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Chapter 8: Problem 2

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?

Short Answer

Expert verified
Heat capacity at constant volume for an ideal gas (HCN) is \(C_v[J /(mol . 掳C)] = 26.986 + 0.0291*T[掳C]\). The change in enthalpy and internal energy for the given processes are approximately 11055.6525 J/mol and 8214.4225 J/mol, respectively. The enthalpy change \(\Delta H\) for a process is independent of the path taken, as it depends only on the initial and final states, hence the pressure variations during the process will not affect the \(\Delta H\) calculated assuming a constant pressure.

Step by step solution

01

Calculate heat capacity at constant volume (C_v) for ideal gas

For an ideal gas, we note that the heat capacity at constant volume (C_v) is given by the formula \(C_v = C_p - R\), where R is the gas constant (8.314 J/(mol.K) in SI units). Substituting given values, \(C_v = 35.3 + 0.0291*T - 8.314 = 26.986 + 0.0291*T\). Hence, \(C_v[J /(mol . 掳C)] = 26.986 + 0.0291*T[掳C]\) is the expression for the heat capacity at constant volume for HCN.
02

Calculate change in Enthalpy \((\Delta H)\)

The change in enthalpy (\(\Delta H\)) under the constant-pressure process is given by the integral of the heat capacity at constant pressure with respect to temperature, \(\Delta H = \int_{T1}^{T2} C_p dT\). Substituting the given limits (from 25掳C to 200掳C) and the expression for \(C_p\), we obtain \(\Delta H = \int_{25}^{200} (35.3 + 0.0291*T) dT\). Solving this integral yields \(\Delta H = [35.3*T + 0.01455*T^2]_{25}^{200} \approx 11055.6525 J/mol\).
03

Calculate change in internal energy \((\Delta U)\)

The change in internal energy (\(\Delta U\)) for the constant-volume process is given by the integral of the heat capacity at constant volume with respect to temperature, \(\Delta U = \int_{T1}^{T2} C_v dT\). Substituting the given limits (from 25掳C to 200掳C) and the expression for \(C_v\) from Step 1, we obtain \(\Delta U = \int_{25}^{200} (26.986 + 0.0291*T) dT\). Solving this integral yields \(\Delta U = [26.986*T + 0.01455*T^2]_{25}^{200} \approx 8214.4225 J/mol\).
04

Explain why \(\Delta H\) is constant

The enthalpy change \(\Delta H\) for a process is the heat exchanged by the system at constant pressure. Hence, even if the pressure varied during the process, as long as the initial and final pressures remained at 0.80 atm, the enthalpy change would remain the same, since it depends only on the initial and final states and not on the path taken.

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Key Concepts

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

Heat Capacity
Heat capacity is a key concept in thermodynamics. It describes the amount of heat required to change a substance's temperature by one degree Celsius. For gases, there are two main types:
  • Heat capacity at constant pressure \(C_p\)
  • Heat capacity at constant volume \(C_v\)
For ideal gases, these are related by \(C_v = C_p - R\), where \(R\) is the gas constant. Understanding the relationship between \(C_p\) and \(C_v\) helps predict how a gas will behave under changes in temperature or pressure. It is crucial for calculating quantities like \(\Delta H\) and \(\Delta U\) in various processes.
Enthalpy Change
Enthalpy change, represented as \(\Delta H\), is the heat absorbed or released at constant pressure. It's calculated using the integral of the heat capacity at constant pressure over a temperature range: \[ \Delta H = \int_{T1}^{T2} C_p \, dT \] This measures the energy involved in heating or cooling a substance, accounting for how the heat capacity changes with temperature. It's important to understand \(\Delta H\) because it provides insight into energy transfers in processes like chemical reactions or phase changes. Even when pressure changes during a process, if initial and final pressures are consistent, \(\Delta H\) remains the same.
Internal Energy
Internal energy change, represented as \(\Delta U\), involves energy changes within a system at constant volume. It's determined by integrating the heat capacity at constant volume: \[ \Delta U = \int_{T1}^{T2} C_v \, dT \] This tells us about the energy changes not only due to heat but also due to changes in the system's internal structure, like molecular rotations or vibrations. Knowing \(\Delta U\) helps us understand how energy distributes within a system without doing mechanical work, crucial for thermodynamic calculations.
Ideal Gas Law
The Ideal Gas Law is a fundamental equation in chemistry and physics. It's expressed as \[ PV = nRT \] where \(P\) is pressure, \(V\) is volume, \(n\) is the amount of substance in moles, \(R\) is the ideal gas constant, and \(T\) is temperature in Kelvin. The law assumes no intermolecular forces and that the gas molecules occupy no volume, making it applicable to ideal situations. It's essential for understanding the behavior of gases and for calculations involving changes in temperature, volume, or pressure. It's a cornerstone of thermodynamics and underlies equations for heat capacities, enthalpy, and internal energy.

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Most popular questions from this chapter

The specific internal energy of formaldehyde (HCHO) vapor at 1 atm and moderate temperatures is given by the formula $$\hat{U}(\mathrm{J} / \mathrm{mol})=25.96 T+0.02134 T^{2}$$ where \(T\) is in \(^{\circ} \mathrm{C}\) (a) Calculate the specific internal energies of formaldehyde vapor at \(0^{\circ} \mathrm{C}\) and \(200^{\circ} \mathrm{C}\). What reference temperature was used to generate the given expression for \(\hat{U} ?\) (b) The value of \(\hat{U}\) calculated for \(200^{\circ} \mathrm{C}\) is not the true value of the specific internal energy of formaldehyde vapor at this condition. Why not? (Hint: Refer back to Section 7.5a.) Briefly state the physical significance of the calculated quantity. (c) Use the closed system energy balance to calculate the heat (J) required to raise the temperature of 3.0 mol HCHO at constant volume from 0^0 C to 200^'C. List all of your assumptions. (d) From the definition of heat capacity at constant volume, derive a formula for \(C_{v}(T)\left[\mathrm{J} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\right]\) Then use this formula and Equation \(8.3-6\) to calculate the heat \((\) J) required to raise the temperature of 3.0 mol of HCHO(v) at constant volume from 0^ C to 200^'C. [You should get the same result you got in Part (c).]

In gas adsorption a vapor is transferred from a gas mixture to the surface of a solid. (See Section \(6.7 .\) ) An approximate but useful way of analyzing adsorption is to treat it simply as condensation of vapor on a solid surface. Suppose a nitrogen stream at \(35^{\circ} \mathrm{C}\) and 1 atm containing carbon tetrachloride with a \(15 \%\) relative saturation is fed at a rate of \(10.0 \mathrm{mol} / \mathrm{min}\) to a \(6-\mathrm{kg}\) bed of activated carbon. The temperature and pressure of the gas do not change appreciably from the inlet to the outlet of the bed, and there is no \(\mathrm{CCl}_{4}\) in the gas leaving the adsorber. The carbon can adsorb 40\% of its own mass of carbon tetrachloride before becoming saturated, at which point it must be either regenerated (remove the carbon tetrachloride) or replaced with a fresh bed of activated carbon. Neglect the effect of temperature on the heat of vaporization of \(\mathrm{CCl}_{4}\) when solving the following problems: (a) Estimate the rate ( \(\mathrm{kJ} / \mathrm{min}\) ) at which heat must be removed from the adsorber to keep the process isothermal, and the time (min) it will take to saturate the bed. (b) The surface-to-volume ratio of spherical particles is \((3 / r)\left(\mathrm{cm}^{2} \text { outer surface }\right) /\left(\mathrm{cm}^{3} \text { volume }\right)\) First, derive that formula. Second, use it to explain how decreasing the average diameter of the particles in the carbon bed might make the adsorption process more efficient. Third, since most of the area on which adsorption takes place is provided by pores penetrating the particle, explain why the surface-to-volume ratio, as calculated by the above expression, might be relatively unimportant.

The heat capacities of a substance have been defined as $$C_{v}=\left(\frac{\partial \hat{U}}{\partial T}\right)_{V}, \quad C_{p}=\left(\frac{\partial \hat{H}}{\partial T}\right)_{P}$$ Use the defining relationship between \(\hat{H}\) and \(\hat{U}\) and the fact that \(\hat{H}\) and \(\hat{U}\) for ideal gases are functions only of temperature to prove that \(C_{p}=C_{v}+R\) for an ideal gas (Eq. \(8.3-12\) ).

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.

Fish and wildlife managers have determined that a sudden temperature increase greater than \(5^{\circ} \mathrm{C}\) would be harmful to the marine ecosystem of a river. Warmer waters contain less dissolved oxygen and cause organisms in a river to increase their metabolism; if the temperature increase is sudden, the organisms do not have time to adapt to the new environment and likely will die. (Changes in river temperatures of five degrees and more due to seasonal temperature variations are common, but those temperature changes are gradual.) A proposed chemical plant plans to use river water for process cooling. The river flows at a rate of \(15.0 \mathrm{m}^{3} / \mathrm{s}\) at a temperature of \(15^{\circ} \mathrm{C}\), and a fraction of it will be diverted to the plant. Preliminary calculations reveal that the cooling water will remove \(5.00 \times 10^{5} \mathrm{kJ} / \mathrm{s}\) of heat from the plant. A portion of the extracted water will evaporate from the plant into the atmosphere, and the remainder will be returned to the river at a temperature of \(35^{\circ} \mathrm{C}\). (a) Draw and completely label a flowchart of the process and prove that there is enough information available to calculate all of the unknown stream flow rates on the chart. (b) Estimate the fraction of the river flow that must be diverted to the plant and the percentage of the cooling water that evaporates. Assume that water has a constant heat capacity of \(4.19 \mathrm{kJ} /\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\) and a heat of vaporization roughly that of water at the normal boiling point, and also assume that the specific enthalpy of the water vapor relative to liquid water at \(15^{\circ} \mathrm{C}\) equals the heat of vaporization. (c) Write (but don't evaluate) an expression for the enthalpy change neglected by the assumption about the specific enthalpy of the steam.
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