Find an expression for the change in entropy when two blocks of the same substance and of equal mass, one at the temperature $T_{h}$ the other at $T_{e},$ are brought into thermal contact and allowed to reach equilibrium. Evaluate the change in entropy for two blocks of copper, each of mass 500 g, with $C_{pm}=24.4\ \mathrm{JK}^{-1}\ \mathrm{mol}^{-1},$ taking $T_{\mathrm{h}}=500 \mathrm{K}$ and $T_{e}=250 \mathrm{K}.$

Given that $S_{n}^{\circ}=29791 K^{-1} \mathrm{mol}^{-1}$ for bismuth at 100 K and the following tabulated heat capacity data (D.G Archer) Chem. Eng.Data 40, 1015 (1995), determine the standard molar entropy of bismuth at 200 K.

$$\begin{matrix} \text{$T / \mathrm{K}$} & \text{100} & \text{120} & \text{140} & \text{150} & \text{160} & \text{180} & \text{200}\\ \text{$C_{pm} /\left(1 \mathrm{K}^{-1} \mathrm{mol}^{-1}\right)$} & \text{23} & \text{23.74} & \text{24.25} & \text{24.44} & \text{24.61} & \text{24.89} & \text{25.11}\\ \end{matrix}$$

Compare the value to the value that would be obtained by taking the heat Capacity to be constant at $24.44 JK^{-1} mol^{-1}$ over this range.

(a) Suppose that 2.5 mmol of perfect gas molecules initially occupies $42\ \mathrm{cm}^{3}$ at 300 K and then expands isothermally to $600\ \mathrm{cm}^{3}.$ Calculate $\Delta G$ for the process. (b) Suppose that 6.0 mmol of perfect gas molecules initially occupies $52\ \mathrm{cm}^{3}$ at 298 K and then expands isothermally to $122\ \mathrm{cm}^{3}.$ Calculate $\Delta G$ for the process.

Distinguish between specific heat capacity, molar heat capacity, and heat capacity.

Calculate the difference in molar entropy (a) between liquid water and ice at $- 5 ^ { \circ } \mathrm { C }$, (b) between liquid water and its vapour at $95 ^ { \circ } \mathrm { C }$ and 1.00 atm. The differences in heat capacities on melting and on vaporization are 37.3 J $K^{−1} mol^{−1}$ and –41.9 $K^{−1} mol^{−1}$, respectively. Distinguish between the entropy changes of the sample, the surroundings, and the total system, and discuss the spontaneity of the transitions at the two temperatures.

The heat capacity of chloroform (trichloromethane, ${CHCl}_{3}$ ) in the range 240 K to 330 K is given by $C_{p, {~m}} /\left({J} {K}^{-1} {~mol}^{-1}\right)=91.47+7.5 \times 10^{-2}(T / {K})$. Calculate the change in molar entropy of the sample.

A jury consists of 8 men and 4 women. Four jurors are selected at random for an interview. How many different groups of 4 jurors are there? Find the probability that all 4 jurors chosen are women.

Synthetic division is a process for dividing a polynomial by x - c. The coefficient of x in the divisor is 1. How might synthetic division be used if you are dividing by 2x - 4?

Is a noise with a sound wave modeled by

$$y = \sin ( 1,500 \pi x )$$

cancelled out by another noise with a sound wave modeled by

$$y = \sin [ 1,500 \pi \left( x - \frac { 1 } { 250 } \right) ?$$

Explain.

Show that a process in which liquid water at $5.0^{\circ} \mathrm{C}$ solidifies to ice at the same temperature is not spontaneous.

The molar heat capacity of $\mathrm{N}_{2}(\mathrm{g})$ in the range 200 K to 400 K is given by $C_{p m} /\left(\mathrm{JK}^{-1}\ \mathrm{mol}^{-1}\right) =28.58+3.77 \times 10^{-3} (\mathrm{T} / \mathrm{K}).$ Given that the standard molar entropy of $\mathrm{N}_{2}(\mathrm{g})$ at 298 K is $191.6\ \mathrm{JK}^{-1}\ \mathrm{mol}^{-1},$ calculate the value at 373 K. Repeat the calculation but this time assuming that $C_{p m}$ is independent of temperature and takes the value $29.13\ \mathrm{JK}^{-1}\ \mathrm{mol}^{-1}.$ Comment on the difference between the results of the two calculations.

A sample consisting of 100 mol of perfect gas molecules at $27^{\circ} \mathrm{C}$ is expanded isothermally from an initial pressure of 3.00 atm to a final pressure of 1.00 atm in two ways: (a) reversibly, and (b) against a constant external pressure of 1.00 atm. Evaluate $q, w, \Delta U, \Delta H, \Delta S, \Delta S_{m r r}$ and $\Delta S_{un}$ in each case.

Consider the entropy changes that occur during the Carnot cycle sketched in the given figure earlier. For each stage of the cycle, is the entropy change of the engine positive, negative, or zero? What is the total entropy change of the engine, and of the environment, during one complete cycle? Is there anything that would prevent a real engine from having the same entropy changes for each stage of its cycle?

Sketch a Carnot cycle on a temperature-volume diagram.