91Ó°ÊÓ

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?

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, \(C_v[J /(mol . °C)] = 26.986 + 0.0291*T[°C]\) 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°C to 200°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°C to 200°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.