### First Law of Thermodynamics

Statement

• The first law of thermodynamics states that- “during any cycle a closed system undergoes, the cyclic integral of heat is always proportional to cyclic integral of the work” and is expressed as-

$\oint dQ\alpha\oint dW$

First Law for a Closed System Undergoing a Cycle [Joule's Experiment]

Figure: Adiabatic Work  and State Diagram

• Let us consider a closed system, which consists of a known mass of water contained in an adiabatic vessel having a thermometer and a paddle wheel as shown in figure. Let a certain amount of work W1-2 be done upon the system by the paddle wheel. The quantity of work can be measured by the fall of weight which drives the paddle wheel through a pulley. The system was initially at temperature T1, the same as that of atmosphere and after work transfer let the temperature rise to T2. The pressure is always 1 atmosphere. The process 1-2 undergone by the system is represented on a state diagram with generalized thermodynamic coordinates x and y.

• Let the insulation now be removed. The system and surroundings interact by heat transfer till the system returns to original temperature T1, attaining the condition of thermal equilibrium with the atmosphere. The system thus executes a cycle which consists of a definite amount of work input W1-2 to the system followed by the transfer of an amount of heat Q2-1 from the system.

• It has been found that this W1-2 is always proportional to heat Q2-1 and the constant of proportionality is called the Joule’s equivalent or constant or the mechanical equivalent of heat.

• Here there are only two energy transfer quantities as the system performs a thermodynamic cycle. If the cycle involves many more heat and work quantities and is expressed algebraically as,

• (ƩW)cycle = J (ƩQ)cycle
Where: J is the Joule’s equivalent.

• This is also expressed in the form,

$\oint dQ=J\oint dW$
Where: symbol denotes the cyclic integral for the closed path.

• This is the first law for a closed system undergoing a cycle.