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# Mathematical statement of the first law

The mathematical statement of the first law of thermodynamics is âˆ†U = q + w

Mathematical statement of the first law

The mathematical statement of the first law of thermodynamics is

âˆ†U = q + w  --------- 7.7

Case 1 : For a cyclic process involving isothermal expansion of an ideal gas,

âˆ†U = 0.

eqn (7.7) â‡’âˆ´ q = -w

In other words, during a cyclic process, the amount of heat absorbed by the system is equal to work done by the system.

Case 2 : For an isochoric process (no change in volume) there is no work of expansion. i.e. Î”V = 0

Î”U = q + w

= q - PÎ”V

Î”V   =0

Î”U = qv

In other words, during an isochoric process, the amount of heat supplied to the system is converted to its internal energy.

Case 3 : For an adiabatic process there is no change in heat. i.e. q= 0. Hence

q = 0

eqn (7.7) â‡’ Î”U = w

In other words, in an adiabatic process, the decrease in internal energy is exactly equal to the work done by the system on its surroundings.

Case 4 : For an isobaric process. There is no change in the pressure. P remains constant. Hence

âˆ†U = q + w

âˆ†U = q - P âˆ†V

In other words, in an isobaric process a part of heat absorbed by the system is used for PV expansion work and the remaining is added to the internal energy of the system.

Problem: 7.1

A gas contained in a cylinder fitted with a frictionless piston expands against a constant external pressure of 1 atm from a volume of 5 litres to a volume of 10 litres. In doing so it absorbs 400 J of thermal energy from its surroundings. Determine the change in internal energy of system.

Solution:

Given data

q = 400 J

V1=5L

V2 = 10L

âˆ†u = q-w (heat is given to the system (+q); work is done by the system(-w)

âˆ†u    q - PdV

= 400 J - 1 atm (10-5)L

=400 J - 5 atm L

[âˆ´ 1 L atm = 101.33 J]

=400 J - 5 Ã— 101.33 J

=400 J - 506.65 J

=- 106.65 J

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11th Chemistry : UNIT 7 : Thermodynamics : Mathematical statement of the first law |