calorie to joule. Since in SI units, we use the unit joule for heat, work or any other form of energy, the term mechanical equivalent is now superfluous and need not be used. As already remarked, the specific heat capacity depends on the process or the conditions under which heat capacity transfer takes place. For gases, for example, we can define two specific heats : specific heat capacity at constant volume and specific heat capacity at constant pressure .
For an ideal gas, we have a simple relation. C p – C v = R ( . ) where C p and C v are molar specific heat capacities of an ideal gas at constant pressure and volume respectively and R is the universal gas constant. To prove the relation, we begin with Eq.
( . ) for mole of the gas : ∆ Q = ∆ U + P ∆ V If ∆ Q is absorbed at constant volume, ∆ V = U U v v v = = = ( . ) where the subscript v is dropped in the last step, since U of an ideal gas depends only on temperature. (The subscript denotes the quantity kept fixed.) If, on the other hand, ∆ Q is absorbed at constant pressure, U P p p p p + ( .
) The subscript p can be dropped from the first term since U of an ideal gas depends only on T . Now, for a mole of an ideal gas PV = RT which gives P R p ( . ) Equations ( . ) to ( .
) give the desired relation, Eq. ( . ). .
THERMODYNAMIC STATE VARIABLES AND EQUATION OF STATE Every equilibrium state of a thermodynamic system is completely described by specific values of some macroscopic variables, also called state variables. For example, an equilibrium state of a gas is completely specified by the values of pressure, volume, temperature, and mass (and composition if there is a mixture of gases). A thermodynamic system is not always in equilibrium. For example, a gas allowed to expand freely against vacuum is not an equilibrium state [Fig.