Problem 2/a. The energy of a particular gaseous system was found to obey the following equation under certain conditions:

The system is taken through a cycle (represented by the continuous straight line in the figure), beginning and ending at the point A. Calculate
a)	the internal energies at the points A, B, C and D;
b)	the value of heat and work for all four processes of the cycle;
c)	and the value of heat and work, while the system undergoes the process between C and B alongside the dashed hyperbola h that is described by the following equation:

d)	and a parametric equation of all the adiabats in the P–V plane.

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Problem 2/b. Later on, in Chapter 10, by the incorporation of quantum consdierations, we shall state that the following equation (namely, the Sackur–Tetrode equation) is fundamental for a monatomic ideal gas:

Prove the validity of the statement. (Check whether the equation meets all those requirements concerning the entropy that are imposed by the postulates of thermodynamics). 
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Problem 3/a. There are two containers, A and B, having the same volume of 2 m3, and they are both filled with a monatomic ideal gas. The containers are separated from each other by a rigid and adiabatic wall; and a wall disallowing the transport of every possible extensive quantities separates the containers from their environment. The container A holds 3 moles, the container B holds 5 moles of the gas. Container A initially has a temperature of 600 K and the total energy in the composite system is 35 kJ.
a)	Calculate the internal energies of the two subsystems and the temperature of container B.
b)	Calculate the pressure in the two containers.
c)	What would be the temperature of the system, if we would let the wall separating the two containers be diathermal? How would the pressure change in each of the containers?
d)	What would be the pressure in the two containers, if we would let the wall separating them be flexible? How would this affect the tempreature measured in the two subsystems, and how would the volume of the containers change?
e)	What would be the common temperature and pressure of the two subsystems, if we would let the wall be diathermal and flexible at the same time?
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Problem 3/b. We have a reactor with one chamber, having a volume of 1 ℓ, completely isolated from the environment. The reactor is filled with Ne gas. We inject a mixture of a small amount of hydrogen gas and evaporated iodine and bromine into the vessel, then we use an electrically heated filament to set the temperature of the system in such a way that the following reactions would occur:

Define the condition of equilibrium!
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Problem 4/a. The fundamental equation of a particular system is determined to be

Calculate the thermodynamic potentials S, H, F, G. Find the intensive parameters of the system, and express the heat capacities  and  as well as the coefficient of thermal expansion α and the isothermal compressibility κ.
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Problem 4/b. The fundamental equation of an ideal van der Waals fluid, as we have shown earlier in Chapter 3, is

with a, b, and c positive real constants. Show how you can use this equation in order to calculate the thermodynamic quantities T, p, μ, F and CV of the system.
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Problem 5/a. We are consuming electric energy from the power company that we use to provide heating to our appartement having a temperature of 25 °C in an environment of –2 °C . Show that at which factor we can improve our cost-benefit ratio if we use the energy to drive a reverse Carnot engine rather than to produce Joule heat.
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Problem 5/b. The so-called air-standard Otto cycle is a rough approximation model of the gasoline engines of cars. In an Otto cycle the working fluid (which we would now consider as a monatomic ideal gas) is
a)	first compressed adiabatically;
b)	then it is heated at constant volume;
c)	then it is expanded adiabatically;
d)	and finally it is cooled down isochorically to the initial state.

Illustrate the operation of the Otto cycle by plotting its P–V and T–S diagrams. Show that the efficiency of the Otto cycle can be expressed as

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Problem 6/a. By using a pycnometer, we have measured the density of 60 and 82.5 g/dm3 aqueous NaCl solutions at 25 °C. The measured densities were 1.0315 and 1.0463 g/cm3, respectively. Calculate the partial molar volume of NaCl and of water in the solution!
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Problem 6/b. We have to gas containers, one containing 1 g of protium, the other 1 g of deuterium. If we open the valve separating the containers, the two gases would be mixed. Calculate the change of entropy!
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Problem 6/c. We have a thermodynamic system containing two components, A and B,  In the close vicinity of 25 °C, the relative activity coefficients of the components are given by empirical formulas:

Calculate the excess entropy, enthalpy and Gibbs free energy of the system.
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Problem 7/a. The equilibrium vapor pressure over a single component fluid was found to be 0.2 atm at 300 K. Assuming that the vapor can be treated as an ideal gas, and that its density is much less than the density of the fluid, estimate the temperature at which the equilibrium vapor pressure would be doubled. The heat of vaporization is 40 kJ/mol.
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Problem 7/b. Sketch the phase diagram of methanol at the vicinity of its triple point (T = 175.610 K, p = 0.0002 Pa by making use of the following data:    
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Problem 7/c. The melting points of the pure compounds A and B are 120 K and 150 K, respectively. The two substances undergo a chemical reaction in both solid and liquid phases, to form a compound AB:

The melting point of the AB compound is 140 K. The reacting system has two eutectic points at atmospheric pressure, one at the absolute mole fraction of A of 0.25 (110 K) and another at 0.80 (90 K). Draw a sketch of the phase diagram for the reacting system, and another of the cooling curves at the absolute mole fraction values of A of 0.25, 0.50 and 0.75. 
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Problem 7/d. It was found that the boiling point of a binary solution of A and B, if the composition of the liquid is  is 100 °C. At the given temperature, the vapor pressure of pure A and B are 120.1 kPa and 89.0 kPa, respectively. Prove that the solution is ideal and calculate the composition of the vapor as boiling begins.
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Problem 8/a. The equilibrium constant of a particular chemical reaction was found to obey the following formula in the close vicinity of 

Calculate the reaction enthalpy and entropy at the given temperature.
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Problem 8/b. The equilibrium constant of a particular chemical reaction gets doubled if we elevate the temperature from 200 K to 300 K. Calculate the reaction enthalpy and the reaction entropy at 250 K.
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Problem 9/a. At 18 °C, water rises at a height of 5 cm in a glass capillary with a radius of 0,3 mm. The contact angle of glass and water is approximately zero, and the density of water is 998.6 kg/m3. Calculate the equilibrium vapor pressure of water above a water drop with a radius of 10 nm based on the given data.
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Problem 9/b. A preliminary assumption of the Debye–Hückel theory is that the concentration distribution of a kind of a charged particle around a chosen ion is given as a function of distance as

where  is the average concentration of the ith kind of charged particles in the solution, e is the elementary charge,  is the charge number of the particle,  is the electric potential and k is the Boltzmann constant. The theory applies the above function in a form of power series expansion up to the linear term, and sums it up to all kinds of particles.
Based on these considerations, argue why the theory works better for a solution of KCl than of CaBr2.
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Problem 10/a. Express the molar heat capacity of an Einstein solid as a function of temperature. Calculate the  and  limits. Draw a plot of the molar heat capacity as a function of temperature between 0 K and 10 K (use  as a constant value).