Download presentation

Presentation is loading. Please wait.

Published byNoelle Durden Modified over 4 years ago

2
Lecture 7: Thermo and Entropy Reading: Zumdahl 10.2, 10.3 Outline –Isothermal processes –Isothermal gas expansion and work –Reversible Processes

3
Isothermal Processes Recall: Isothermal means T = 0. Since E = nC v T, then E = 0 for an isothermal process. Since E = q + w: q = -w (isothermal process)

4
Example: Isothermal Expansion Consider a mass connected to a ideal gas contained in a “piston”. Piston is submerged in a constant T bath such that T = 0.

5
Isothermal Expansion (cont.) Initially,V = V 1 P = P 1 Pressure of gas is equal to that created by mass: P 1 = force/area = M 1 g/A where A = piston area g = gravitational acceleration (9.8 m/s 2 ) kg m -1 s -2 = 1 Pa

6
Isothermal Expansion (cont.) One-Step Expansion. We change the weight to M 1 /4, then P ext = (M 1 /4)g/A = P 1 /4 The mass will be lifted until the internal pressure equals the external pressure. In this case V final = 4V 1 w = -P ext V = -P 1 /4 (4V 1 - V 1 ) = -3/4 P 1 V 1

7
Two Step Expansion In this expansion we go in two steps: Step 1: M 1 to M 1 /2 Step 2: M 1 /2 to M 1 /4 In first step: P ext = P 1 /2, V final = 2V 1 w 1 = -P ext V = -P 1 /2 (2V 1 - V 1 ) = -1/2 P 1 V 1

8
Two Step Expansion (cont.) In Step 2 (M 1 /2 to M 1 /4 ): P ext = P 1 /4, V final = 4V 1 w 2 = -P ext V =- P 1 /4 (4V 1 - 2V 1 ) = -1/2 P 1 V 1 w total = w 1 + w 2 = -P 1 V 1 /2 - P 1 V 1 /2 = -P 1 V 1 w total,2 step > w total,1 step

9
Two Step Expansion (cont.) Graphically, we can envision this two-step process on a PV diagram: Work is given by the area under the “PV” curve.

10
Infinite Step Expansion Imagine that we perform a process in which we change the weight “infinitesimally” between expansions. Instead of determining the sum of work performed at each step to get w total, we integrate:

11
Infinite Step Expansion (cont.) Graphically: Two StepReversible

12
Infinite Step Expansion (cont.) If we perform the integration from V 1 to V 2:

13
Two Step Compression Now we will do the opposite….take the gas and compress: V init = 4V 1 P init = P 1 /4 Compression in two steps: first place on mass = M 1 /2 second, replace mass with one = M 1

14
Two Step Compression (cont.) In first step: w 1 = -P ext V = -P 1 /2 (2V 1 - 4V 1 ) = P 1 V 1 w total = w 1 + w 2 = 2P 1 V 1 (see table 10.3) In second step: w 2 = -P ext V = -P 1 (V 1 - 2V 1 ) = P 1 V 1

15
Compression/Expansion In two step example: w expan. = -P 1 V 1 w comp. = 2P 1 V 1 w total = P 1 V 1 q total = -P 1 V 1 We have undergone a “cycle” where the system returns to the starting state. Now, E = 0 (state fxn) But, q = -w ≠ 0

16
Defining Entropy Let’s consider the four-step cycle illustrated: –1: Isothermal expansion –2: Isochoric cooling –3: Isothermal compression –4: Isochoric heating

17
Defining Entropy (cont) Step 1: Isothermal Expansion at T = T high from V 1 to V 2 Now T = 0; therefore, E = 0 and q = -w Do expansion reversibly. Then:

18
Defining Entropy (cont) Step 2: Isochoric Cooling to T = T low. Now V = 0; therefore, w = 0 q 2 = E = nC v T = nC v (T low -T high )

19
Defining Entropy (cont) Step 3: Isothermal Compression at T = T low from V 2 to V 1. Now T = 0; therefore, E = 0 and q = -w Do compression reversibly, then

20
Defining Entropy (cont) Step 4: Isochoric Heating to T = T high. Now V = 0; therefore, w = 0 q 4 = E = nC v T = nC v (T high -T low ) = -q 2

21
Defining Entropy (cont)

22
Defining Entropy (end) The thermodynamic definition of entropy(finally!)

23
Calculating Entropy T = 0 V = 0 P = 0

24
Calculating Entropy Example: What is S for the heating of a mole of a monatomic gas isochorically from 298 K to 350 K? 3/2R

25
Connecting with Lecture 6 From this lecture: Exactly the same as derived in the previous lecture!

Similar presentations

Presentation is loading. Please wait....

OK

First Law of Thermodynamics

First Law of Thermodynamics

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google