(i) Explain Kohlrausch's law. Write its applications.
Calculate the e.m.f. of the following cell: \[ \text{Cu(s)} | \text{Cu}^{2+} (1M) || \text{Ag}^{+} (0.01M) | \text{Ag(s)} \] Given: \[ E^0_{\text{Cu}^{2+}/\text{Cu}} = +0.34V, \quad E^0_{\text{Ag}^{+}/\text{Ag}} = +0.80V \]
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).