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### The concept of Enthalpy of System:

- The enthalpy of a system is defined as the sum of the internal energy of the system and energy that arises due to pressure and volume. It is denoted by letter H
- Mathematically,

H = U + PV

where, H = Enthalpy U = Internal Energy P = Pressure V = Volume

- As the internal energy E, Pressure P and Volume V are state functions, the enthalpy of the system is a state function.

### Change in Enthalpy of System:

- Let H
_{1}and H_{2}be the enthalpies of the system in the initial state and final state respectively. Let P_{1}, V_{1}and U_{1}be the pressure, volume and internal energy of the system in the initial state. Let P_{2}, V_{2}and U_{2}be the pressure, volume and internal energy of the system in the final state.

The change in enthalpy is given by ΔH = H_{1} – H_{2} ………………. (1)

By definition of enthalpy H = U + PV

For initial state, H1 = U_{1} + P _{1}V_{1}

For final state, H_{2} = U_{2} + P_{2} V_{2}

Substituting these values in equation (1)

Δ H = (U_{2} + P_{2} V_{2} ) – (U_{1} + P_{1}V_{1})

∴ Δ H = (U_{2} – U_{1}) + (P_{2} V_{2} – P_{1} V_{1})

At constant pressure P_{1} = P_{2} = P

∴ Δ H = (U2 – U_{1}) + (P V 2 – P V_{1})

∴ Δ H = (U_{2} – U_{1}) + P( V_{2} – V_{1})

∴ Δ H = ΔU + PΔ V

This is mathematical expression for change in enthalpy of system.

- Thus the change in enthalpy of the system is equal to the sum of the increase in the internal energy of the system and the mechanical work due to expansion.
- If Δ H is positive then the reaction is endothermic and when ΔH is negative the reaction is exothermic.

### Change in Enthalpy of System at Constant Pressure (Isobaric Process ):

- The expression for the change in enthalpy of a system at constant pressure is

Δ H = Δ U + PΔ V ………… (1)

Where, ΔH = Change in enthalpy ΔU = Change in internal energy P = Pressure ΔV = Change in volume

- The mathematical statement of the first law of thermodynamics is

ΔU = q + PΔ V

Where, q = Heat supplied to the system ΔU = Change in internal energy P = Pressure ΔV = Change in volume

- Let us denote q = q
_{p}= heat absorbed by the system at constant pressure.

q_{p} = ΔU + PΔV ………. (2)

From equations (1) and (2) we have

ΔH = qp

Thus at constant pressure, the change in enthalpy is equal to heat absorbed at constant pressure.

### Change in enthalpy of System at Constant Volume (Isochoric Process ):

- The expression for the change in enthalpy of a system is

Δ H = Δ U + PΔ V

Where, ΔH = Change in enthalpy ΔU = Change in internal energy P = Pressure ΔV = Change in volume

In isochoric process ΔV = 0

∴ Δ H = Δ U + P(0)

Δ H = Δ U ………….. (1)

- The mathematical statement of the first law of thermodynamics is

Δ U = q + PΔV ………….. (2)

Where, q = Heat supplied to the system ΔU = Change in internal energy P = Pressure ΔV = Change in volume

Substituting ΔV = 0 in equation (2)

q = ΔU

- let us denote q = q
_{v}= heat absorbed by system at constant volume.

q_{v} = ΔE ………….. (3)

From equations (1) and (3) we have

ΔH = ΔU = q_{v}

- Thus at constant volume, the change in enthalpy is equal to heat absorbed at constant volume and also equal to change in internal energy. Thus in isochoric process heat supplied to the system is used for increasing internal energy of the system.

### The Relation between ΔH and ΔU:

- Change in the enthalpy of a system is given by

Δ H = ΔU + PΔV

Where, ΔU = Change in internal energy P = Pressure of the system ΔV = Change in volume

- But for reactions involving gases

PV = nRT

Where P = Pressure of a gas V = Volume of the gas n = Number of moles of the gas

R = Universal gas constant T = Absolute temperature of the gas

For initial state, P V_{1} = n_{1}RT

For Final state, P V_{2} = n_{2}RT

PV_{2} – PV_{1} = n_{2}RT – n_{1}RT

∴ P( V_{2} – V_{1}) = (n_{2} – n_{1} )RT

∴ PΔV = ΔnRT

WhereΔH = Change in enthalpy or heat of reaction at constant pressure.

ΔU = Change in internal energy or heat

Δn = Difference in the number of moles of gaseous products and reactants.

R = Universal gas constant 8.314 J/ mol / k, T = Absolute temperature.

### Work Done in a Chemical Reaction :

- The work done by a system at constant pressure and temperature is given by

W = – P_{ext}ΔV

Assuming P_{ext} = P

W = – P( V2 – V1)

∴ W = – PV2 + PV1

But for reactions involving gases PV = nRT

Where, P = Pressure of a gas V = Volume of the gas n = Number of moles of the gas

R = Universal gas constant T = Absolute temperature of the gas

For initial state, P V_{1} = n_{1}RT

For Final state, P V_{2} = n_{2}RT

W = – PV2 + PV1 = – n_{2}RT + n_{1}RT

∴ W = – (n_{2} – n_{1} )RT

∴ W = – ΔnRT

This is an expression for work done in chemical reaction.

### Conditions Under Which Change in Enthalpy of System (ΔH) is Equal to Change in Internal Energy of the System(ΔU):

- When the reaction is carried out in closed vessel, there is no change in volume. ΔV = 0, hence Δ H = ΔU + PΔV gives Δ H = ΔU.
- When the reaction involves only solids and liquids, then change in volume is negligible. ΔV = 0, hence Δ H = ΔU + PΔV gives Δ H = ΔU.In
- In a chemical reaction in which the number of gaseous reactants consumed is equal to the number of moles of gaseous product formed then

Δn = 0, hence ΔH =ΔU + ΔnRT gives ΔH = ΔU.

#### Note:

- Work has significance only when the number of gaseous reactants consumed and the number of moles of gaseous product formed is different and there is a change in volume.
- Heat Supplied q is not thermodynamic function but heat supplied at constant pressure (q
_{p}) and heat supplied at constant volume (q_{v}) are thermodynamic functions: - By the first law of thermodynamics, ΔU = q + W. In this relation ΔU is a state function as well as thermodynamic function. Work is path function as well as non-thermodynamic function. For algebraic addition q on the right-hand side of the equation, heat supplied is path function as well as non-thermodynamic function.
- For constant pressure process, ΔH = q
_{p}Where, ΔH is enthalpy of a system which is state function as well as thermodynamic function. Hence heat supplied at constant pressure q_{p}is a thermodynamic function. - For constant volume process, ΔU = q
_{p}Where, ΔU is the internal energy of a system which is state function as well as thermodynamic function. Hence heat supplied at constant volume q_{v}is a thermodynamic function.

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