According to first law of thermodynamics,

q= ΔU + PΔV

If the process is carried out at constant volume , ΔV=0

q_{v} = ΔU

where v indicates constant volume.

**Internal energy change** is the heat absorbed or evolved at constant volume.

ΔU is a state function, therefore q_{v} ,is also a state function.

If the process is carried out at constant pressure, the work of expansion is given by

w = –P ΔV

where ΔV is the increase in volume and P is the constant pressure.

q=ΔU – w

where q is the heat absorbed by the system, ΔU is the increase in internal energy of the system and w is the work done by the system.

Under conditions of constant pressure, putting w= –P ΔV and representing the heat absorbed by q_{p} we get,

q_{p} = ΔU + PΔV

When the system absorbs q_{p} joules of heat, its internal energy increases from U_{1} to U_{2} and the volume increases from V_{1} to V_{2}.Then, we have,

ΔU = U_{2} – U_{1}

ΔV = V_{2} -V_{1}

q_{p} = ( U_{2} – U_{1} ) + P ( V_{2} -V_{1})

q_{p}= ( U_{2} + PV_{2} ) – ( U_{1} + PV_{1})

The thermodynamic quantity U + PV is called **heat content or enthalpy of the system** and is represented by symbol H.

H= U + PV

If H_{2} is the enthalpy of the system in the final state and H1 is the value in the initial state, then

H_{2} = U_{2} + PV_{2}

H_{1} = U_{1} + PV_{1}

q_{p} =H_{2} -H_{1}

q_{p} =ΔH

**Enthalpy change** of a system is equal to the heat absorbed or evolved by the system at constant pressure.

As most of the reactions are carried out at constant pressure ,the measured value of the heat evolved or absorbed is the enthalpy change enthalpy.

ΔH= ΔU + PΔV

Enthalpy change accompanying a process may be defined as the sum of the increase in internal energy of the system and the pressure- volume work done i.e. the work of expansion.

The energy stored within the substance or the system that is available for conversion into heat is called the **heat content or enthalpy** of the system or substance.

The absolute value of heat content or enthalpy of a substance or a system cannot be measured.

U and V are extensive properties therefore the **enthalpy is also an extensive property.**

**Relation between heat of reaction at constant pressure and at constant volume**

q_{p} =ΔH and q_{v} = ΔU

ΔH= ΔU + PΔV

ΔH= ΔU + P(V_{2} – V_{1})

ΔH= ΔU + (PV_{2} – PV_{1})

where V_{1} is the initial volume and V_{2} is the final volume of the system.

For ideal gas,

PV = nRT

PV_{1} =n_{1} RT

PV_{2} =n_{2} RT

where n_{1} is the number of moles of gaseous reactants and n_{2} is the number of moles of gaseous products.

ΔH= ΔU + (n_{2}RT -n_{1}RT)

ΔH= ΔU + (n_{2} – n_{1} )RT

ΔH= ΔU + Δn_{g} RT

where Δn_{g }= n_{2} -n_{1} is the difference between the number of moles of the gaseous products and those of the gaseous reactants.

q_{p} = q_{v} – Δn_{g}RT

Conditions under which q_{p} =q_{v} or ΔH = ΔU

1) When the reaction is carried out in closed vessel so that volume remains constant, i.e. ΔV= 0.

2)When reaction involves only solids or liquids or solution but no gaseous reactant and product. This is because the volume changes of the solids and liquids during a chemical reaction and negligible .

3)When reaction involves gaseous reactants and products but their number of moles are equal in the reactions.

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