol}\right) \rightarrow \mathrm{HCN}\left(\mathrm{v}, 200^{\circ} \mathrm{C}, 50 \mathrm{m}^{3} / \mathrm{kmol}\right)$$ (d) If the process of Part (b) were carried out in such a way that the initial and final pressures were each 0.80 atm but the pressure varied during the heating, the value of \(\Delta \hat{H}\) would still be what you calculated assuming a constant pressure. Why is this so?

An adiabatic membrane separation unit is used to dry (remove water vapor from) a gas mixture containing 10.0 mole \(\% \mathrm{H}_{2} \mathrm{O}(\mathrm{v}), 10.0\) mole \(\% \mathrm{CO},\) and the balance \(\mathrm{CO}_{2} .\) The gas enters the unit at \(30^{\circ} \mathrm{C}\) and flows past a semipermeable membrane. Water vapor permeates through the membrane into an air stream. The dried gas leaves the separator at \(30^{\circ} \mathrm{C}\) containing \(2.0 \mathrm{mole} \% \mathrm{H}_{2} \mathrm{O}(\mathrm{v})\) and the balance \(\mathrm{CO}\) and \(\mathrm{CO}_{2}\). Air enters the separator at \(50^{\circ} \mathrm{C}\) with an absolute humidity of \(0.002 \mathrm{kg} \mathrm{H}_{2} \mathrm{O} / \mathrm{kg}\) dry air and leaves at \(48^{\circ} \mathrm{C}\). Negligible quantities of \(\mathrm{CO}, \mathrm{CO}_{2}, \mathrm{O}_{2},\) and \(\mathrm{N}_{2}\) permeate through the membrane. All gas streams are at approximately 1 atm. (a) Draw and label a flowchart of the process and carry out a degree of freedom analysis to verify that you can determine all unknown quantities on the chart. (b) Calculate (i) the ratio of entering air to entering gas (kg humid air/mol gas) and (ii) the relative humidity of the exiting air. (c) List several desirable properties of the membrane. (Think about more than just what it allows and does not allow to permeate.)

A mixture of \(n\) -hexane vapor and air leaves a solvent recovery unit and flows through a \(70-\mathrm{cm}\) diameter duct at a velocity of \(3.00 \mathrm{m} / \mathrm{s}\). At a sampling point in the duct the temperature is \(40^{\circ} \mathrm{C}\), the pressure is \(850 \mathrm{mm}\) Hg, and the dew point of the sampled gas is \(25^{\circ} \mathrm{C}\). The gas is fed to a condenser in which it is cooled at constant pressure, condensing \(70 \%\) of the hexane in the feed. (a) Perform a degree-of-freedom analysis to show that enough information is available to calculate the required condenser outlet temperature \(\left(^{\circ} \mathrm{C}\right)\) and cooling rate \((\mathrm{kW})\) (b) Perform the calculations. (c) If the feed duct diameter were \(35 \mathrm{cm}\) for the same molar flow rate of the feed gas, what would be the average gas velocity (volumetric flow rate divided by cross-sectional area)? (d) Suppose you wanted to increase the percentage condensation of hexane for the same feed stream. Which three condenser operating variables might you change, and in which direction?

A sheet of cellulose acetate film containing 5.00 wt\% liquid acetone enters an adiabatic dryer where \(90 \%\) of the acetone evaporates into a stream of dry air flowing over the film. The film enters the dryer at \(T_{\mathrm{f} 1}=35^{\circ} \mathrm{C}\) and leaves at \(T_{\mathrm{f} 2}\left(^{\circ} \mathrm{C}\right) .\) The air enters the dryer at \(T_{\mathrm{al}}\left(^{\circ} \mathrm{C}\right)\) and 1.01 atm and exits the dryer at \(T_{\mathrm{a} 2}=49^{\circ} \mathrm{C}\) and 1 atm with a relative saturation of \(40 \% . C_{p}\) may be taken to be \(1.33 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) for dry film and \(0.129 \mathrm{kJ} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\) for liquid acetone. Make a reasonable assumption regarding the heat capacity of dry air. The heat of vaporization of acetone may be considered independent of temperature. Take a basis of \(100 \mathrm{kg}\) film fed to the dryer for the requested calculations. (a) Estimate the feed ratio [liters dry air (STP)/kg dry film]. (b) Derive an expression for \(T_{\mathrm{al}}\) in terms of the film temperature change, \(\left(T_{\mathrm{f} 2}-35\right),\) and use it to answer Parts (c) and (d). (c) Calculate the film temperature change if the inlet air temperature is \(120^{\circ} \mathrm{C}\). (d) Calculate the required value of \(T_{\mathrm{al}}\) if the film temperature falls to \(34^{\circ} \mathrm{C},\) and the value if it rises to \(36^{\circ} \mathrm{C}.\) (e) If you solved Parts (c) and (d) correctly, you found that even though the air temperature is consistently higher than the film temperature in the dryer, so that heat is always transferred from the air to the film, the film temperature can drop from the inlet to the outlet. How is this possible?

Saturated propane vapor at \(2.00 \times 10^{2}\) psia is fed to a well- insulated heat exchanger at a rate of \(3.00 \times 10^{3} \mathrm{SCFH}\) (standard cubic feet per hour). The propane leaves the exchanger as a saturated liquid (i.e., a liquid at its boiling point) at the same pressure. Cooling water enters the exchanger at \(70^{\circ} \mathrm{F},\) flowing cocurrently (in the same direction) with the propane. The temperature difference between the outlet streams (liquid propane and water) is \(15^{\circ} \mathrm{F}\). (a) What is the outlet temperature of the water stream? (Use the Antoine equation.) Is the outlet water temperature less than or greater than the outlet propane temperature? Briefly explain. (b) Estimate the rate (Btu/h) at which heat must be transferred from the propane to the water in the heat exchanger and the required flow rate \(\left(1 \mathrm{b}_{\mathrm{m}} / \mathrm{h}\right)\) of the water. (You will need to write two separate energy balances.) Assume the heat capacity of liquid water is constant at \(1.00 \mathrm{Btu} /\left(\mathrm{lb}_{\mathrm{m}} \cdot^{\circ} \mathrm{F}\right)\) and neglect heat losses to the outside and the effects of pressure on the heat of vaporization of propane.
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