THERMODYNAMICS

average energy of × ½ k B T = k B T . In three dimensions, the average energy is k B T . For a mole of a solid, the total energy is U = k B T × N A = RT ( ∵ k B T × N A = R ) Now, at constant pressure, ∆ Q = ∆ U + P ∆ V ≅ ∆ U , since for a solid ∆ V is negligible. Therefore, U R ( .

) Table . Specific and molar heat capacities of some solids at room temperature and atmospheric pressure As Table . shows, the experimentally measured values which generally agrees with Substance Speci"c heat –v (J kg K ) – – Molar speci"c heat (J mol K ) – – predicted value 3R at ordinary temperatures. (Carbon is an exception.) The agreement is known to break down at low temperatures.

Specific heat capacity of water The old unit of heat was calorie. One calorie was earlier defined to be the amount of heat required to raise the temperature of 1g of water by °C. With more precise measurements, it was found that the specific heat of water varies slightly with temperature. Figure .

shows this variation in the temperature range to ° C. Fig. . Variation of specific heat capacity of water with temperature.

For a precise definition of calorie, it was, therefore, necessary to specify the unit temperature interval. One calorie is defined to be the amount of heat required to raise the temperature of 1g of water from . ° C to . ° C.

Since heat is just a form of energy, it is preferable to use the unit joule, J. In SI units, the specific heat capacity of water is J kg – K – i.e. . J g – K – .

The so called mechanical equivalent of heat defined as the amount of work needed to produce cal of heat is in fact just a conversion factor between two different units of energy : 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.

. (a)]. During the rapid expansion, pressure of the gas may not be uniform throughout. Similarly, a mixture of gases undergoing an explosive chemical reaction (e.g.

a mixture of petrol vapour and air when ignited by a spark) is not an equilibrium state; again its temperature and pressure are not uniform [Fig. . (b)]. Eventually, the gas attains a uniform temperature and pressure and comes to thermal and mechanical equilibrium with its surroundings.

Fig. . (a) The partition in the box is suddenly removed leading to free expansion of the gas. (b) A mixture of gases undergoing an explosive chemical reaction.

In both situations, the gas is not in equilibrium and cannot be described by state variables. In short, thermodynamic state variables describe equilibrium states of systems. The various state variables are not necessarily independent. The connection between the state variables is called the equation of state.

For example, for an ideal gas, the equation of state is the ideal gas relation P V = µ R T For a fixed amount of the gas i.e. given µ , there are thus, only two independent variables, say P and V or T and V . The pressure-volume curve for a fixed temperature is called an isotherm . Real gases may have more complicated equations of state.

The thermodynamic state variables are of two kinds: extensive and intensive . Extensive variables indicate the ‘size’ of the system. Intensive variables such as pressure and temperature do not. To decide which variable is extensive and which intensive, think of a relevant system in equilibrium, and imagine that it is divided into two equal parts.

The variables that remain unchanged for each part are intensive. The variables whose values get halved in each part are extensive. It is easily seen, for example, that internal energy U , volume V , total mass M are extensive variables. Pressure P , temperature T , and density ρ are intensive variables.

It is a good practice to check the consistency of thermodynamic equations using this classification of variables. For example, in the equation ∆ Q = ∆ U