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First Law of Thermodynamics for a Control Mass Undergoing a Cycle

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Quick
The first law of thermodynamics states that during any cycle a system (control mass) undergoes, the cyclic integral of the heat is proportional to the cyclic integral of the work.


Equation

(Eq2)      ∮ δQ = ∮ δW

basic statement of the first law of thermodynamics

Nomenclature

∮δQ	the cyclic integral of the heat transfer, represents the net heat transfer during the cycle
∮δW	the cyclic integral of the work, represents the net work during the cycle
J	proportionality factor that depends on the units used for work and heat

Details

Consider as a control mass the gas in the container as shown:



Let this system go through a cycle that is made up of two processes. In the first process work is done on the system by the paddle that turns as the weight is lowered. Let the system then return to its initial state by transferring heat from the system until the cycle has been completed.

Originally, work was measured in mechanical units of force times distance, such as foot pounds force or joules, and heat was measured in thermal units, such as the British thermal unit (Btu) or the calorie. Measurements of work and heat were made during a cycle for a wide variety of systems and for various amounts of work and heat. When the amounts of work and heat were compared, it was found that they were always proportional. Such observations led to the formulation of the first law of thermodynamics, which in equation form is written:

(Eq1)

J ∮ δQ = ∮ δW

The basis of every law of nature is experimental evidence, and this is true also of the first law of thermodynamics. Many different experiments have been conducted on the first law, and every one thus far has verified it either directly or indirectly. The first law has never been disproved.

The units for work and heat or for any other form of energy either are the same or are directly proportional. In SI units, the joule is used as the unit for both work and heat ad for any other energy unit. In English units, the basic unit for work is the foot pound force, and the basic unit for heat is the British thermal unit (Btu). The Btu is defined in terms of the basic SI metric units:

1 Btu = 778.17 ft lbf

This unit is termed the International British thermal unit. For much engineering work, the accuracy of other data does not warrant more accuracy than the relation 1 Btu = 778 ft lbf. Because these units are equivalent, it is not necessary to include the factor J explicitly in Eq1, but simply to recognize that for any system of units, each equation must have consistent units throughout. Therefore, Eq1 may be written as:

(Eq2)

∮ δQ = ∮ δW

which can be considered the basic statement of the first law of thermodynamics.