Using the equation of state P(U-a)=R T, verify (a) the cyclic relation and (b) the reciprocity r

Using the equation of state P(U-a)=R T, verify (a) the...

Key Concepts

Reciprocity Relation

The reciprocity relation is a property of partial derivatives stating that the derivative of one variable with respect to another (with a third held constant) is the reciprocal of the inverse derivative, i.e., (?X/?Y)_Z = 1/(?Y/?X)_Z. This concept simplifies the manipulation of thermodynamic equations by allowing certain derivatives to be easily interconverted.

Equation of State

An equation of state is a relation among state variables (such as pressure, temperature, and internal energy) that characterizes the macroscopic behavior of a thermodynamic system. It provides a fundamental framework from which one can derive other thermodynamic identities and relationships, allowing the prediction and analysis of how the system responds to changes in its conditions.

Cyclic Relation in Thermodynamics

The cyclic relation (or triple product rule) is a mathematical identity linking partial derivatives of three interdependent variables, typically expressed as (?X/?Y)_Z · (?Y/?Z)_X · (?Z/?X)_Y = -1. This relation ensures consistency among the derivatives and is an essential tool in thermodynamics for verifying and deriving interrelations among state variables.

Transcript

00:01 So let's start the solution for discussion, but before that, let's take a look at the given information that we have.

00:10 Right.

00:11 So after the question statement, it says i use the relation that b times v negative a is equal to r times d.

00:24 Right.

00:25 And it says that we need to verify.

00:27 Now, there are two parts that we need to verify.

00:29 The first part is the cyclic relation, right? and the b part is the reciprocity relation at constant u, right? so we will start with at constant u.

00:53 Now we will start our solution from part a.

00:57 So for part a, we need to verify the cyclic relation for the state, equation that is p times v negative a is equal to r times t so we will need to calculate the derivatives of each term in the equation right so first of all we are going to rearrange this equation for p right so we would have p is equal to r times t divided by v negative a then we need to calculate the derivatives right so we have the partial differential change in pressure with respect to v under the constant t right and this would be equal to negative r divided by v negative a whole square is equal to negative b divided by v negative a right now what we need to do is that we need to solve this given equation right this given equation for v right so when we solve it for v we we get that v is equal to r divided by r times t divided by p plus a right so it's derivative partial derivative with respect to t under the constant p it would be equal to r divided by p right then again we need to rearrange this equation for t right first of all we did it for p then for v and then we are going to rearrange it for t right so t is equal to p times the specific volume um negative a divided by r so it's partial derivative it would be equal to v negative a divided by r right now uh what we are going to do is that we are going to we can now substitute the calculated terms into the cyclic equation to prove it, right? so the cyclic equation is the partial derivative of p with respect to v under the constant t, times the partial derivative of v with respect to t under the constant p.

03:21 And over here we have the partial derivative of t with respect to p under the constant v.

03:27 It's equal to negative 1.

03:29 We need to plug in the values of all of these partial derivatives, right? we just calculated them, right? so you would get that's equal to negative p divided by v negative a.

03:40 Over here we have r divided by b.

03:43 And over here we have v negative a divided by r, right? it's equal to negative one...

Please give Ace some feedback