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 = q_{v}

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

V_{1}=5L

V_{2} = 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 |

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