Question

\(10 \mathrm{kmol}\) of methane gas are heated from 1 atm and \(298 \mathrm{K}\) to \(1 \mathrm{atm}\) and \(1000 \mathrm{K}\). Calculate the total amount of heat transfer required when \((a)\) disassociation is neglected and \((b)\) when disassociation is considered. The natural logarithm of the equilibrium constant for the reaction \(\mathrm{C}+2 \mathrm{H}_{2}\) \(\rightleftharpoons \mathrm{CH}_{4}\) at \(1000 \mathrm{K}\) is \(2.328 .\) For the solution of part (a) use empirical coefficients of Table \(A-2 c\). For the solution of part ( \(b\) ) use constant specific heats and take the constantvolume specific heats of methane, hydrogen and carbon at \(1000 \mathrm{K}\) to be \(63.3,21.7,\) and \(0.711 \mathrm{kJ} / \mathrm{kmol} \cdot \mathrm{K},\) respectively. The constant-volume specific heat of methane at \(298 \mathrm{K}\) is \(27.8 \mathrm{kJ} / \mathrm{kmol} \cdot \mathrm{K}\).

Short answer

Question: Calculate the total amount of heat transfer required to heat 10 kmol of methane gas from 1 atm and 298 K to 1 atm and 1000 K, considering both without and with dissociation. Use the given empirical coefficients for methane and constant specific heats of methane, hydrogen, and carbon at 1000 K. Answer: To calculate the heat transfer required without considering dissociation, integrate the specific heat capacity expression using the given empirical coefficients for methane. For the heat transfer calculation with dissociation, use the equilibrium constant and given specific heats of methane, hydrogen, and carbon to find the number of moles before and after dissociation at 1000 K, and calculate the total heat transfer.

Step-by-step solution

(a) Calculate Heat Transfer Without Considering Dissociation

We have the initial conditions: \(T_1 = 298 K, P_1 = 1 \, \text{atm} \, (=\, 101.3 \, kPa)\) and the final conditions: \(T_2 = 1000 K, P_2 = 1 \, \text{atm}\). We need to use empirical coefficients for heat calculations without considering dissociation. The expression for the constant pressure specific heat capacity \(c_p\) for the substance is: $$c_p = a + bT + cT^2 + dT^3$$ where \(a, b, c,\) and \(d\) are empirical coefficients. According to Table A-2c, coefficients for methane are: $$ a = 1.702, \, b = 9.081 \times 10^{-3}, \, c = 5.996 \times 10^{-6}, \, d = 0 $$ Integrating \(c_p\) from \(T_1\) to \(T_2\), we get the enthalpy change \(\Delta h\): $$\Delta h = \int_{T_1}^{T_2} c_p dT$$ By substituting the given coefficients, we can calculate the integral and multiply it by the number of kmols of methane. Finally, we will find the heat transfer required.

(b) Calculate Heat Transfer Considering Dissociation

For this part, we have the equilibrium constant for the reaction, the dissociation reaction and the specific heats of methane, hydrogen, and carbon at 1000 K. We will first calculate the number of moles for each component before and after dissociation using the equilibrium constant. From the reaction \(C + 2H_2 \rightleftharpoons CH_4\), let \(x\) kmol of methane dissociate. Then, $$n(CH_4, after) = 10 - x$$ $$n(H_2, after) = 2x$$ $$n(C, after) = x$$ Now, using the equilibrium constant \(K(T)\), we have: $$K(T) = \frac{n(C, after) \cdot [n(H_2, after)]^2}{n(CH_4, after)} = \frac{x(2x)^2}{10-x}$$ We are given \(\ln K(T_2) = 2.328\). Thus, we need to solve for \(x\) by first calculating \(K(T_2)\) and then using the obtained equation to find the number of moles before and after dissociation. Finally, we will use the specific heat values given at 1000 K and the initial specific heat value at 298 K to calculate the heat transfer required considering dissociation.

Key concepts

Specific Heat Capacity

Understanding specific heat capacity is crucial when solving problems related to heat transfer. It tells us how much heat energy is required to change the temperature of a unit mass of a substance by one degree Celsius. The specific heat capacity depends on the material and, sometimes, the temperature itself. For gases like methane, the specific heat can be presented using empirical coefficients which provide a more accurate representation over a larger range of temperatures. In our exercise, specific heats are given for methane at two different temperatures to help us calculate the heat transfer required in both cases.

When the textbook exercise asks you to calculate the heat transfer without considering dissociation, it means you'll use the specific heat capacity of methane as a whole molecule throughout the entire temperature change. However, when dissociation is included, you'll need to consider the different specific heat values for each component gas after dissociation: methane, hydrogen, and carbon.

Thermodynamic Equilibrium

When we discuss thermodynamic equilibrium, we’re talking about a condition where all parts of a system are at the same temperature and no heat transfer occurs within the system. In the context of our exercise, we assume the system reaches thermodynamic equilibrium at the final temperature of 1000 K.

Why is this important? It ensures that the calculations for heat transfer take into account a stable final state, as the equilibrium represents a balance in the molecular energy distribution. In the second part of the exercise where chemical dissociation occurs, the equilibrium constant provided will be instrumental in assessing the new equilibrium state after some of the methane has broken down into hydrogen and carbon.

Chemical Dissociation

In thermodynamics, chemical dissociation refers to the breakdown of a compound into its components due to the application of heat. This concept is essential for part (b) of our heat transfer problem. Methane (\( CH_4 \)) can dissociate into hydrogen (\( H_2 \)) and carbon (\( C \)) at high temperatures. This process significantly impacts the heat capacity and consequently the heat transfer calculated.

For each \( \text{kmol} \) of methane that dissociates, we produce 2 \( \text{kmol} \) of hydrogen and 1 \( \text{kmol} \) of carbon. The equilibrium condition given in the exercise indicates the extent to which this dissociation occurs at the final temperature—information that lets us adjust the final amount of each species present in the system. Our calculations then must account not just for heating the substances, but also for the energy involved in breaking the chemical bonds as part of the dissociation process.

Enthalpy Change

The enthalpy change, represented by \( \Delta h \), is a way of expressing the heat transfer at constant pressure and is used to calculate the overall heat required in our exercise. Enthalpy change is often found by integrating the specific heat over the temperature range of interest, provided the process does not include a phase change or chemical reaction. In this problem's context, the enthalpy change without dissociation can be calculated directly by integrating the temperature-dependent specific heat of methane from its initial to the final temperature.

However, when considering dissociation, we also have to include the enthalpy changes associated with breaking the methane into hydrogen and carbon. This requires adjusting our approach to include the enthalpies of individual components and the energy absorbed or released due to the dissociation process. The calculation of enthalpy change for the dissociation scenario can be more complex and necessitates a solid understanding of thermodynamics and reaction equilibrium.