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Ideal Gases & Constant Heat Capacity

1. An ideal gas with Cp/Cv=1.5 where Cp and Cv are respectively the heat capacities at constant pressure and constant volume, is used as the working substance of a Carnot engine. The temp of the source is 600K and that of the sink is 300K.

The volume of gas changes from 4 to 1 liter at the low temp and the pressure at the volume of 4 liters is one atm.

calculate the volumes and pressures at the extreme points of the high temp expansion.

calculate the heat absorbed there

Calculate the heat given out at the low temp

what is the efficiency ?

2. Two identical bodies of constant heat capacity are initially at temperatures T1 and T2. They are used as the sink and source of a Carnot engine and no other source of heat is available.

Calculate the final, common temperature when all possible work has been extracted from the system.

Calculate the maximum work obtainable.

Solution Preview

1. An ideal gas with Cp/CV=1.5 where Cp and CV are respectively the heat capacities at constant pressure and constant volume, is used as the working substance of a Carnot engine. The temp of the source is 600K and that of the sink is 300K.

The volume of gas changes from 4 to 1 liter at the low temp and the pressure at the volume of 4 liters is one atm.

• Calculate the volumes and pressures at the extreme points of the high temp expansion.
• Calculate the heat absorbed there
• Calculate the heat given out at the low temp
• What is the efficiency

In a standard Carnot cycle the processes are as follows:

A to B is an isothermal expansion at high temperature TH
B to C is an adiabatic expansion
C to D is an isothermal compression at low temperature TL
C to to A is an adiabatic compression.

We know that VC, PC and VD

Since C and D are states of the system that share the same temperature (the line that connect them is an isotherm), we can write the following equation of states:

(1)
In addition we can write:
(2)

Also note
From state D to state A the gas undergoes an adiabatic compression. In any adiabatic process the relation between the pressure and the volume is:

Where
This relation holds in state D as well as in state A (and at any point along the process), hence:

Since we showed in equation (1) that , we can obtain one equation with two unknowns (PA,VA):

(3)

Where do we get the second equation? This is simply the state equation:

Using ...

Solution Summary

Word file and PDF attached give all the steps to finding the temperature and efficiency in an ideal gas and Carnot engine question.

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