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Faculty: Science Department: Chemistry Course: B.Sc. Sem.: II (All Four Sections) Topic: Thermodynamics) Unit IV Teacher: Dr

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Faculty: Science Department: Chemistry Course: B.Sc. Sem.: II (All four sections) Topic: Thermodynamics) Unit IV Teacher: Dr. V. Mushran Dr. S. Sahai Dr. K.K. Tungala State function and Path function The state function and path function are two ways of expressing different thermodynamic properties of systems. The key difference between state function and path function is that state functions do not depend on the path or process whereas path functions depend on path or process. State function is a thermodynamic term that is used to name a property whose value does not depend on the path taken to reach that specific value. State functions are also known as point functions. A state function only depends on the current state of the thermodynamic system and its initial state (independent from the path). The state function of a thermodynamic system describes the equilibrium state of that system irrespective of how the system arrived at that state. Examples of State Functions: Mass, Energy – enthalpy, internal energy, Gibbs free energy, etc., Entropy, Pressure, Temperature, Volume, Chemical composition and Altitude. A state function depends on three things: the property, initial value and final value. Enthalpy is a state function. It can be given as a mathematical expression as given below. In which, t1 is the final state, t0 is the initial state and h is the enthalpy of the system. Path Function Path function is a thermodynamic term that is used to name a property whose value depends on the path taken to reach that specific value. In other words, a path function depends on the path taken to reach a final state from an initial state. Path function is also called a process function. A path function gives different values for different paths. Hence path functions have variable values depending on the route. Therefore, when expressing the path function mathematically, multiple integrals and limits are required to integrate the path function. Examples of Path Functions: Mechanical work, Heat, Arc length, etc. The internal energy is given by the following equation: ∆U = q + w In which ∆U is the change in internal energy, q is the heat and w is the mechanical work. The internal energy is a state function, but heat and work are path functions. Note: Here, work is equal to the area under the curve. More area under the curve= More work done. Difference between State Function and Path Function State Function Path Function State function is a thermodynamic term that is Path function is a thermodynamic term that is used to name a property whose value does not used to name a property whose value depends depend on the path taken to reach that specific on the path taken to reach that specific value. value. State functions are also called point functions. Path functions are also called process functions. State functions do not depend on the path or Path functions are also called process process. functions. State functions do not depend on the path or Path functions depend on the path or process. process. State function can be integrated using the Path function requires multiple integrals and initial and final values of the thermodynamic limits of integration to integrate the property. property of the system. The value of state function remains the same The value of path function of a single step regardless of the number of steps. process is different from a multiple step process. State functions include entropy, enthalpy, Path functions include heat and mechanical mass, volume, temperature, etc. work. Exact and Inexact differentials The internal energy U is a state function, like V, because it depends only on the state of the system. The integral of the differential of a state function along any arbitrary path is simply the difference between values of the function at two limits. For example, if a system goes from state a to state b , we can write 푏 ∫ 푑푈 = 푈푏 − 푈푎 = ∆푈 푎 Since the integral is path independent, the differential of a state function is called an exact differential. The quantities q and w are not state functions. The integrals of their differentials in going from state a to state b depend on path chosen. Therefore, their differentials are called inexact differentials. We use δ instead of d to indicate inexact differentials. In going from state a to state b the work done is represented by 푏 ∫ 휹풘 = 푤 푎 Note that the result of the integration is not written as wb-wa., because the amount of work done depends on the particular path that is followed between state a and state b. If an infinitesimal quantity of heat δq is absorbed by a system, and an infinitesimal amount of work δw is done on the system, the infinitesimal change in the internal energy is given by dU = δq + δw Exact differential represent, the given function is independent of path. The properties such as temperature, pressure, density, mass, volume, enthalpy, entropy and internal energy are exact differentials. They depend upon their initial and final states, and therefore they are called state or point functions. On the other hand, inexact differential represents the given function is dependent on path and hence called path function. Examples heat, work, etc. In Figure 1, there are two points S1 and S2. These points represent the initial and final states. From the Figure 1, one can not calculate the work done because there is no path. There could be any path from state S1 and S2. In Figure 2, 3 and 4, the shaded area under the curve is the work done. For the same initial and final states, different path can have different shaded area and hence different work done. Therefore quantities like heat and work are path functions since they depend upon path and they are inexact differential. Test for Exactness For an exact differential By the identity of mixed partial differentials, This is called Euler reciprocal relation. Thus du does not depend on path as the mixed partial derivatives are equal. ‘u’ can be any thermodynamic function like H, U and G; and x and y may be thermodynamic variables p, V, and T. Example: Note: Change in V is independent of path but the work done is dependent. dV = Mdp +ndT with, M = R/P and N = − RT/p2 Thus, dw is not an exact differential. .
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