Questions and Answers of Thermodynamics Fundamentals

A system consists of a thin film of surface area A, of internal energy U(S, A), whereHence, the surface tension is given byExpress the heat Qif to provide to the film for a variationΔAif = Af−Ai
To establish the positivity of the specific heat at constant pressure Cp compressibility coefficient at constant temperature κT (see relations (6.31)), follow the steps given here [65]:a) Show that
An isolated system consisting of two closed subsystems A and B is separated by a diathermal wall. Initially, they are held at temperatures TiA and TiB . Subsystem A contains NA moles of gas. The
A phase diagram is drawn for a mixture of two substances at a fixed pressure p with a liquid phase and a gaseous phase (Fig. 6.18). The substances are labelled 1 and 2 and the diagram is shown as a
The process whereby molecules bind to a metallic surface is called adsorption. Here, molecules are adsorbed on Pt nanoparticles. The specific heat of an average Pt nanoparticle is CV. The heat
A metallic block of mass M is brought to a temperature T0 and plunged into a calorimeter filled with a mass M of water. The system consisting of the metallic block and the water container is
A gas is characterised by the enthalpy H(S, p) = Cp T, where Cp is a constant (called heat capacity and defined in § 5.2), and by pV = NR T, where p is its pressure, V its volume, T its temperature
Establish the expression of the differential of the internal energy dU S (T, V), V as a function of the temperature T and the volume V. In the particular case of a gas that satisfies the relation pV
The grand potential Φ(T, V, {μA}), also known as the Landau free energy, is a thermodynamical potential obtained by performing Legendre transformations ofthe internal energy U(S, V, {NA}). Use
James Atkinson was a British engineer who designed several combustion engines. The thermodynamic cycle bearing his name is a modification of the Otto cycle intended to improve its efficiency. The
The surface tension modifies the melting point of particles. The effect is important when the diameter is in the nanometer range. A differential equation has to be written for Tm (r), the melting
A system is made up of a vertical cylinder which is sealed at the top and closed by a piston at the bottom. A valve A controls the intake of gas at the top and an exhaust valve B (also at the top) is
A thermoelectric refrigerator becomes cold by expelling heat into the environment at a temperature T+. The power supplied to the device is PW and the thermal power corresponding to the rejected heat
An ideal gas characterised by the coefficient c found in relation (5.62) and the coefficient γ = (c + 1) /c undergoes a Rankine engine cycle consisting of four reversible processes:• 1−→ 2 :
The Otto cycle is a model for a spark ignition engine and represents the mode of operation of most non-diesel car engines. It consists of four processes when the system is closed, and of two
For an ideal gas, show that at any point on a Clapeyron (p, V) diagram, the absolute value of the slope is greater for an adiabatic process (A) than an isothermalprocess (I).
An ideal gas characterised by the coefficient c found in relation (5.62) and the coefficient γ = (c + 1) /c undergoes a refrigeration cycle consisting of four reversible processes (Fig. 7.24):•
The chemical reaction of ammonia oxidation reads,Consider that initially this reaction is taking place with NNH3 (0) moles of NH3 and NO2 (0) moles of O2. Find the amount of NH3, O2, NO and H2O at
a) There are two isomers of butane: butane (C4H10) and isobutane (methylpropane) (iso-C4H10). Determine the standard enthalpy of isomerisation Δh◦ of butane to isobutane in terms of the enthalpies
Acetylene (C2H2) can be produced through a chemical reaction between water (H2O) and calcium carbide (CaC2):where (s) and (l) indicate whether the substance is solid or liquid. A cave explorer
The oxidation of methane can take place according to either one of the following reactions:When the reactions stop at time tf because all the methane is burned, the total mass of the products (CO2,
Apply the general definition of the battery potential,to the Daniell cell (§ 8.7.4) and show that it yields relation (8.108). Show that the battery potential can be written as,where ΔΕΞ
Steel wool is placed inside a cylinder filled with molecular oxygen O2, considered as an ideal gas. A piston ensures a constant pressure of the gas. The steel wool reacts with the molecular oxygen to
The Fischer esterification reaction is given by,Determine the equilibrium constant K of this reaction in terms of the concentrations of the reactants cR-(C=O)-OH, cR-OH and of the products
The variance v is the number of degrees of freedom of a system consisting of r substances in m phases taking part in n chemical reactions. The variance v is obtained by subtracting n constraints from
A container at a pressure p and temperature T contains two substances 1 and 2, present both in liquid and gas phases. Estimate the partial pressure pA of substanceEntropy of mixing, as a function of
At sea level, water from the outlet of a river is diverted to a power plant that operates on the principle of osmosis. A turbine is installed in the pipe that brings the river water to an osmotic
At constant temperature T, show that the vapour pressure p (T,B) of a paramagnetic liquid has the following dependence on the magnetic induction field B,in the small field limit, i.e. cB0 RT2. The
An isolated system consists of two rigid subsystems of volumes V1 and V2 separated by a rigid and porous membrane. Helium (He) can diffuse through the membrane, but oxygen (O2) cannot.We label the
Two parallel plates are made of two metals 1 and 2 inside a vacuum chamber (Fig. 9.11). The chemical potentials of electrons in metals 1 and 2 are μ1 and μ2. When contact is established between the
Consider an electrochemical cell made of two half-cells that are identical except that they are maintained at different temperatures. This is called a thermogalvanic cell. Determine the
A U-shaped tube contains a solution of paramagnetic salt, such as CoSO4. One side of the U-shaped tube, considered as a subsystem 1, is in a magnetic induction field B, the other side, considered as
The magnetocaloric effect is defined by the temperature variation observed in a material under adiabatic conditions when the applied magnetic induction field B varies. The effect becomes particularly
Model the temperature profile T(z) of the Earth’s atmosphere as a function of height z. Ignore winds, clouds and the many effects due to the presence of moisture in the air and treat the air as an
Some magnetic materials undergo a first-order phase transition when a magnetic induction field B is applied. These materials are of interest because some of them have a strong magnetocaloric effect,
Consider a mixture of water and salt with a low salt concentration. Use the ideal mixture law (8.68) to evaluate the chemical potential of water in the salt solution. Recall that according to
In the electromechanical system presented here (Fig. 9.12), a metallic plate 1 is fixed mechanically and a metallic plate 2 vibrates under the effect of a spring of elastic constant k, of length z
A container with a long spout is filled with oil and water in such a way that the spout is filled to its top (Fig. 10.5). The water and the oil surfaces are in contact with the atmosphere. The height
A spherical weight of radius R is blocking a horizontal circular hole at the bottom of a container filled with a liquid (Fig. 10.6). The liquid of mass density m is at a height H above the hole (H
Electrowetting is a means of driving microfluidic motion by modifying the surface tension [133]. Here, we determine the effect of an electrostatic potential difference Δφ applied between the
Consider a homogeneous and uniform fluid which consists of different reactive chemical substances.1. Determine the variation rate n˙A of the chemical substance density A.2. When the system is in a
1. Determine the expression of the pressure time derivative.2. Determine the expression of the pressure gradient.
A transport equation such as Ohm’s law (11.74),relates two vectors, which are the conductive electric current density jq and electric potential gradient ∇φ , through a linear application, which
The transport properties of a thermoelectric material of cross-section area A and length L are defined by the transport equations,in conformity with relations (11.92), where ∇μe = 0, and (11.95).
A Peltier generator is made up of two thermoelectric elements connected in series (Fig. 11.13). One side of the Peltier generator is maintained at temperature T+ and the other temperature T−. The
A long copper rod of thermal diffusivity λ is heated at one end with a flame passing underneath it periodically, while the other end is located so far away that it remains at room temperature T0.
The heat diffusion equation was established in § 11.4.2, in the absence of any source term due to a conductive electric current density, i.e. jq = qe je = 0. Show that for an electric conductor in
In electrochemistry, the observed current is generally determined by ion diffusion in the electrolyte. It was found that diffusion-limited currents can be avoided by using very small electrodes,
Show that the temperature profile (11.44),where T is the temperature and x the spatial coordinate, is a solution of the heat diffusion equation (11.37). T(x, t) = с ст exp *² 4Xt
In this chapter, several examples of a current density in one direction inducing the gradient of an intensive quantity in another direction were shown. These effects are referred to by the name of
A biological medium consists of two substances 1 and 2 of densities n1 and n2. This medium is generating both substances by processes characterised by the matter source densities π1 (n1, n2) and
A fluid flows through a pipe which is shaped in such a way that the velocity field depends linearly on the position x along the pipe (Fig. 10.7). At the inlet (x = 0), the velocity is v0. At the
Establish the continuity equation (10.34) for the mass density by working out the change in mass inside an infinitesimal cubic box centred at a position written in Cartesian coordinates as (x, y, z).
A heat exchanger is made up of two identical pipes separated by an impermeable diathermal wall of section area A, thickness h and thermal conductivity κ. In both pipes, a liquid flows at uniform
An isotropic conductor is in the presence of a magnetic induction field B. The electric resistivity rank-2 tensor is a function of the magnetic induction field B and Ohm’s law is written as,The
A rod is connected at both ends to electrodes. The electric wires that connect the rod to each electrode are strong enough to carry an electric current flowing through the rod and yet thin enough for
A crystal consists of a honeycomb lattice. It is invariant under a rotation of angle θ = π/6 in the horizontal plane around the vertical axis. This means that thephysical properties of the crystal
Estimate the temperature profile of a wire of length L and radius r when an electric current I is driven through it, from the left end to the right end, causing it to heat up. The wire has an
Estimate the temperature profile of a wire of length L and radius r when an electric current I is driven through it, from the left end to the right end, causing it to heat up. The wire has an
Two long blocks, made up of different homogeneous materials, are at temperatures T1 and T2 when they are brought into contact with one another. The interface quickly reaches a temperature T0 given
Calculate the density of gaseous CO2 at each of the following states:a. 10 bar, 290 K,b. 30 bar, 333 K,c. 100 bar, 410 K,Using the perfect gas model. Compare the values obtained with the values
Propane is in a 5 m3 container as a saturated vapor at 300 K. The sealed tank is then cooled to 260 K by fresh snow. What are the pressure (Pa) and liquid volume in this 260 K state, and how much
A constant volume vessel of 100 liters holds 1 kg of water at an initial pressure of 30 bar. The container is then cooled to 305 K.Determine the initial temperature of the system as well as the
Refrigerant 134a in a heat pump is compressed in an adiabatic piston?cylinder system from 0.14 MPa, 280 K to 1 MPa, 400 K. Calculate the work done on the fluid by the piston. Use the tables in
5 kg of refrigerant 134a at a pressure of 7.5 bar and 0.7 quality is to be cooled inside a sealed rigid container until the pressure reaches 2.1 bar. Referring to the tables in Appendix A.4,
In order to illustrate the nature of the critical point, one can place CO2 in a quartz vial and seal the device. Assuming that vc = 0.002 m3/kg, and that the device is filled at room temperature
Determine the energy transfer as heat (J) required to change the state of 2 kg of methane from a saturated liquid at 1 atm to a gas at 300 K and 1 atm, assuming that the methane is confined at
A 2 m3?container for heating water is initially filled with H2O liquid and vapor at 400 K, with 20% of the volume being liquid. Use the aatables in Appendix A.3 to work this exercise. Check your
Carbon dioxide at P1?and T1 is contained inside a piston?cylinder. It undergoes polytropic compression according to the process equation PVn = constant.?a. Derive the formula for the energy transfer
Two adjacent tanks are connected by a valve. The tank on the left contains 0.5 kg of water at a pressure of 11 bar and a temperature of 600 K. The tank on the right contains 0.25 kg of saturated
A heat pump consists of the hardware shown, with refrigerant 134a as the working fluid. The valve and compressor are well insulated and may be treated as adiabatic. Measurements during operation are
A steady flow of air at the rate M? will enter a steady-state device at T1, P1 and emerge at T2, P2. The device requires shaft power W? and is cooled by heat transfer at the rate Q?0 to the
Liquid methane is to be converted into gas for insertion into the natural gas pipeline by a steady-state gasification plant. The methane will enter the plant as a saturated liquid at 1 atmosphere
The inlet of a supersonic aircraft engine is a supersonic diffuser. Air enters at high speed ?1?at state 1 (T1, P1), and exits at high pressure P2 at much lower speed 2 . The diffuser is adiabatic
An alkali metal, potassium, is the working fluid in a nuclear power system that was proposed in the 1960s for space application, either on board of a spacecraft or in power stations on the Moon or
An adiabatic device is to be designed that will take in two steady flows of gas at given states 1 and 2, with T2 > T1, and discharge the gas at a given pressure P3 and undetermined temperature T3.
Thermal power systems for use in space normally reject energy as heat by radiating it into space. Since fluid-filled radiators are generally quite heavy, it has been suggested that this energy be
Referring to Appendix A.4, compute the minimum amount of power required by an adiabatic compressor that handles 5 kg/min of R134a, compressing it from a temperature and pressure of 320 K and 0.6 bar
The average amount of solar energy received at the earth’s surface in a certain location is approximately 1350 W/m2. This energy can be used in a solar power plant, because of the reradiation of
Steam enters an adiabatic diffuser at 800 m/s as a saturated vapor at 0.4 MPa. What is the maximum possible discharge pressure? Use Appendix A.3 for data.Data From A.3 SIV / T/ piat / v/ $ 1 UV/
Steam enters the nozzle of a turbine with a velocity of 3 m/s at a pressure of 35 bar and a temperature of 800 K. At the exit of the nozzle the pressure and temperature are found to be 1 bar and 400
A control mass consists of a mixture of fluids in a piston–cylinder system. The weight of the piston maintains a constant pressure P on the fluid mixture, which is also maintained at a constant
Compute the maximum quality of oxygen that can be achieved by expanding it adiabatically in a piston?cylinder system from the saturated vapor state at 10 bar to twice the volume. Use Appendix A.8 for
Calculate the thermodynamic pressure of saturated water vapor at 100?C and compare the result with what can be obtained from tabulated values of vapor?liquid equilibrium pressures in Appendix
Consider a control mass undergoing a cyclic process, thus returning to its initial state after a time interval. Prove that, if there is a net positive input of energy as heat dQ into the control
An inventor claims to have built a device that receives 1 kW of energy as heat and converts it into 750 W of electric power. The rest of the energy is put out as mechanical work and dissipated
It has been proposed that energy be taken as heat from the atmosphere around Madagascar and used to run a power plant. Energy would be rejected as heat to the Indian Ocean. Estimate the maximum
In cases where viscous stresses and thermal conduction can be neglected, the Euler equations describe the motion of a fluid. The one dimensional form of the Euler equations (mass, momentum, and
Assume that for an ideal gas, within a temperature interval, the dependence of the isobaric specific heat on temperature can be approximated by a quadratic equation cP - (T) = K1 + K2 T + K3T2,
Calculate the entropy difference for the same substance and conditions of Exercise 6.2(b), and compare it with the values you can obtain with GASMIX. Why is therea small difference?Data From Exercise
Starting from the Gibbs equation, show that for a polytropic (constant specific heat) gas undergoing an isentropic process where ? ? cP/c?. T2 T₁ P1 P₂(x-1)/y =
Consider steam flowing through an adiabatic valve for the two different process conditions:(a) Inlet at Pin = 70 bar and Tin = 730 K, and outlet at Pout = 20 bar;(b) Inlet at Pin = 4 bar and
When designing the air-conditioning system of an airplane, the work required to is entropically compress air must be calculated. Consider the process at steady state, the inlet compressor pressure is
The vapor pressure of a simple compressible fluid can be roughly approximated as an exponential function of temperature. Show how such approximation is suggested by the Clapeyron equation.
Consider an idealized power system operating according to the Carnot cycle. The working fluid can be modeled as a polytropic ideal gas.(a) Draw qualitatively the cycle in the P?v diagram of the
A piston–cylinder device contains 1 kg of a gaseous refrigerant. The gas undergoes a compression from P1 = 1.25 bar and ν1 = 42.77 m3/kg to P2 = 10 bar and ν1 = 10.67 m3/kg. The compression
Determine the change in specific entropy in kJ/(kg ? K) between the states:?(a) Air, P1 = 0.1 MPa, T1 = 20?C, P2?= 0.1 MPa, T2?= 120?C;(b) Air, P1 = 1 bar, T1 = 27 ?C, P2?= 3 bar, T2?= 377?C;(c)
Determine the reference state for the entropy and enthalpy of R134a chosen for the model implemented in STANMIX by inspecting the thermodynamic T ?? s or P ? h diagram in Appendix A. Determine the
Obtain the Maxwell relations (6.38c) and (6.38d) starting from the Gibbs equation.Data From ?(6.38c) and (6.38d)These are known as the Maxwell relations and are valid for simple compressible
Given the truncated virial equation forcalculate the pressure for T = 180 K and v = 0.03 m3/kg and calculate the specific volume for P = 0.3 MPa and T = 320 K. Compare the values with the values

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