Mathematical
statement of the first law
The mathematical statement of the first law of thermodynamics
is
∆U = q + w
--------- 7.7
Case 1 : For a cyclic process involving isothermal expansion of an
ideal gas,
∆U = 0.
eqn (7.7) ⇒∴ 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. ΔV = 0
ΔU = q + w
= q - PΔV
ΔV =0
ΔU = 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) ⇒ ΔU = 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
∆U = q + w
∆U = q - P ∆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
∆u = q-w (heat is given to the system (+q); work is done
by the system(-w)
∆u q - PdV
= 400 J - 1 atm (10-5)L
=400 J - 5 atm L
[∴ 1 L atm = 101.33 J]
=400 J - 5 × 101.33 J
=400 J - 506.65 J
=- 106.65 J
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