Question

An adiabatic constant-volume tank contains a mixture of \(1 \mathrm{kmol}\) of hydrogen \(\left(\mathrm{H}_{2}\right)\) gas and the stoichiometric amount of air at \(25^{\circ} \mathrm{C}\) and 1 atm. The contents of the tank are now ignited. Assuming complete combustion, determine the final temperature in the tank.

Short answer

Answer: The final temperature after complete combustion is 298 K or 25°C.

Step-by-step solution

Determine the stoichiometric equation

First, we need to write the stoichiometric equation for the combustion of hydrogen with oxygen (from air). The balanced equation for the combustion of hydrogen in oxygen is given by: 2H2 + O2 -> 2H2O

Determine the initial moles of each component and the final moles after the combustion process

We are given 1 kmol of hydrogen (H2), and the stoichiometric amount of air. Since air contains roughly 21% oxygen (O2) and 79% nitrogen (N2) by volume according to the problem, we can calculate the initial moles of O2 and N2: Initial amount of O2 needed = (1 kmol H2) * 1/2 = 0.5 kmol O2 Initial amount of N2 in the mixture = (0.5 kmol O2) * (79% N2/21% O2) = 1.88 kmol N2 All O2 will be consumed in the combustion process, and the number of moles of N2 will be conserved. After the combustion process, the moles of other components in the mixture are: Final amount of H2O = (1 kmol H2) * 1 = 1 kmol H2O Final amount of N2 = 1.88 kmol N2

Apply the energy conservation principle

Since the combustion process is adiabatic and constant-volume, we can apply the energy conservation principle: ΔU = Q - W = 0 where ΔU is the change in internal energy of the gas mixture, Q is the heat transfer, and W is the work done. The change in internal energy (ΔU) of the mixture can be expressed as: ΔU = nCv(T_final - T_initial) where n is the total moles of the mixture, Cv is the molar specific heat at constant volume, T_initial is the initial temperature, and T_final is the final temperature. Using the ideal gas law and the known properties of each component, we can determine the final temperature.

Calculate the final temperature

First, let's find the total initial moles in the mixture: n_initial = n_H2 + n_O2 + n_N2 = 1 + 0.5 + 1.88 = 3.38 kmol Now, let's find the total final moles in the mixture after the combustion process: n_final = n_H2O + n_N2 = 1 + 1.88 = 2.88 kmol After this, we can calculate the molar specific heat at constant volume (Cv) for each component of the mixture. For simplicity, we will assume the following values: - Cv for H2 = 2.40R - Cv for O2 = 2.79R - Cv for N2 = 2.49R - Cv for H2O = 3.00R Now, using the energy conservation principle: ΔU = (n_final * Cv_final - n_initial * Cv_initial) * (Tf - Ti) From the above equation, we only have Tf as the unknown value. We can now solve for Tf: Tf = Ti + (ΔU / (n_final * Cv_final - n_initial * Cv_initial)) Considering given initial temperature Ti = 25°C: Tf = 298 K + (0 / ( (1 kmol * 3.00R) + (1.88 kmol * 2.49R) - ((0.5 kmol * 2.79R) + (1.88 kmol * 2.49R) + (1 kmol * 2.40R)))) Tf = 298 K + 0 = 298 K The final temperature in the adiabatic constant-volume tank after complete combustion is 298 K or 25°C.