Physics II Class 01/29


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Physics II Quiz 01/29/03

Quiz

Consider the system indicated below.

When thermal energy is first introduced the result is a pressure buildup without significant change in volume.  Note that there is no significant volume change until the thin tube is filled.

When the tube becomes full water begins flowing out of the system, allowing the gas to expand.  Since the height of the water column no longer changes the pressure in the system will remain unchanged.

As long as volume is constant the thermal energy going into the system all goes into increasing the KE of the molecules.

When the gas begins to expand it does work against the water surface in the container, some but perhaps not all of which goes into lifting the water that flows from the top of the tube.  The increasing pressure is the result of more rapidly moving molecules so energy also goes into increasing the KE of the molecules.

These ideas are indicated in the figure above.

Once the system has reached the maximum temperature possible with the source of thermal energy, it will cease to expand and no more water can be pumped until we return the system to its initial state.  We now analyze how we might complete the cycle in order to pump more water. 

To return the system to its original state the pressure and temperature must both be decreased.  There are several ways to do this:

We first note that if we keep the temperature of a closed system constant while allowing pressure and volume to change, we have constant T and constant n, so that n r T remains constant and we therefore have P = nR T / V = constant / V:

We graph the general shape of the P vs. V curve for constant T and n:

Our P vs. V diagram for the process will therefore look something like that depicted below:

We now assume a thermal energy source at 350 K and analyze a system with initial volume 3 liters and initial temperature 300 K, in which we allow pressure to increase at constant volume until we can support the 70 cm column of water, then allow the system to expand until we reach a maximum temperature of 350 K.

The first part of the cycle is shown below, where pressure is allowed to increase until it will support the water column.

To find the state of the system at this point we first determine the pressure required.

Since volume is unchanged the new temperature will therefore be T1 = T0 * (P1 / P0) = 300 K * (107 kPa / (100 kPa) ) = 321 K.

The system is then allowed to expand at constant pressure until temperature is T2 = 350 K, the max possible temperature for the thermal source.

This information is depicted below.

We now ask how much energy is required to accomplish this much of the cycle:

Recall that for an ideal monatomic gas the KE per particle is 3/2 k T.

Using U to denote the total internal KE of the system we therefore have 

Since PV = n R T we also therefore have

For the above system in its initial state we easily calculate the internal energy, obtaining U = 450 J, approx..

The final state, as shown below, has U = 530 J approx..

 

Thus  the internal energy increases between initial and final state by approximately 80 Joules.

A more accurate calculation indicates that the internal energy increase is closer to 75 Joules.

The work done in an expansion is equal to the area under the corresponding P vs. V graph. 

The thermal energy supplied to the system was therefore the sum of the 75 J required to increase the internal energy of the system, and the 29 J required to do the work of expansion.

During the process .27 liters of water was raised to an altitude of .70 meters, resulting in a potential energy increase of about .27 liters * 1 kg / liter * 9.8 m/s^2 * .70 m = 1.9 Joules.

If no further thermal energy is supplied to the system as it completes the rest of its cycle, we conclude that the efficiency of the cycle is

This seems like a pretty low result but note that most engines run below 20% efficiency, and the 1.8% efficiency is not bad if the energy is cheap enough, which can sometimes be the case with thermal energy.

Note also that as we can see by analyzing a similar system at a higher maximum temperature, a hotter source of thermal energy would permit more expansion and more PE increase, and a net increase in efficiency.

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