The conversion of sulfur dioxide \( \text{SO}_2 \) to sulfur trioxide \( \text{SO}_3 \) in the Contact Process is a key step in the production of sulfuric acid. This reaction occurs in the presence of a vanadium(V) oxide catalyst.
Step 1: Reaction Characteristics:
The reaction:
\[ 2\ \text{SO}_2 (g) + \text{O}_2 (g) \rightarrow 2\ \text{SO}_3 (g) \]
is known to be exothermic, releasing heat. The standard enthalpy change \( \Delta H^\circ \) for this reaction is negative, indicating that heat is released to the surroundings.
Step 2: Reversibility:
This reaction is also reversible. Under industrial conditions, it does not go to completion, and the formation of \( \text{SO}_3 \) can be reversed back to \( \text{SO}_2 \) and \( \text{O}_2 \) under certain conditions.
The use of a catalyst and optimized conditions such as temperature and pressure help shift the equilibrium towards the formation of more \( \text{SO}_3 \).
Step 3: Industrial Significance:
The exothermic nature of the reaction helps in maintaining operational efficiency, while its reversibility is critical for maximizing \( \text{SO}_3 \) yield through the recirculation of unreacted gases and careful control of reaction conditions.
Consider a process with transfer function: \[ G_p = \frac{2e^{-s}}{(5s + 1)^2} \] A first-order plus dead time (FOPDT) model is to be fitted to the unit step process reaction curve (PRC) by applying the maximum slope method. Let \( \tau_m \) and \( \theta_m \) denote the time constant and dead time, respectively, of the fitted FOPDT model. The value of \( \frac{\tau_m}{\theta_m} \) is __________ (rounded off to 2 decimal places).
Given: For \( G = \frac{1}{(\tau s + 1)^2} \), the unit step output response is: \[ y(t) = 1 - \left(1 + \frac{t}{\tau}\right)e^{-t/\tau} \] The first and second derivatives of \( y(t) \) are: \[ \frac{dy(t)}{dt} = \frac{t}{\tau^2} e^{-t/\tau} \] \[ \frac{d^2y(t)}{dt^2} = \frac{1}{\tau^2} \left(1 - \frac{t}{\tau}\right) e^{-t/\tau} \]
Methanol is produced by the reversible, gas-phase hydrogenation of carbon monoxide: \[ {CO} + 2{H}_2 \rightleftharpoons {CH}_3{OH} \] CO and H$_2$ are charged to a reactor, and the reaction proceeds to equilibrium at 453 K and 2 atm. The reaction equilibrium constant, which depends only on the temperature, is 1.68 at the reaction conditions. The mole fraction of H$_2$ in the product is 0.4. Assuming ideal gas behavior, the mole fraction of methanol in the product is ____________ (rounded off to 2 decimal places).