Solution 44P

Introduction

We have to calculate the pressure and volume at the all points of the pv diagram and then we can draw the final PV diagram.

Knowing the energy input and work output we can calculate the efficiency of the engine, from the efficiency and the temperature of the hot reservoir we can calculate the temperature of the cold reservoir, From the same information we can calculate the final waste of energy per cycle.

Step 1

The temperature of the hot reservoir is

And the work output per cycle is the work done to increase the height of the stone, which is given by

And the heat input, that is the heat supplied to the engine from the hot reservoir is

To draw PV diagram we have to calculate the pressure and volume of all the points in the cycle. The PV diagram of a carnot cycle look like the following diagram,

It consist of 4 process, two isothermal process and two adiabatic process.

In the diagram ab and cd are iso thermal process. ab  represents the higher temperature and cd represents the lower temperature. Also the volume  is maximum which is also given in the problem. Hence we know that

And

So, we know the volume and temperature of the point c, hence let us first calculate the pressure at that point, which is

Now, the segment bc represents a adiabatic expansion, hence we can write that

So the pressure at the point b is

Now we will calculate the volume at the point a.

We know that the energy supplied in the isothermal expansion will only be used for work done. And the energy supplied in the isothermal expansion is equal to the energy supplied to the system from the hot reservoir. Hence we can write that

So the pressure at the point a is

Now to calculate the volume at d, we will consider the segment ad in the pv diagram. Since this is the adiabatic process we can write that

So the pressure at the point d is

So the PV diagram will look like