# First Law of Thermodynamics statement & derivation

## Statement of First Law of Thermodynamics

The first law of thermodynamics also known as the ‘Law of Conservation of Energy’.

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“ Energy can neither be created nor destroyed, but only be changed from one form to another form”.

Or

“ The total energy of a system and its surrounding remain constant”.

## Mathematical Form and derivation

Consider a system whose initial system is ‘E_{1}’ let a quantity of heat ‘q’ absorbed by the system from the surroundings and does some work ‘W’ on the surroundings while the internal energy change to

‘E_{2}’.

Then, change in internal energy is given by

ΔE = E_{1} – E_{2}

According to the first law of Thermodynamics

ΔE = q – W

Value of ΔE is negative when the system loses energy and positive when system gains energy.

### Pressure-Volume Work

Consider a gas enclosed in a cylinder, fitted with a frictionless and weightless piston. Let the area of the piston is ‘A’ external pressure acting on the piston ‘P’, the force exerted by the gas on the piston ‘F’.

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Now, if the system expands, and the piston moves against the surroundings travelling a distance ‘ΔI’. Then,

Work = Force x Displacement

W = F x ΔI ———– 1

Since,

P = F/A

Therefore,

F = P x A ———— 2

Put the value of ‘f’ into equation # 1, we get

W = P x A x ΔI ———— 3

Since,

A x ΔL = ΔV

So, equation # 3 we can write as

W = P x ΔV

**Or **

**W = P **ΔV

According to the first law of thermodynamics

ΔE = q – w ———– 4

Put the value of ‘W’ in equation # 4 we get

ΔE = q – P ΔV

**Or **

**q = **Δ**E + p **Δ **V**

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## Application of First Law of Thermodynamics

### Process At Constant Volume

According to the first law of thermodynamics

ΔE = q – P Δ V

ΔE = q v – P Δ V ————– 1

Where, qv = heat absorbed at constant volume when the volume of the system not change

Δ V = 0

Under this condition, no work is done

P Δ V = p x 0 = 0

Hence, equation # 1 becomes

Δ E = q v ————- 2

This equation shows that at constant volume, the heat evolved is equal to the change in the internal energy and no work is done.

**Process At Constant Pressure**

When heat is given to system at constant pressure the internal energy of the system increases, as a result of which the system performs work on the surroundings:

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Thus,

qp = ΔE + PΔV ———- 1

where qp is the heat absorbed at the constant pressure

we know that

ΔE = E_{1} – E_{2}

ΔV = V_{1} – V_{2}

Then equation # 1 can be written as:

= (E_{2} – E_{1}) + P(V_{2} – V_{1})

qp = (E_{2} + PV_{2}) – (E_{1} + PV_{1}) ———- 2

the enthalpy ‘H’ is mathematically defined as:

H = E + PV

At initial state

H_{1} = E_{1} + PV_{1}

At final state

H_{2} = E_{2} + PV_{2}

Substituting the value in equation # 2 we get

qp = H_{2} – H_{1} = Δ H

qp = ΔH ———– 3

Equation #1 can be written as:

ΔH = Δe + PΔV ———- 4

Equation #3 shows that heat absorbed or evolved from a system at constant pressure is equal to the change in enthalpy of the system.

Thus, change in enthalpy is the heat absorbed or evolved by a system at constant pressure.