Isothermal process - wikipedia Isothermal process `` Isothermal '' redirects here. For other uses, see Isotherm. Thermodynamics The classical Carnot heat engine Branches Classical Statistical Chemical Quantum thermodynamics Equilibrium / Non-equilibrium Laws Zeroth First Second Third Systems State Equation of state Ideal gas Real gas State of matter Equilibrium Control volume Instruments Processes Isobaric Isochoric Isothermal Adiabatic Isentropic Isenthalpic Quasistatic Polytropic Free expansion Reversibility Irreversibility Endoreversibility Cycles Heat engines Heat pumps Thermal efficiency System properties Note : Conjugate variables in italics Property diagrams Intensive and extensive properties Process functions Work Heat Functions of state Temperature / Entropy ( introduction ) Pressure / Volume Chemical potential / Particle number Vapor quality Reduced properties Material properties Property databases Specific heat capacity c = ( displaystyle c = ) T ( displaystyle T ) ∂ S ( displaystyle partial S ) N ( displaystyle N ) ∂ T ( displaystyle partial T ) Compressibility β = − ( displaystyle \beta = - ) 1 ( displaystyle 1 ) ∂ V ( displaystyle partial V ) V ( displaystyle V ) ∂ p ( displaystyle partial p ) Thermal expansion α = ( displaystyle alpha = ) 1 ( displaystyle 1 ) ∂ V ( displaystyle partial V ) V ( displaystyle V ) ∂ T ( displaystyle partial T ) Equations Carnot 's theorem Clausius theorem Fundamental relation Ideal gas law Maxwell relations Onsager reciprocal relations Bridgman 's equations Table of thermodynamic equations Potentials Free energy Free entropy Internal energy U ( S, V ) ( displaystyle U ( S, V ) ) Enthalpy H ( S, p ) = U + p V ( displaystyle H ( S, p ) = U + pV ) Helmholtz free energy A ( T, V ) = U − T S ( displaystyle A ( T, V ) = U-TS ) Gibbs free energy G ( T, p ) = H − T S ( displaystyle G ( T, p ) = H - TS ) History Culture History General Heat Entropy Gas laws `` Perpetual motion '' machines Philosophy Entropy and time Entropy and life Brownian ratchet Maxwell 's demon Heat death paradox Loschmidt 's paradox Synergetics Theories Caloric theory Theory of heat Vis viva ( `` living force '' ) Mechanical equivalent of heat Motive power Key publications `` An Experimental Enquiry Concerning... Heat '' `` On the Equilibrium of Heterogeneous Substances '' `` Reflections on the Motive Power of Fire '' Timelines Thermodynamics Heat engines Art Education Maxwell 's thermodynamic surface Entropy as energy dispersal Scientists Bernoulli Boltzmann Carnot Clapeyron Clausius Carathéodory Duhem Gibbs von Helmholtz Joule Maxwell von Mayer Onsager Rankine Smeaton Stahl Thompson Thomson van der Waals Waterston Book : Thermodynamics An isothermal process is a change of a system, in which the temperature remains constant : ΔT = 0. This typically occurs when a system is in contact with an outside thermal reservoir ( heat bath ), and the change will occur slowly enough to allow the system to continue to adjust to the temperature of the reservoir through heat exchange. In contrast, an adiabatic process is where a system exchanges no heat with its surroundings ( Q = 0 ). In other words, in an isothermal process, the value ΔT = 0 and therefore ΔU = 0 ( only for an ideal gas ) but Q ≠ 0, while in an adiabatic process, ΔT ≠ 0 but Q = 0. Contents 1 Examples 2 Details for an ideal gas 3 Calculation of work 4 Entropy changes 5 Etymology 6 See also 7 References Examples Isothermal processes can occur in any kind of system that has some means of regulating the temperature, including highly structured machines, and even living cells. Some parts of the cycles of some heat engines are carried out isothermally ( for example, in the Carnot cycle ). In the thermodynamic analysis of chemical reactions, it is usual to first analyze what happens under isothermal conditions and then consider the effect of temperature. Phase changes, such as melting or evaporation, are also isothermal processes when, as is usually the case, they occur at constant pressure. Isothermal processes are often used and a starting point in analyzing more complex, non-isothermal processes. Isothermal processes are of special interest for ideal gases. This is a consequence of Joule 's second law which states that the internal energy of a fixed amount of an ideal gas depends only on its temperature. Thus, in an isothermal process the internal energy of an ideal gas is constant. This is a result of the fact that in an ideal gas there are no intermolecular forces. Note that this is true only for ideal gases ; the internal energy depends on pressure as well as on temperature for liquids, solids, and real gases. In the isothermal compression of a gas there is work done on the system to decrease the volume and increase the pressure. Doing work on the gas increases the internal energy and will tend to increase the temperature. To maintain the constant temperature energy must leave the system as heat and enter the environment. If the gas is ideal, the amount of energy entering the environment is equal to the work done on the gas, because internal energy does not change. For details of the calculations, see calculation of work. For an adiabatic process, in which no heat flows into or out of the gas because its container is well insulated, Q = 0. If there is also no work done, i.e. a free expansion, there is no change in internal energy. For an ideal gas, this means that the process is also isothermal. Thus, specifying that a process is isothermal is not sufficient to specify a unique process. Details for an ideal gas Figure 1. Several isotherms of an ideal gas on a p-V diagram For the special case of a gas to which Boyle 's law applies, the product pV is a constant if the gas is kept at isothermal conditions. The value of the constant is nRT, where n is the number of moles of gas present and R is the ideal gas constant. In other words, the ideal gas law pV = nRT applies. This means that p = n R T V = constant V ( displaystyle p = ( nRT over V ) = ( ( \text ( constant ) ) over V ) ) holds. The family of curves generated by this equation is shown in the graph in Figure 1. Each curve is called an isotherm. Such graphs are termed indicator diagrams and were first used by James Watt and others to monitor the efficiency of engines. The temperature corresponding to each curve in the figure increases from the lower left to the upper right. Calculation of work Figure 2. The purple area represents the work for this isothermal change In thermodynamics, the reversible work involved when a gas changes from state A to state B is W A → B = − ∫ V A V B p d V ( displaystyle W_ ( A \to B ) = - int _ ( V_ ( A ) ) ^ ( V_ ( B ) ) p , dV ) For an isothermal, reversible process, this integral equals the area under the relevant pressure - volume isotherm, and is indicated in purple in Figure 2 for an ideal gas. Again, p = nRT / V applies and with T being constant ( as this is an isothermal process ), the expression for work becomes : W A → B = − ∫ V A V B p d V = − ∫ V A V B n R T V d V = − n R T ∫ V A V B 1 V d V = − n R T ln ⁡ V B V A ( displaystyle W_ ( A \to B ) = - int _ ( V_ ( A ) ) ^ ( V_ ( B ) ) p , dV = - int _ ( V_ ( A ) ) ^ ( V_ ( B ) ) ( \frac ( nRT ) ( V ) ) dV = - nRT int _ ( V_ ( A ) ) ^ ( V_ ( B ) ) ( \frac ( 1 ) ( V ) ) dV = - nRT ln ( \frac ( V_ ( B ) ) ( V_ ( A ) ) ) ) By convention, work is defined as the work on the system by its surroundings. If, for example, the system is compressed, then the work is positive and the internal energy of the system increases. Conversely, if the system expands, it does work on the surroundings and the internal energy of the system decreases. It is also worth noting that for ideal gases, if the temperature is held constant, the internal energy of the system also is constant, and so ΔU = 0. Since the First Law of Thermodynamics states that ΔU = Q + W ( IUPAC convention ), it follows that Q = − W for the isothermal compression or expansion of ideal gases. Entropy changes Isothermal processes are especially convenient for calculating changes in entropy since, in this case, the formula for the entropy change, ΔS, is simply Δ S = Q rev T ( displaystyle Delta S = ( \frac ( Q_ ( \text ( rev ) ) ) ( T ) ) ) where Q is the heat transferred reversibly to the system and T is absolute temperature. This formula is valid only for a hypothetical reversible process ; that is, a process in which equilibrium is maintained at all times. A simple example is an equilibrium phase transition ( such as melting or evaporation ) taking place at constant temperature and pressure. For a phase transition at constant pressure, the heat transferred to the system is equal to the enthalpy of transformation, ΔH, thus Q = ΔH. At any given pressure, there will be a transition temperature, T, for which the two phases are in equilibrium ( for example, the normal boiling point for vaporization of a liquid at one atmosphere pressure ). If the transition takes place under such equilibrium conditions, the formula above may be used to directly calculate the entropy change Δ S tr = Δ H tr T tr ( displaystyle Delta S_ ( \text ( tr ) ) = ( \frac ( Delta H_ ( \text ( tr ) ) ) ( T_ ( \text ( tr ) ) ) ) ). Another example is the reversible isothermal expansion ( or compression ) of an ideal gas from an initial volume V and pressure P to a final volume V and pressure P. As shown in Calculation of work, the heat transferred to the gas is Q = − W = n R T ln ⁡ V B V A ( displaystyle Q = - W = nRT ln ( \frac ( V_ ( \text ( B ) ) ) ( V_ ( \text ( A ) ) ) ) ). This result is for a reversible process, so it may be substituted in the formula for the entropy change to obtain Δ S = n R ln ⁡ V B V A ( displaystyle Delta S = nR ln ( \frac ( V_ ( \text ( B ) ) ) ( V_ ( \text ( A ) ) ) ) ). Since an ideal gas obey 's Boyle 's Law, this can be rewritten, if desired, as Δ S = n R ln ⁡ P A P B ( displaystyle Delta S = nR ln ( \frac ( P_ ( \text ( A ) ) ) ( P_ ( \text ( B ) ) ) ) ). Once obtained, these formulas can be applied to an irreversible process, such as the free expansion of an ideal gas. Such an expansion is also isothermal and may have the same initial and final states as in the reversible expansion. Since entropy is a state function, the change in entropy of the system is the same as in the reversible process and is given by the formulas above. Note that the result Q = 0 for the free expansion can not be used in the formula for the entropy change since the process is not reversible. The difference between the reversible and free expansions is found in the entropy of the surroundings. In both cases, the surroundings are at a constant temperature, T, so that ΔS = − Q / T ; the minus sign is used since the heat transferred to the surroundings is equal in magnitude and opposite in sign to the heat, Q, transferred to the system. In the reversible case, the change in entropy of the surroundings is equal and opposite to the change in the system, so the change in entropy of the universe is zero. In the free expansion, Q = 0, so the entropy of the surroundings does not change and the change in entropy of the universe is equal to ΔS for the system. Etymology The adjective `` isothermal '' is derived from the Greek words ἴσος ( isos ) meaning `` equal '', and θέρμη ( therme ) meaning `` heat. '' See also Joule -- Thomson effect Joule expansion ( also called free expansion ) Adiabatic process Cyclic process Isobaric process Isochoric process Polytropic process References Jump up ^ Keenan, J.H. ( 1970 ). `` Chapter 12 : Heat - engine cycles ''. Thermodynamics. Cambridge, Massachusetts : MIT Press. Jump up ^ Rock, P.A. ( 1983 ). `` Chapter 11 : Thermodynamics of chemical reactions ''. Chemical Thermodynamics. Mill Valley, CA : University Science Books. ISBN 0 - 935702 - 12 - 1. ^ Jump up to : Petrucci, R.H. ; Harwood, W.S. ; Herring, F.G. ; Madura, J.D. ( 2007 ). `` Chapter 12 ''. General Chemistry. Upper Saddle River, NJ : Pearson. ISBN 0 - 13 - 149330 - 2. ^ Jump up to : Klotz, I.M. ; Rosenberg, R.M. ( 1991 ). `` Chapter 6, Application of the first law to gases ''. Chemical Thermodynamics. Meno Park, CA : Benjamin. Jump up ^ Adkins, C.J. ( 1983 ). Equilibrium Thermodynamics. Cambridge : Cambridge University Press. Jump up ^ Atkins, Peter ( 1997 ). `` Chapter 2 : The first law : the concepts ''. Physical Chemistry ( 6th ed. ). New York, NY : W.H. Freeman and Co. ISBN 0 - 7167 - 2871 - 0. ^ Jump up to : Atkins, Peter ( 1997 ). `` Chapter 4 : The second law : the concepts ''. Physical Chemistry ( 6th ed. ). New York, NY : W.H. Freeman and Co. ISBN 0 - 7167 - 2871 - 0. 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Question: the process in which the value of ∆u=0 is
Answer: isothermal process