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(CENG131)[2008](s)hw3~625^_10025.pdf
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CENG131

CHEMICAL ENG GINEERING THERMODYN NAMICS

HONG KONG UNIVERSSITY OF SCIENCE AND TECHNOLOGY,SPRING G 2009

HOMEWORK #3
PROBLEM 1

A well-insulated gas tank is separa rated into two partitions by a fixed, adiabatic, impe ermeable diaphragm. On one side of the diap phragm is moles of nitrogen at temperature and pressure . On the other side is moles of nitrogen at the same temperature but a diffe erent pressure . The diaphragm suddenly rupttures, allowing the two volumes of gas to mix, fina ally reaching equilibrium at temperature and d pressure . Derive expressions for the final tem mperature, , and pressure, , in terms of ,, , , and . Then show that, for the simple e case of . ,
the total entropy of the gas tank co ontent must increase or remain the same. Assumme nitrogen is an ideal gas. You will need to use the inequality . . . . (i.e. arithmetic mean n >= geometric mean), where and are positive e numbers.
SOLUTION
SYSTEM


We will treat the two bodies o of nitrogen gas as separate systems, A and B. Both h are closed system, with possibility of hea at and work exchange. Initially, they are separated d by a diaphragm. In the final state, they togetherr make up a homogeneous mixture of nitrogen, so o they must share the same intensive properties..

SYMBOL DEFINITIONS AND GIVEN INFORMATION
R = gas constant = 8.314 Jmol-1K-1
.
= The constant-volume heat capacity of nitrogen

.
= The constant-pressure heat capacity of nitrogen
. = Initial pressure of system X


.
= Final pressure of both systems


= Initial temperature of both systems
. = Final pressure of both systems
. = Initial total volume of system X
. = Final total volume of systems X

=
Number of moles of nitrogen in system A = 1 mol

=
Number of moles of nitrogen in system B = 1 mol

=
Total internal energy of system X
= Initial total entropy of system X
= Final total entropy of system X
= Total entropy change of the content of the gas tank


=
Heat transfer into system X

=
Work done on system X


GOALS
Prove that the total entropy of the gas tank content (i.e., A+B) increases or remains constant for this process.
ASSUMPTIONS
1.
The nitrogen behaves as an ideal gas at all times.

2.
The nitrogen has constant heat capacities.

3.
The kinetic and potential energy of the systems can be ignored.

4.
The gas tank is an adiabatic enclosure of the gas inside.


DETERMINING THE FINAL STATE
Our first task is to determine the final state of the two systems. If we can do that, using the fact that entropy is a state function, we can evaluate the entropy change. We need two intensive properties to specify the final state, and we have two equations: the ideal gas equation and the First Law. So in principle we should be able to do so.
The final temperature can be easily obtained by the First Law applied on both (closed) systems (neglecting the kinetic and potential energy terms)