Five kilograms of liquid carbon tetrachloride undergo a mechanically reversible, isobaric change of state at 1 bar during which the temperature changes from $0^{\circ} \mathrm{C} \text { to } 20^{\circ} \mathrm{C}$ Determine $\Delta V^{t}, W, Q, \Delta H^{t}, \text { and } \Delta U^{t}.$ The properties for liquid carbon tetrachloride at 1 bar and $0^\circ C$ may be assumed independent of temperature: $\beta=1.2 \times 10^{-3} \mathrm{K}^{-1}, C_{P}=0.84 \mathrm{kJ} \cdot \mathrm{kg}^{-1} \cdot \mathrm{K}^{-1}, \text { and } \rho=1590 \mathrm{kg} \cdot \mathrm{m}^{-3}$

Generally, volume expansivity $\beta$ and isothermal compressibility $\kappa$ depend on T and P. Prove that: $\left(\frac{\partial \beta}{\partial P}\right)_{T}=-\left(\frac{\partial \kappa}{\partial T}\right)_{P}$

One mole of an ideal gas with $C_P = (5/2)R$ and $C_V = (3/2)R$ expands from $P_1 = 6$ bar and $T_1 = 800 K$ to $P_2 = 1$ bar by each of the following paths: (a) Constant volume (b) Constant temperature (c) Adiabatically. Assuming mechanical reversibility, calculate W, Q, $\Delta U, \text { and } \Delta H$ for each process. Sketch each path on a single PV diagram.

Maximize the Shannon entropy $S=-k \sum_{i} P_{i} \log P_{i}$ subject to the constraints that $\sum P_{i}=1 \text { and }\langle f(x)\rangle=\sum P_{i} f\left(x_{i}\right)$ and show that $P_{i}=\frac{1}{Z(\beta)} \mathrm{e}^{-\beta f\left(x_{i}\right)}$, $Z(\beta)=\sum \mathrm{e}^{-\beta f\left(x_{i}\right)}$, $\langle f(x)\rangle=-\frac{\mathrm{d}}{\mathrm{d} \beta} \ln Z(\beta)$.

Water flows through a horizontal coil heated from the outside by high-temperature flue gases. As it passes through the coil, the water changes state from liquid at 200 kPa and $80^\circ C$ to vapor at 100 kPa and $125^\circ C.$ Its entering velocity is $3 m \cdot s^{−1}.$ and its exit velocity is $200 m \cdot s^{−1}.$ Determine the heat transferred through the coil per unit mass of water. Enthalpies of the inlet and outlet streams are: $\text { Inlet: } 334.9 \mathrm{kJ} \cdot \mathrm{kg}^{-1} ; \text { Outlet: } 2726.5 \mathrm{kJ} \cdot \mathrm{kg}^{-1}$

An incompressible $(\rho = constant)$ liquid flows steadily through a conduit of circular cross-section and increasing diameter. At location 1, the diameter is 2.5 cm and the velocity is $2 \mathrm{m} \cdot \mathrm{s}^{-1};$ at location 2, the diameter is 5 cm. (a) What is the velocity at location 2? (b) What is the kinetic-energy change $\left(\mathrm{J} \cdot \mathrm{kg}^{-1}\right)$ of the fluid between locations 1 and 2?

The Tait equation for liquids is written for an isotherm as: $V=V_{0}\left(1-\frac{A P}{B+P}\right)$ where V is molar or specific volume, $V_0$ is the hypothetical molar or specific volume at zero pressure, and A and B are positive constants. Find an expression for the isothermal compressibility consistent with this equation.

Express the volume expansivity and the isothermal compressibility as functions of density $\rho$ and its partial derivatives. For water at $50^\circ C$ and 1 bar, $\kappa=44.18 \times 10^{-6}$ $bar^{−1}.$ To what pressure must water be compressed at $50^\circ C$ to change its density by 1%? Assume that $\kappa$ is independent of P.

A substance for which $\kappa$ is a constant undergoes an isothermal, mechanically reversible process from initial state $(P_1, V_1)$ to final state $(P_2, V_2),$ where V is molar volume. (a) Starting with the definition of $\kappa,$ show that the path of the process is described by: $V=A(T) \exp (-\kappa P)$ (b) Determine an exact expression which gives the isothermal work done on 1 mol of this constant-$\kappa$ substance.

Rough Stuff makes 2 products: khaki shorts and khaki pants for men. Each product passes through the cutting machine area, which is the chief constraint during production. Khaki shorts take 15 minutes on the cutting machine and have a contribution margin per pair of shorts of $16. Khaki pants take 24 minutes on the cutting machine and have a contribution margin per pair of pants of$32. If it is assumed that Rough Stuff has 4,800 hours available on the cutting machine to service a minimum demand for each product of 3,000 units, how much will profits increase if 100 more hours of machine time can be obtained?

A car leaving a parking space has velocity (in metres per second) given by the equation

$$v = \frac { 1 } { 2 } t ^ { 2 } - 2 t$$

, where t is the time in seconds. The car is assumed to move along a straight line at all times. (a) Show that the car is stationary when t = 0 and when t = 4. (b) Describe the car’s motion during the first eight seconds. (c) Find the distance travelled in the first four seconds. (d) Find the distance travelled in the first eight seconds. (e) Find the displacement after eight seconds. (f) At what time is the acceleration

$$5 m / s ^ { 2 }$$

? (g) Find the maximum speed in the first eight seconds.

A system comprised of chloroform, 1,4-dioxane, and ethanol exists as a two-phase vapor/liquid system at $50^\circ C$ and 55 kPa. After the addition of some pure ethanol, the system can be returned to two-phase equilibrium at the initial T and P. In what respect has the system changed, and in what respect has it not changed?

Liquid water at $180^{\circ} \mathrm{C}$ and 1002.7 kPa has an internal energy (on an arbitrary scale) of $762.0 \mathrm{kJ} \cdot \mathrm{kg}^{-1}$ and a specific volume of $1.128 \mathrm{cm}^{3} \cdot \mathrm{g}^{-1}.$ (a) What is its enthalpy? (b) The water is brought to the vapor state at $300^{\circ} \mathrm{C}$ and 1500 kPa, where its internal energy is $2784.4 \mathrm{kJ} \cdot \mathrm{kg}^{-1}$ and its specific volume is $169.7 \mathrm{cm}^{3} \cdot \mathrm{g}^{-1}.$ Calculate $\Delta U$ and $\Delta H$ for the process.

Westerville Company reported the following results from last year's operations:

$$\begin{array}{lr} \text{Sales} & \$1,000,000\\ \text{Variable expenses} & \underline{\hspace{10pt} 300,000}\\ \text{Contribution margin} & 700,000\\ \text{Fixed expenses} & \underline{\hspace{10pt} 500,000}\\ \text{Net operating income} & \underline{\underline{\$\hspace{10pt} 200,000}}\\ \text{Average operating assets} & \underline{\underline{\$\hspace{10pt} 200,000}}\\ \end{array}$$

At the beginning of this year, the company has a $\$ 120,000$ investment opportunity with the following cost and revenue characteristics:

$$\begin{array}{lr} \text{Sales} & \$200,000\\ \text{Contribution margin ratio} & 60\% \text{ of sales}\\ \text{Fixed expenses} & \$90,000\\ \end{array}$$

The company's minimum required rate of return is 15%.

If the company pursues the investment opportunity and otherwise performs the same as last year, what turnover will it earn this year?