This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 3 More applications of derivatives 3.1 Exact & inexact differentials in thermodynamics So far we have been discussing total or exact differentials du = u x y dx + u y x dy, (1) but we could imagine a more general situation du = M ( x, y ) dx + N ( x, y ) dy. (2) If the differential is exact, M = ( u x ) y and N = u y x . By the identity of mixed partial derivatives, we have M y x = 2 u xy = N x y (3) Ex: Ideal gas pV = RT (for 1 mole), take V = V ( T, p ), so dV = V T p dT + V p T dp = R p dT RT p 2 dp (4) Now the work done in changing the volume of a gas is dW = pdV = RdT RT p dp. (5) Lets calculate the total change in volume and work done in changing the system between two points A and C in p, T space, along paths AC or ABC . 1. Path AC: dT dp = T 2 T 1 p 2 p 1 T p so dT = T p dp (6) & T T 1 p p 1 = T p T T 1 = T p ( p p 1 ) (7) so (8) dV = R p T p dp R p 2 [ T 1 + T p ( p p 1 )] dp = R p 2 ( T 1 T p p 1 ) dp (9) dW = R p ( T 1 T p p 1 ) dp (10) 1 A B C (p,T) (p 2 ,T 2 ) (p 1 ,T 1 ) T p Figure 1: Path in p,T plane for thermodynamic process. Now we can calculate the change in volume and the work done in the process: V 2 V 1 = Z AC dV = R ( T 1 T p p 1 ) 1 p fl fl fl fl p 2 p 1 = R T 2 p 1 T 1 p 2 p 1 p 2 (11) W 1 W 2 = Z AC pdV = R ( T 1 T p p 1 ) ln p fl fl fl fl p 2 p 1 = R T 2 p 1 T 1 p 2 p 2 p 1 ln p 2 p 1 (12) 2. Path ABC: Note along AB dT = 0, while along BC dp = 0. V 2 V 1 = Z ABC V T p dT + V p T dp = Z p 2 p 1 V p T dp + Z T 2 T 1 V T p dT = Z p 2 p 1 RT 1 p 2 dp + Z T 2 T 1 R p 2 dT = R T 2 p 1 T 1 p 2 p 1 p 2 (13) W 2 W 1...
View
Full
Document
 Fall '07
 HIRSHFELD

Click to edit the document details