A certain semiconductor material has a conductivity of $0.0124 \mathrm{~W} / \mathrm{cm} \cdot{ }^{\circ} \mathrm{C}$. A rectangular bar of the material has a cross-sectional area of $1 \mathrm{~cm}^2$ and a length of $3 \mathrm{~cm}$. One end is maintained at $300^{\circ} \mathrm{C}$ and the other end at $100^{\circ} \mathrm{C}$, and the bar carries a current of $50 \mathrm{~A}$. Assuming the longitudinal surface is insulated, calculate the midpoint temperature in the bar. Take the resistivity as $1.5 \times 10^{-3} \Omega \cdot \mathrm{cm}$.

The feed to an ammonia synthesis reactor contains 25 mole% nitrogen and the balance hydrogen. The flow rate of the stream is 3000 kg/h. Calculate the rate of flow of nitrogen into the reactor in kg/h. (Suggestion: First calculate the average molecular weight of the mixture.)

Specify fin efficiency.

Derive an expression for the temperature distribution in a hollow cylinder with heat sources that vary according to the linear relation

$$\dot{q}=a+b r$$

with $\dot{q}_i$ the generation rate per unit volume at $r=r_i$. The inside and outside temperatures are $T=T_i$ at $r=r_i$ and $T=T_o$ at $r=r_o$.

Discover the heat transfer per unit area through the composite wall. Assume one-dimensional heat flow.

$$\begin{aligned} k_A & =150 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C} \\ k_B & =30 \\ k_C & =50 \\ k_D & =70 \\ A_B & =A_D \end{aligned}$$

A plane wall of thickness $2 L$ has an internal heat generation that varies according to $\dot{q}=\dot{q}_0 \cos a x$, where $\dot{q}_0$ is the heat generated per unit volume at the center of the wall $(x=0)$ and $a$ is a constant. If both sides of the wall are maintained at a constant temperature of $T_w$, derive an expression for the total heat loss from the wall per unit surface area.

(a) The wavenumber of the fundamental vibrational transition of $^{32} \mathrm{C}_{2}$ is $564.9 \mathrm{cm}^{-1}.$ Calculate the force constant of the bond. (b) The wavenumber of the fundamental vibrational transition of $^{71} B r^{81} Br$ is $323.2 \mathrm{cm}^{-1}.$ Calculate the force constant of the bond.

(a) Calculate the minimum excitation energy (i.e. the difference in energy between the first excited state and the ground state) of a proton constrained to rotate in a circle of radius 100 pm around a fixed point. (b) Calculate the value of $|m_{t}|$ for the system described in the preceding Exercise corresponding to a rotational energy equal to the classical average energy at $25^{\circ} \mathrm{C}$ (which is equal to $\frac{1} {2} kT).$

Show that Φ(t)=Ψ(t)Ψ−1(t0), where Φ(t) and Ψ(t)are as defined in this section.

The rotation of an ${ }^1 \mathrm{H}^{127} \mathrm{I}$ molecule can be pictured as the orbital motion of an $\mathrm{H}$ atom at a distance $160 \mathrm{pm}$ from a stationary $\mathrm{I}$ atom. (This picture is quite good; to be precise, both atoms rotate around their common centre of mass, which is very close to the I nucleus.) Suppose that the molecule rotates only in a plane.
Calculate the energy needed to excite the molecule into rotation.
What, apart from 0 , is the minimum angular momentum of the molecule?

(a) The rotation of a molecule can be represented by the motion of a particle moving over the surface of a sphere. Calculate the magnitude of its angular momentum when l = 1 and the possible components of the angular momentum along the z-axis. Express your results as multiples of ft. (b) The rotation of a molecule can be represented by the motion of a particle moving over the surface of a sphere with angular momentum quantum number l = 2. Calculate the magnitude of its angular momentum and the possible components of the angular momentum along the z-axis. Express your results as multiples of h.

(a) How many nodes are there in the wavefunction of a harmonic oscillator with (i) v = 3; (ii) v = 4? (b) How many nodes are there in the wavefunction of a harmonic oscillator with (i) v = 5, (ii) v = 35?

Heat is generated in a 2.5-cm-square copper rod at the rate of $35.3 \mathrm{MW} / \mathrm{m}^3$. The rod is exposed to a convection environment at $20^{\circ} \mathrm{C}$, and the heat-transfer coefficient is $4000 \mathrm{~W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Calculate the surface temperature of the rod.

A counterflow double-pipe heat exchanger operates with hot water flowing inside the inner pipe and a polymer fluid flowing in the annular space between the two pipes. The water-flow rate is $2.0 \mathrm{~kg} / \mathrm{s}$ and it enters at a temperature of $90^{\circ} \mathrm{C}$. The polymer enters at a temperature of $10^{\circ} \mathrm{C}$ and leaves at a temperature of $50^{\circ} \mathrm{C}$ while the water leaves the exchanger at a temperature of $60^{\circ} \mathrm{C}$. Determine the value of the overall heat-transfer coefficient expressed in $\mathrm{W} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$, if the area for the heat exchanger is $20 \mathrm{~m}^2$.