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A refrigerator or heat pump is basically a heat engine run in reverse direction. It essentially consists of three parts (1) Source : At higher temperature \[{{T}_{1}}\]. (2) Working substance : It is called refrigerant liquid ammonia and freon works as a working substance. (3) Sink : At lower temperature \[{{T}_{2}}\]. The working substance takes heat \[{{Q}_{2}}\] from a sink (contents of refrigerator) at lower temperature, has a net amount of work done W on it by an external agent (usually compressor of refrigerator) and gives out a larger amount of heat \[{{Q}_{1}}\] to a hot body at temperature \[{{T}_{1}}\] (usually atmosphere). Thus, it transfers heat from a cold to a hot body at the expense of mechanical energy supplied to it by an external agent. The cold body is thus cooled more and more. The performance of a refrigerator is expressed by means of "coefficient of performance" \[\beta \] which is defined as the ratio of the heat extracted from the cold body to the work needed to transfer it to the hot body. i.e.  \[\beta =\frac{\text{Heat extracted}}{\text{Work done}}=\frac{{{Q}_{2}}}{W}=\frac{{{Q}_{2}}}{{{Q}_{1}}-{{Q}_{2}}}\] A perfect refrigerator is one which transfers heat from cold to hot body without doing work i.e. \[W=0\] so that \[{{Q}_{1}}={{Q}_{2}}\] and hence \[\beta =\infty \] (1) Carnot refrigerator : For Carnot refrigerator \[\frac{{{Q}_{1}}}{{{Q}_{2}}}=\frac{{{T}_{1}}}{{{T}_{2}}}\] \[\Rightarrow \] \[\frac{{{Q}_{1}}-{{Q}_{2}}}{{{Q}_{2}}}=\frac{{{T}_{1}}-{{T}_{2}}}{{{T}_{2}}}\] or \[\frac{{{Q}_{2}}}{{{Q}_{1}}-{{Q}_{2}}}=\frac{{{T}_{2}}}{{{T}_{1}}-{{T}_{2}}}\] So coefficient of performance \[\beta =\frac{{{T}_{2}}}{{{T}_{1}}-{{T}_{2}}}\] where \[{{T}_{1}}=\] temperature of surrounding, \[{{T}_{2}}=\] temperature of cold body. It is clear that \[\beta =0\] when \[{{T}_{2}}=0\] i.e. the coefficient of performance will be zero if the cold body is at the temperature equal to absolute zero. (2) Relation between coefficient of performance and efficiency of refrigerator  We know \[\beta =\frac{{{Q}_{2}}}{{{Q}_{1}}-{{Q}_{2}}}\]\[=\frac{{{Q}_{2}}/{{Q}_{1}}}{1-{{Q}_{2}}/{{Q}_{1}}}\]                                 ... (i) But the efficiency \[\eta =1-\frac{{{Q}_{2}}}{{{Q}_{1}}}\] or \[\frac{{{Q}_{2}}}{{{Q}_{1}}}=1-\eta \]            ...(ii) From (i) and (ii) we get, \[\beta =\frac{1-\eta }{\eta }\]

Heat engine is a device which converts heat into work continuously through a cyclic process. The essential parts of a heat engine are (1) Source : It is a reservoir of heat at high temperature and infinite thermal capacity. Any amount of heat can be extracted from it. (2) Working substance : Steam, petrol etc. (3) Sink : It is a reservoir of heat at low temperature and infinite thermal capacity. Any amount of heat can be given to the sink. The working substance absorbs heat \[{{Q}_{1}}\] from the source, does an amount of work W, returns the remaining amount of heat to the sink and comes back to its original state and there occurs no change in its internal energy. By repeating the same cycle over and over again, work is continuously obtained. The performance of heat engine is expressed by means of ?efficiency? \[\eta \] which is defined as the ratio of useful work obtained from the engine to the heat supplied to it. \[\eta =\frac{\text{Work done}}{\text{Heat input}}=\frac{W}{{{Q}_{1}}}\] For cyclic process \[\Delta U=0\] hence from FLOT  \[\Delta Q=\Delta W\] So  \[W={{Q}_{1}}-{{Q}_{2}}\] \[\Rightarrow \] \[\eta =\frac{{{Q}_{1}}-{{Q}_{2}}}{{{Q}_{1}}}=1-\frac{{{Q}_{2}}}{{{Q}_{1}}}\] A perfect heat engine is one which converts all heat into work i.e. \[W={{Q}_{1}}\] so that \[{{Q}_{2}}=0\] and hence\[\eta =1\]. But practically efficiency of an engine is always less than 1.

(1) PV-graphs
	1 \[\xrightarrow{\,}\] Isobaric (P-constant) 2\[\xrightarrow{\,}\] Isothermal (Because\[P\propto \frac{1}{V}\]) 3 \[\xrightarrow{\,}\]Adiabatic (Because \[P\propto \frac{1}{{{V}^{\gamma }}}\]) 4 \[\xrightarrow{\,}\] Isochoric (V-constant)
(2) PT-graphs
	1 \[\xrightarrow{\,}\] Isobaric (P-constant) 2 \[\xrightarrow{\,}\] Isothermal (T-constant) 3 \[\xrightarrow{\,}\] Adiabatic (Because\[P\propto {{T}^{\frac{\gamma }{\gamma -1}}}\]) 4 \[\xrightarrow{\,}\] Isochoric (In isochoric \[P\propto T\])
(3) VT-graphs
	1 \[\xrightarrow{\,}\] Isochoric (V-constant) 2 \[\xrightarrow{\,}\] Adiabatic (Because\[V\propto {{T}^{\frac{1}{1-\gamma }}}\]) 3 \[\xrightarrow{\,}\] Isothermal (T-constant) 4 \[\xrightarrow{\,}\] Isobaric (In isobaric \[V\propto T\])

                                                                                                                                                                                                                                                                 

(1) Reversible process : A reversible process is one which can be reversed in such a way that all changes occurring in the direct process are exactly repeated in the opposite order and inverse sense and no change is left in any of the bodies taking part in the process or in the surroundings. For example if heat is absorbed in the direct process, the same amount of heat should be given out in the reverse process, if work is done on the working substance in the direct process then the same amount of work should be done by the working substance in the reverse process. The conditions for reversibility are (i) There must be complete absence of dissipative forces such as friction, viscosity, electric resistance etc. (ii) The direct and reverse processes must take place infinitely slowly. (iii) The temperature of the system must not differ appreciably from its surroundings. Some examples of reversible process are (a) All isothermal and adiabatic changes are reversible if they are performed very slowly. (b) When a certain amount of heat is absorbed by ice, it melts. If the same amount of heat is removed from it, the water formed in the direct process will be converted into ice. (c) An extremely slow extension or contraction of a spring without setting up oscillations. (d) When a perfectly elastic ball falls from some height on a perfectly elastic horizontal plane, the ball rises to the initial height. (e) If the resistance of a thermocouple is negligible there will be no heat produced due to Joules heating effect. In such a case heating or cooling is reversible. At a junction where a cooling effect is produced due to Peltier effect when current flows in one direction and equal heating effect is produced when the current is reversed. (f) Very slow evaporation or condensation. It should be remembered that the conditions mentioned for a reversible process can never be realised in practice. Hence, a reversible process is only an ideal concept. In actual process, there is always loss of heat due to friction, conduction, radiation etc. (2) Irreversible process : Any process which is not reversible exactly is an irreversible process. All natural processes such as conduction, radiation, radioactive decay etc. are irreversible. All practical processes such as free expansion, Joule-Thomson expansion, electrical heating of a wire are also irreversible. Some examples of irreversible processes are given below (i) When a steel ball is allowed to fall on an inelastic lead sheet, its kinetic energy changes into heat energy by friction. The heat energy raises the temperature of lead sheet. No reverse transformation of heat energy occurs. (ii) The sudden and fast stretching of a spring may produce vibrations in it. Now a part of the energy is dissipated. This is the case of irreversible process. (iii) Sudden expansion or contraction and rapid evaporation or condensation are examples of irreversible processes. (iv) Produced by the passage of an electric current through a resistance is irreversible. (v) more...

When we perform a process on a given system, its state is, in general, changed. Suppose the initial state of the system is described by the values \[{{P}_{1}},\,{{V}_{1}},\,{{T}_{1}}\] and the final state by \[{{P}_{2}},\,{{V}_{2}},\,{{T}_{2}}\]. If the process is performed in such a way that at any instant during the process, the system is very nearly in thermodynamic equilibrium, the process is called quasi-static. This means, we can specify the parameters P, V, T uniquely at any instant during such a process. Actual processes are not quasi-static. To change the pressure of a gas, we can move a piston inside the enclosure. The gas near the piston is acted upon by piston. The pressure of the gas may not be uniform everywhere while the piston is moving. However, we can move the piston very slowly to make the process as close to quasi-static as we wish. Thus, a quasi-static process is an idealised process in which all changes take place infinitely slowly.

A cyclic process consists of a series of changes which return the system back to its initial state. In non-cyclic process the series of changes involved do not return the system back to its initial state. (1) In case of cyclic process as \[{{U}_{f}}={{U}_{i}}\]\[\Rightarrow \] \[\Delta U={{U}_{f}}-{{U}_{i}}=0\]  i.e. change in internal energy for cyclic process is zero and also \[\Delta U\propto \Delta T\]\[\Rightarrow \]\[\Delta T=0\] i.e. temperature of system remains constant. (2) From FLOT  \[\Delta Q=\Delta U+\Delta W\]\[\Rightarrow \]\[\Delta Q=\Delta W\] i.e. heat supplied is equal to the work done by the system. (3) For cyclic process P-V graph is a closed curve and area enclosed by the closed path represents the work done. If the cycle is clockwise work done is positive and if the cycle is anticlockwise work done is negative. (4) Work done in non cyclic process depends upon the path chosen or the series of changes involved and can be calculated by the area covered between the curve and volume axis on PV diagram.                

When a thermodynamic system undergoes a change in such a way that no exchange of heat takes place between System and surroundings, the process is known as adiabatic process. In this process P, V and T changes but \[\Delta Q=0\]. (1) Essential conditions for adiabatic process (i) There should not be any exchange of heat between the system and its surroundings. All walls of the container and the piston must be perfectly insulating. (ii) The system should be compressed or allowed to expand suddenly so that there is no time for the exchange of heat between the system and its surroundings. Since, these two conditions are not fully realised in practice, so no process is perfectly adiabatic. (2) Some examples of adiabatic process (i) Sudden compression or expansion of a gas in a container with perfectly non-conducting walls. (ii) Sudden bursting of the tube of bicycle tyre. (iii) Propagation of sound waves in air and other gases. (iv) Expansion of steam in the cylinder of steam engine. (3) FLOT in adiabatic process : From \[\Delta Q=\Delta U+\Delta W\] For adiabatic process \[\Delta Q=0\] \[\Rightarrow \] \[\Delta U=-\,\Delta W\] If \[\Delta W=\] positive then \[\Delta U=\] negative so temperature decreases i.e. adiabatic expansion produce cooling. If \[\Delta W=\] negative then \[\Delta U=\] positive so temperature increases i.e. adiabatic compression produce heating. (4) Equation of state : In adiabatic change ideal gases do not obeys Boyle's law but obeys Poisson's law. According to it \[P{{V}^{\gamma }}=\] constant; where \[\gamma =\frac{{{C}_{P}}}{{{C}_{V}}}\] (i) For temperature and volume  \[T{{V}^{\gamma -1}}=\]constant \[\Rightarrow \] \[{{T}_{1}}{{V}_{1}}^{\gamma -1}={{T}_{2}}{{V}_{2}}^{\gamma -1}\] or \[T\propto {{V}^{1-\gamma }}\] (ii) For temperature and pressure  \[\frac{{{T}^{\gamma }}}{{{P}^{\gamma -1}}}\]= const. \[\Rightarrow \]\[{{T}_{1}}^{\gamma }{{P}_{1}}^{1-\gamma }={{T}_{2}}^{\gamma }{{P}_{2}}^{1-\gamma }\]or\[T\propto {{P}^{\frac{\gamma -1}{\gamma }}}\]or \[P\propto {{T}^{\frac{\gamma }{\gamma -1}}}\]   Special cases of adiabatic process

Type of gas	\[P\propto \frac{1}{{{V}^{\gamma }}}\]	\[P\propto {{T}^{\frac{\gamma }{\gamma -1}}}\]	\[T\propto \frac{1}{{{V}^{\gamma -1}}}\]
Monoatomic \[\gamma =5/3\]	\[P\propto \frac{1}{{{V}^{5/3}}}\]	\[P\propto {{T}^{5/2}}\]	\[T\propto \frac{1}{{{V}^{2/3}}}\]
Diatomic \[\gamma =7/5\]	\[P\propto \frac{1}{{{V}^{7/5}}}\]	\[P\propto {{T}^{7/2}}\]	\[T\propto \frac{1}{{{V}^{2/5}}}\]
Polyatomic \[\gamma =4/3\]	\[P\propto \frac{1}{{{V}^{4/3}}}\]	\[P\propto {{T}^{4}}\]	\[T\propto \frac{1}{{{V}^{1/3}}}\]

(5) Indicator diagram (i) Curve obtained on PV graph are called adiabatic curve. (ii) Slope of adiabatic curve : From \[P{{V}^{\gamma }}=\text{constant}\] By differentiating, we get          \[dP\,{{V}^{\gamma }}+P\gamma {{V}^{\gamma -1}}\,dV=0\] \[\frac{dP}{dV}=-\gamma \frac{P{{V}^{\gamma -1}}}{{{V}^{\gamma }}}=-\gamma \left( \frac{P}{V} \right)\] \[\therefore \] Slope of adiabatic curve\[\tan \varphi =-\gamma \left( \frac{P}{V} \right)\] (iii) But we also know that slope of isothermal curve \[\tan \theta =\frac{-P}{V}\] Hence \[{{\text{(Slope)}}_{\text{Adi}}}=\gamma \times {{\text{(Slope)}}_{\text{Iso}}}\] or \[\frac{{{\text{(Slope)}}_{\text{Adi}}}}{{{\text{(Slope)}}_{\text{Iso}}}}>1\] (6) Specific heat : Specific heat of a gas during adiabatic change is zero As \[C=\frac{Q}{m\Delta T}=\frac{0}{m\Delta T}=0\] [As Q = 0] (7) Adiabatic elasticity \[({{E}_{\phi }}):\] \[P{{V}^{\gamma }}=\text{constant}\] Differentiating both sides \[d\,P{{V}^{\gamma }}+P\gamma {{V}^{\gamma -1}}dV=0\] \[\gamma P=\frac{dP}{-dV/V}=\frac{\text{Stress}}{\text{Strain}}={{E}_{\varphi }}\]\[\Rightarrow \]\[{{E}_{\varphi }}=\gamma P\] i.e. adiabatic elasticity is g times that more...

When a thermodynamic system undergoes a physical change in such a way that its temperature remains constant, then the change is known as isothermal changes. (1) Essential condition for isothermal process (i) The walls of the container must be perfectly conducting to allow free exchange of heat between the gas and its surrounding. (ii) The process of compression or expansion should be so slow so as to provide time for the exchange of heat. Since these two conditions are not fully realised in practice, therefore, no process is perfectly isothermal. (2) Equation of state : In this process, P and V change but T = constant i.e. change in temperature \[\Delta T=0\]. Boyleís law is obeyed i.e. PV= constant \[\Rightarrow \,\,{{P}_{1}}{{V}_{1}}={{P}_{2}}{{P}_{2}}\] (3) Example of isothermal process : Melting of ice (at \[{{0}^{o}}C\]) and boiling of water (at \[{{100}^{o}}C\]) are common example of this process. (4) Indicator diagram : According to PV = constant, graph between P and V is a part of rectangular hyperbola. The graphs at different temperature are parallel to each other are called isotherms. \[{{T}_{1}}

When a thermodynamic process undergoes a physical change in such a way that its volume remains constant, then the change is known as isochoric process. (1) Equation of state : In this process P and T changes but V = constant. Hence Gay-Lussac?s law is obeyed in this process i.e.  \[P\propto T\Rightarrow \frac{{{P}_{1}}}{{{T}_{1}}}=\frac{{{P}_{2}}}{{{T}_{2}}}=\] constant (2) Indicator diagram : Graph 1 and 2 represent isometric increase in pressure at volume \[{{V}_{1}}\] and isometric decrease in pressure at volume \[{{V}_{2}}\] respectively and slope of indicator diagram\[\frac{dP}{dV}=\infty \] (i) Isometric heating (a) Pressure \[\xrightarrow{\,}\] increases (b) Temperature \[\xrightarrow{\,}\] increases (c) \[\Delta Q\] \[\xrightarrow{\,}\]positive (d) \[\Delta U\] \[\xrightarrow{\,}\]positive (ii) Isometric cooling  (a) Pressure \[\xrightarrow{\,}\] decreases (b) Temperature \[\xrightarrow{\,}\] decreases  (c) \[\Delta Q\] \[\xrightarrow{\,}\]negative (d) \[\Delta U\]\[\xrightarrow{\,}\]negative   (3) Specific heat : Specific heat of gas during isochoric process \[{{C}_{V}}=\frac{f}{2}R\] (4) Bulk modulus of elasticity : \[K=\frac{\Delta P}{\frac{-\Delta V}{V}}=\frac{\Delta P}{0}=\infty \] (5) Work done in isochoric process \[\Delta W=P\Delta V=P[{{V}_{f}}-{{V}_{i}}]=0\]       [As V = constant] (6) FLOT in isochoric process : From \[\Delta Q=\Delta U+\Delta W\] \[\because \] \[\Delta W=0\Rightarrow {{(\Delta Q)}_{V}}=\Delta U=\mu {{C}_{V}}\,\Delta T=\mu \frac{R}{\gamma -1}\Delta T=\frac{{{P}_{f}}{{V}_{f}}-{{P}_{i}}{{V}_{i}}}{\gamma -1}\]

When a thermodynamic system undergoes a physical change in such a way that its pressure remains constant, then the change is known as isobaric process. (1) Equation of state : In this process V and T changes but P remains constant. Hence Charle?s law is obeyed in this process. Hence if pressure remains constant  \[V\propto T\Rightarrow \frac{{{V}_{1}}}{{{T}_{1}}}=\frac{{{V}_{2}}}{{{T}_{2}}}\] (2) Indicator diagram : Graph 1 represent isobaric expansion, graph 2 represent isobaric compression. Slope =\[\frac{dP}{dV}=0\]            Slope = \[\frac{dP}{dV}=0\] (i) In isobaric expansion (Heating)  Temperature \[\xrightarrow{\,\,\,\,}\] increases so \[\Delta U\] is positive Volume \[\xrightarrow{\,\,\,\,}\] increases so \[\Delta W\] is positive Heat \[\xrightarrow{\,\,\,\,}\] flows into the system so \[\Delta Q\] is positive (ii) In isobaric compression (Cooling) Temperature \[\xrightarrow{\,\,\,\,}\] decreases so \[\Delta U\] is negative Volume \[\xrightarrow{\,\,\,\,}\] decreases so \[\Delta W\] is negative Heat \[\xrightarrow{\,\,\,\,}\] flows out from the system so \[\Delta Q\] is negative (3) Specific heat : Specific heat of gas during isobaric process \[{{C}_{P}}=\left( \frac{f}{2}+1 \right)R\] (4) Bulk modulus of elasticity : \[K=\frac{\Delta P}{\frac{-\Delta V}{V}}=0\] [As \[\Delta P=0\]] (5) Work done in isobaric process \[\Delta W=\int_{{{V}_{i}}}^{{{V}_{f}}}{P\,dV}=P\int_{{{V}_{i}}}^{{{V}_{f}}}{dV}=P[{{V}_{f}}-{{V}_{i}}]\]                [As P = constant] \[\Rightarrow \] \[\Delta W=P({{V}_{f}}-{{V}_{i}})=\mu R[{{T}_{f}}-{{T}_{i}}]=\mu R\,\Delta T\] (6) FLOT in isobaric process : From  \[\Delta Q=\Delta U+\Delta W\] \[\because \] \[\Delta U=\mu \,{{C}_{V}}\,\Delta T\]\[=\mu \frac{R}{(\gamma -1)}\Delta T\] and  \[\Delta W=\mu R\,\Delta T\] \[\Rightarrow \] \[{{(\Delta Q)}_{P}}=\mu \frac{R}{(\gamma -1)}\Delta T+\mu R\,\Delta T\]\[=\mu \left( \frac{\gamma }{\gamma -1} \right)R\,\Delta T\]\[=\mu \,{{C}_{P}}\,\Delta T\] (7) Examples of isobaric process : All state changes occurs at constant temperature and pressure. Boiling of water (i) Water \[\xrightarrow{\,}\] vapours (ii) Temperature \[\xrightarrow{{}}\] constant (iii) Volume \[\xrightarrow{\,}\]increases (iv) A part of heat supplied is used to change volume (expansion) against external pressure and remaining part is used to increase it's potential energy (kinetic energy remains constant) (v) From FLOT \[\Delta Q=\Delta U+\Delta W\Rightarrow mL=\Delta U+P({{V}_{f}}-{{V}_{i}})\] Freezing of water (i) Water \[\xrightarrow{\,}\] ice (ii) Temperature \[\xrightarrow{\,}\] constant (iii) Volume \[\xrightarrow{\,}\]increases (iv) Heat is given by water it self. It is used to do work against external atmospheric pressure and to decreases the internal potential energy. (v) From FLOT \[\Delta Q=\Delta U+\Delta W\Rightarrow -mL=\Delta U+P({{V}_{f}}{{V}_{i}})\]