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

step 1 First, let's write down the cyclic relation here. We have del p over del v, t constant, times de

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

Key Concepts

Equation of State

This concept refers to a mathematical relationship connecting state variables such as pressure, temperature, and volume. It provides a basis for describing the thermodynamic state of a system and serves as the starting point for derivations of other important relationships by differentiating with respect to one variable while holding another constant.

Partial Derivatives in Thermodynamics

Partial derivatives capture how one state variable changes when another variable is varied, while keeping a third variable constant. They enable the exploration of interdependencies among thermodynamic variables and are essential in proving properties like cyclic and reciprocity relations through the application of differentiation rules.

Cyclic Relation

The cyclic relation is a fundamental property in thermodynamics where the product of three specific partial derivatives taken in a cyclic manner equals -1. This relation arises from the consistency of variable interdependence in a three-dimensional state space, reflecting the inherent symmetry in the way variables are interrelated.

Reciprocity Relation

The reciprocity relation states that the partial derivative of one variable with respect to a second (keeping the third variable constant) is the reciprocal of the derivative of the second with respect to the first. This principle ensures consistency and symmetry in the mathematical descriptions of thermodynamic systems and is widely used when transforming between different sets of variables.

Transcript

00:01 First, let's write down the cyclic relation here.

00:06 We have del p over del v, t constant, times del v over del t, p constant, and del t over del p constant, and del t over del p, v constant, is equal to minus 1.

00:24 Okay, so to find del p over del v, first let's write down the equation that we have in the question in terms, of p.

00:32 So we have p is equal to r t divided by v minus a.

00:39 So from here we can say del p del v, del v, in t constant is equal to minus r t divided by v minus a squared.

00:58 So here again we know that from the equation in the question we have have p v minus a is equal to r t so from here we can say r t divided by v minus a is equal to so let's use this here now i can say this is minus r t divided by v minus a times v minus a so so minus rt over v minus a would be minus p over v minus a.

01:49 This is the other v minus a.

01:53 Okay, so so far we know del p over del v is minus p, del p over del v t constant is minus p over v minus a.

02:11 Now the next term, let's find del v over del t.

02:15 To find del v over del t, let's write v in terms of the other variables.

02:22 It's rt divided by p plus a.

02:29 So to find del v over del t, we need to differentiate.

02:37 And then we find this is equal to r over.

02:45 And next term, we want to find del t over del p.

02:50 So again, isolate t.

02:52 T is equal to p times v minus a divided by r.

02:59 And therefore, del t over del p, then v is constant, is equal to v minus a over r.

03:13 Now let's put all of this in the cyclic equation...