Heat is defined as the energy that flow into or out of a system because of a difference in temperature between the thermodynamic system and its surrounding. According to IUPAC convention heat given by system is expressed with `-ve` sign heat given to system is expressed with `+ve` sign.

(i) `q_v = nC_v dT` (for constant volume process)

(ii) `q_P = nC_p dT` (for constant pressure process)

(iii) `C_(p, m)- C_(v, m) = R`

(iv) `C_v` & `C_P` depends on temperature even for an ideal gas. (`C = a+ bT + cT^2.....)`

(v) It is a path function.

`C_v`, `C_p` are heat capacity of system and `C_(v, m), C_(p, m)` are heat capacity of one mole system at constant volume and pressure respectively.

Every system having some quantity of matter is associated with a definite amount of energy, called internal energy. Internal energy is stored in different forms inside the molecule.

`U = U_text(Kinetic) + U_text(Potential) + U_text(Electronic) + U_text(nuclear) + .........`

`text(Note)` :
(i) `U` is a state function & is an extensive property.

`DeltaU = U_text(final) - U_text(initial)`

(ii) It does not include kinetic energy of motion of system as a whole or its potential energy due to its position.

(iii) `DeltaE = q_v`, heat supplied to a gas at constant volume, since all the heat supplied goes to increase the internal energy of the gas.

(iv) For a given system, if `U` is a function of `T` and Volume:

`U =f(T, V)`

`dU = ((delU)/(delT))_v dT +((delU)/(delV))_T dV`

(v) For isochoric process : `dV = 0`

`dU = ((delU)/(delT))_v dT`

`dU = C_vdT`

`DeltaU = int C_v dT`

(vi) For an ideal gas, change in internal energy with change in volume at constant temperature is zero. i.e.

`((delU)/(delV))_T = 0`

`dU = C_v dT`

`DeltaU = intC_v dT`

Internal energy is stored in the molecular motion and capacity to store energy depends upon degree of freedom of molecules.

Laws of thermodynamic are deduced from experimental observation with logical reasons. There are four laws :

`text(Zeroth law of thermodynamics)` : It is bases on thermal equation two system in thermal equation with a third system are also in thermal equation with each other.

`text(First law of Thermodynamics)` : Total energy of universe remain constant. It is law of conservation of energy. Let us consider a system whose internal energy is `U_1`. If the system is supplied with heat `q`, the internal energy of the system increases to `U_1 + q`. If work (`w`) is now done on the system, the internal energy in the final state of the system, `U_2` is given by

`U_2 = U_1 + q + w`

or `U_2 - U_1 = q + w`

`Delta U = q + w`

According to lUPAC, heat, added to the system and work done on the system are assigned positive values as both these modes increase the internal energy of the system.

Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make space for it by displacing its environment and establishing its volume and pressure. The enthalpy of a system is defined as:

`H = U + PV`
So, `dH = dU + d(PV)`

where `H` is the enthalpy of the system
`U` is the internal energy of the system

`H = U + PV`

`P` is the pressure at the boundary of the system and its environment, `V` is the volume of the system.

Note that the `U` term is equivalent to the energy required to create the system, and that the `PV` term is equivalent to the energy that would be required to "make space" for the system if the pressure of the envirorunent remained constant.

`text(Property of Enthalpy parameter)`
(i) Enthalpy is a thermodynamic potential. It is a state function and an extensive quantity.

(ii) The total enthalpy, (absolute value) `H`, of a system cannot be measured directly. Thus, change in enthalpy, `DeltaH`, is a more useful quantity than its absolute value.

(iii) The unit of measurement for enthalpy (`SI`) is joule.

(iv) The enthalpy is the preferred expression of system energy changes in many chemical and physical measurements, because it simplifies certain descriptions of energy transfer. This is because a change in enthalpy takes account of energy transferred to the environment through the expansion of the system under study.

(v) The change `DeltaH` is positive in endothermic reactions, and negative in exothermic processes. `DeltaH` of a system is equal to the sum of non-mechanical work done on it and the heat supplied to it.

(vi) For quasi-static processes under constant pressure, `DeltaH` is equal to the change in the internal energy of the system, plus the work that the system has done on its surroundings. This means that the change in enthalpy under such conditions is the heat absorbed (or released) by a chemical reaction.

Chemical reactions are generally carried out at constant pressure (atmospheric pressure) so it has been found useful to define a new state function Enthalpy (`H`) as:

`H = U + PV`

`DeltaH = DeltaU + Delta(PV)`

At constant pressure

`DeltaH = DeltaU +PDeltaV`

combining with first law,

`DeltaH = q_P =` Heat added at constant pressure

`text(Note)` :
(i) Transfer of heat at constant volume brings about a change in the internal energy of the system whereas that at constant pressure brings about a change in the enthalpy of the system.

(ii) For a given system

`H = f(T,P)`

`dH = ((delH)/(delT))_P dT + ((delH)/(delP))_T dP`

(iii) For isobaric process: `dP = 0`

`dH = ((delH)/(delT))_P dT`

`dH = C_p dT`

`DeltaH = intC_p dT`

(iv) For an ideal gas, change in enthalpy at constant temperature with change in pressure is zero. i.e.

`((delH)/(delP))_T = 0 => dH = C_P dT => Delta H = int C_pdT`

(a) `text(Relationship between)` `DeltaH` & `Delta U` : The difference between `DeltaH` & `DeltaU` becomes significant only when gases are involved (insignificant in solids and liquids)

`Delta H = Delta U + Delta (PV)`

If substance is not undergoing chemical reaction or phase change,

`Delta H = Delta U + n R Delta T`

In case of chemical reaction

`Delta H = Delta U +( Delta n_g) RT`

(b) Difference between enthalpy and internal energy : Chemists routinely use `H` as the energy of the system, but the `pV` term is not stored in the system, but rather in the surroundings, such as the atmosphere. When a system, for example, `n` mole of a gas of volume `V` at pressure `P` and temperature `T`, is created or brought to its present state from absolute zero, energy must be supplied equal to its internal energy `U` plus `p V`, where `p V` is the work done in pushing against the ambient (atmospheric) pressure. This additional energy is, therefore, stored in the surroundings and can be recovered when the system collapses back to its initial state. In basic chemistry scientists are typically interested in experiments conducted at atmospheric pressure, and for reaction energy calculations they care about the total energy in such conditions, and therefore typically need to use `H`. In basic physics and thermodynamics, it may be more interesting to study the internal properties of the system and therefore the internal energy is used.

(c) `text(Change in internal energy and enthalpy in phase transition)` : At certain temperature under one atmospheric pressure, one phase change into other phase by taking certain amount of Heat. The temperature at which this happens is called transition temperature and heat absorbed during the process is called Enthalpy of phase transition. Heat absorbed during transition is exchanged at constant pressure and temperature and it is significant to know that the process is reversible.

`text(Fusion)` : Solid ice at `273` `K` and `1` atm pressure reversibly changes into liquid water. Reversibly, isothermally and isobarically, absorbed heat is knows as latent heat of fusion or enthalpy of fusion.

`text(Vaporisation)` : Water at `373` `K` and `1` atm pressure changes into vapors absorbed heat is known as latent heat of vaporisation. The latent heat of vaporisation is heat exchanged isothermally, isobarically and reversibly to convert water into its vapour at boiling point. Internal energy change of phase transitions involving gas phase has no practical significance because it is not possible to carry out `DeltaU` of phase transition directly through an experiment. However `DeltaU` of phase transition can be determined theoretically from experimentally obtained value of `DeltaH` of phase transition.

`H_2O(l) -> H_2O(g)`

`DeltaH_text(vaporisation) = DeltaU_text(vaporisation) + P(V_2 -V_1)`

`DeltaH_text(vaporisation) = DeltaU_text(vaporisation) + {RT//V}{V_g}`

Ignore volume of liquid as it is very less compared to gas under normal pressure.

`=> DeltaH_(vap.) = DeltaU_(vap.) + RT`

where `R` is gas constant and `T` absolute temperature for condensed phase transitions for solid liquid transititons.

`DeltaH_(vap) approx DeltaU_(vap)`

(i) `text(Isothermal process : In isothermal process, work done can be calculated as)`

`w = - int_(V_1)^(V_2) PdV`

Since `dT =0 => dU =0` for an ideal gas

from 1st law `q = -w`

(a) If process is reversible

`w = -n RT ln (V_2/V_1)`

(b) `text(Irreversible isothermal expansion)` : If external pressure over the piston is abruptly changed from the equilibrium value, the mechanical equilibrium of system is disturbed and piston rushes out :

This type of `PV` work is irreversible `PV` work. To calculate irreversible `PV` work, Law of conservation of energy is used. Suppose as a result of difference in pressure a piston moves out and acquire kinetic energy `DeltaKE` and in the process volume increase by `DeltaV` then `W_(irr) = - P_(ext) DeltaV - DeltaKE`

If after sufficient times piston come back to equilibrium state (off course in the process it moves up and down from equilibrium position many times), `DeltaKE = 0` : All the acquired kinetic energy is transferred back to ideal gas. See fig.1.

(c) `text[Irreversible isothermal expansion and compression (Many steps)]` : Consider an irreversible expansion of an ideal gas from initial pressure `P_1` to final pressure `P_1` in four steps. The gas is allowed to expand against constant external pressure of `P_1`, `P_2`, `P_3` and `P_4` and finally `P_1`. Hence the system passes on to final state through four equilibrium states. The work done in the process is shown graphically. The area under the isotherm is the magnitude of reversible work. Clearly the magnitude of reversible work of expansion is greater than irreversible work. As the number of intermediate steps in irreversible expansion is increased, the magnitude of work increases, and as number of steps tend to infinity `w_(irr)` tends to `w_(rev)`. The graphical comparison of irreversible and reversible work is shown in fig.2.

(d) `text(Free expansion of ideal gas)` : When ideal gas is allowed to expand against zero external pressure, the process is called free expansion. `w = 0` for free expansion. During the free expansion, the ideal gas do not lose any energy, and hence temperature of ideal gas remains constant. Hence, free expansion of ideal gas is an example of isothermal, adiabatic irreversible process.

However if a real gas is allowed to expand in vacuum, the gas maybe cooled or heated up depending upon temperature of the real gas. The temperature above which a gas gets heated up upon expansion is called inversion temperature.

`text(Important points)` : If the reversible isothermal expansion is reversed by gradually increasing the pressure the system will return to initial state retracing it's path. This means path of reversible process can be exactly reversed if conditions are reversed. Work done by the system during reversible isothermal expansion is maximum possible work obtainable from system under similar condition.

(ii) `text(Isobaric process)` : In isobaric process, pressure remains constant during the process.

`w= int_(V_2)^(V_1) PdV = -P_text(ext.)(V_2 -V_1) = -nRDeltaT` & `DeltaH = q_p`

(iii) `text(lsochoric process)` : In isochoric process, volume remains constant during the process.

Since `dV = 0` `=>` `w=0`

From 1st law `DeltaU = q`

(A) Graphical comparison between adiabatic reversible and irreversible process : See fig.1.


(B) Comparison between isothermal & adiabatic process : See fig.2.