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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

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 gas stream containing \(n\) -hexane in nitrogen with a relative saturation of \(90 \%\) is fed to a condenser at \(75^{\circ} \mathrm{C}\) and 3.0 atm absolute. The product gas emerges at \(0^{\circ} \mathrm{C}\) and 3.0 atm at a rate of \(746.7 \mathrm{m}^{3} / \mathrm{h}\). (a) Calculate the percentage condensation of hexane (moles condensed/mole fed) and the rate \((\mathrm{kW})\) at which heat must be transferred from the condenser. (b) Suppose the feed stream flow rate and composition and the heat transfer from the condenser are the same as in Part (a), but the condenser and outlet stream pressure is only 2.5 atm instead of 3.0 atm. How would the outlet stream temperatures and flow rates and the percentage condensations of hexane calculated in Parts (a) and (b) change (increase, decrease, no change, no way to tell)? Don't do any calculations, but explain your reasoning.

Molten sodium chloride is to be used as a constant-temperature bath for a high-temperature chemical reactor. Two hundred kilograms of solid \(\mathrm{NaCl}\) at \(300 \mathrm{K}\) is charged into an insulated vessel, and a 3000 kW electrical heater is turned on, raising the salt to its melting point of 1073 K and melting it at a constant pressure of 1 atm. (a) The heat capacity \(\left(C_{p}\right)\) of solid \(\mathrm{NaCl}\) is \(50.41 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\) at \(T=300 \mathrm{K},\) and \(53.94 \mathrm{J} /(\mathrm{mol} \cdot \mathrm{K})\) at \(T=500 \mathrm{K},\) and the heat of fusion of \(\mathrm{NaCl}\) at \(1073 \mathrm{K}\) is \(30.21 \mathrm{kJ} / \mathrm{mol} .\) Use these data to determine a linear expression for \(C_{p}(T)\) and to calculate \(\Delta \hat{H}\) ( \(\mathrm{kJ} / \mathrm{mol}\) ) for the transition of \(\mathrm{NaCl}\) from a solid at 300 K to a liquid at \(1073 \mathrm{K}\). (b) Write and solve the energy balance equation for this closed system isobaric process to determine the required heat input in kilojoules. (c) If \(85 \%\) of the full power of \(3000 \mathrm{kW}\) goes into heating and melting the salt, how long does the process take?

On a cold winter day the temperature is \(2^{\circ} \mathrm{C}\) and the relative humidity is \(15 \% .\) You inhale air at an average rate of \(5500 \mathrm{mL} / \mathrm{min}\) and exhale a gas saturated with water at body temperature, roughly \(37^{\circ} \mathrm{C} .\) If the mass flow rates of the inhaled and exhaled air (excluding water) are the same, the heat capacities \(\left(C_{p}\right)\) of the water-free gases are each \(1.05 \mathrm{J} /\left(\mathrm{g} \cdot^{\circ} \mathrm{C}\right),\) and water is ingested into the body as a liquid at \(22^{\circ} \mathrm{C},\) at what rate in \(\mathrm{J} /\) day do you lose energy by breathing? Treat breathing as a continuous process (inhaled air and liquid water enter, exhaled breath exits) and neglect work done by the lungs.

Saturated steam at \(300^{\circ} \mathrm{C}\) is used to heat a countercurrently flowing stream of methanol vapor from \(65^{\circ} \mathrm{C}\) to \(260^{\circ} \mathrm{C}\) in an adiabatic heat exchanger. The flow rate of the methanol is 6500 standard liters per minute, and the steam condenses and leaves the heat exchanger as liquid water at \(90^{\circ} \mathrm{C}.\) (a) Calculate the required flow rate of the entering steam in \(\mathrm{m}^{3} / \mathrm{min}\). (b) Calculate the rate of heat transfer from the water to the methanol ( \(\mathrm{kW}\) ). (c) Suppose the outlet temperature of the methanol is measured and found to be \(240^{\circ} \mathrm{C}\) instead of the specified value of \(260^{\circ} \mathrm{C}\). List five possible realistic explanations for the \(20^{\circ} \mathrm{C}\) difference. 7 An adiabatic heat exchanger is one for which no heat is exchanged with the surroundings. All of the heat lost by the hot stream is transferred to the cold stream.
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