Which of the following are functions? If f is not a function explain why. \text { i. } f: \mathbb{R} \rightarrow \mathbb{R} \text { with } f(x)=\frac{1}{1-x^{2}} \text { ii. } f: \mathbb{Z} \rightarrow \mathbb{Z} \text { with } f(x)=\frac{x}{2} \text { iii. } f: \mathbb{R} \rightarrow \mathbb{R} \text { with } f(x)=\ln (x) \text { Let } f: \mathbb{R} \rightarrow \mathbb{R} \text { and } g: \mathbb{R} \rightarrow \mathbb{R} \text { with } f(x)=x+2 \text { and } g(x)=-x \text {. Find } g \circ f,(g \circ f)^{-1}, f^{-1} \text { and } g^{-1} \text { Let } f: \mathbb{R}^{*} \rightarrow \mathbb{R} \text { with } f(x)={ }^{x} 1 \text {, where } \mathbb{R}^{*} \text { is the set of all real numbers } different from 0. i. Determine whether or not f is a one to one function ii. Determine whether or not f is an onto function \text { d) Given a function } F: \mathcal{P}(\{a, b, c\}) \rightarrow \mathbb{Z} \text { is defined by } F(A)=|A| \text { for all } A \in \mathcal{P}(\{a, b, c\}) i. Is a one-to-one function? Prove or give a counter-example. ii. Is F an onto function? Prove or give a counter-example. \text { Let } f: A \rightarrow B \text { and } g: B \rightarrow C \text { be functions. Prove that if } g \circ f \text { is one-to-one }

E2A.6(a) A sample of 4.50g of methane occupies 12.7 dm3 at 310 K. (i) Calculate the work done when the gas expands isothermally against a constant external pressure of 200 Torr until its volume has increased by 3.3 dm². (ii) Calculate the work that would be done if the same expansion occurred reversibly. E2A.6(b) A sample of argon of mass 6.56g occupies 18.5 dm3 at 305 K.(i) Calculate the work done when the gas expands isothermally against a constant external pressure of 7.7kPa until its volume has increased by 2.5 dm3.(ii) Calculate the work that would be done if the same expansion occurred reversibly. F=\frac{k T}{2 l} \ln \left(\frac{1+v}{1-v}\right) \quad v=\frac{n}{N} where k is Boltzmann's constant, N is the total number of units, and l= 45 nm for DNA. (a) What is the magnitude of the force that must be applied to extend a DNA molecule with N=200 by 90 nm? (b) Plot the restoring force against v, noting that v can be either positive or negative. How is the variation of the restoring force with end-to-end distance different from that predicted by Hooke's law? (c) Keeping in mind that the difference in end-to-end distance from an equilibrium value is x = nl and, consequently, dx = ldn= Nldv,write an expression for the work of extending a DNA molecule. Hint: You must integrate the expression for w. The task can be accomplished best with mathematical software.

A gas stream containing 3 mol% Ammonia (NH3) in Air is to be passed to a packed absorption columnat a rate of 5 kg s1. The column is to use Water (H20) as the solvent to reduce the ammonia content in the air leaving the column to 0.01 mol%. The gas and water streams can be assumed to be at 25 °C.The column operates at 1 bar pressure. The relationship that describes the equilibrium between Ammonia and Water at these conditions is given by: y = Hx Where the Henry's Law constant, H = 1.3 (mole frac NH3 in gas) (mole frac NH3 in liquid)1 From pilot scale experiments the Overall Mass Transfer Coefficient, KG, has been found to remain constant with a value of 200 × 10--6 kmol m² s-1 Using the protocol outlined on Page 55 of the gas absorption notes, specify an absorption column to achieve the required separation using 1 inch Raschig Rings (See Table 1 in attached data sheets). Your Design Specification must clearly show the following: Any assumptions made must be stated clearly. What the minimum liquid rate for the column is. What the liquid rate of solvent is under normal operating conditions. Based on the assumption of operating at 60% of the flooding gas flow rate, what the diameter of the column should be in meters. Confirmation that the wetting rate of your column falls within the acceptable range. The values of HTU, NTU and thus the total height of packing required in the column.

P2E.1 Calculate the final temperature, the work done, and the change of internal energy when 1.00 mol NH,(g) at 298 K is used in a reversible adiabatic expansion from 0.50 dm³ to 2.00dm².

When measuring small pressure differences with a manometer, often one arm of is inclined to improve the accuracy of the reading. The air pressure in a circular duct is to be measured using a manometer whose open arm is inclined 32° from the horizontal, as shown in The figure. The density of the liquid in the manometer is 0.86 kg/L, and the vertical distance between the fluid levels in the two arms of the manometer is 12 cm. Determine the gage pressure of air in the duct (in Pa) and the length of the fluid column in the inclined arm above the fluid level in the vertical arm (in cm).

E1A.3b) A perfect gas undergoes isothermal compression, which reduces its volume by 1.80 dm". The final pressure and volume of the gas are 1.97 bar and 2.14 dm', respectively. Calculate the original pressure of the gas in (i) bar,(ii) torr.

4.36.) A gas mixture of methane and steam at atmospheric pressure and 500°C is fed to a reactor, where the following reactions occur: \mathrm{CH}_{4}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{CO}+3 \mathrm{H}_{2} \quad \text { and } \quad \mathrm{CO}+\mathrm{H}_{2} \mathrm{O} \rightarrow \mathrm{CO}_{2}+\mathrm{H}_{2} The product stream leaves the reactor at 850°C. Its composition (mole fractions) is: y_{\mathrm{CO}_{2}}=0.0275 \quad y_{\mathrm{CO}}=0.1725 \quad y_{\mathrm{H}_{2} \mathrm{O}}=0.1725 \quad y_{\mathrm{H}_{2}}=0.6275 Determine the quantity of heat added to the reactor per mole of product gas.

Consider the reaction of 2-bromo-2-methylpropane with water, to answer the following question(s). (a) Write three-step reaction mechanism. (b) Add curved arrows to indicate electron flow for above reaction mechanism (c) Label the nucleophile, Nu, and the electrophile, E*, in the Step 2 and indicate the overall reactiontype.

Estimate the rate of heat transfer from the compressor. Assume for air that Cp = 7/2 Rand that enthalpy is independent of pressure. 2.27 Fifty (50) kmol per hour of air is compressed from Pt = 1.2 bar to P2 = 6.0 bar in a steady-flow compressor. Delivered mechanical power is 98.8 kW. Temperatures and velocities are: T1 = 300 K T2 = 520 K u1 = 10 m.s^-1 u2 = 3.5 m.s^-1

P The vapour pressure, p, of nitric acid varies with temperature as follows: Determine (a) the normal boiling point and (b) the enthalpy of vaporization of nitric acid. to be perfect and calculate the partial pressures of the two components. Plot them against their respective mole fractions in the liquid mixture and find the Henry's law constants for the two components.

1- Methane is burned with atmospheric air. The analysis of the products on a dry basis is as follows: CO210.00% CO= 0.53% O2= 2.37% Calculate the equivalence ratio.