Heat and Temperature

It is often said that heat is a form of energy. A more precise statement would be that heat is a process in which energy is added to a system. In particular, a system is heated when energy is added to it as a result of a temperature difference. In this experiment, the source of heat will be an electrical hotplate. The system will be a quantity of water, or some other substance. The substance will be heated because the temperature of the hotplate is higher than that of the substance. Temperature will be measured by a digital thermometer that reads up to above 100 degrees Celsius.

The energy of a system can take many forms. In mechanics we study kinetic energy (e.g. the energy of water flowing in a river), and potential energy due to a force that acts on the system (e.g. the potential energy change when water flows over a waterfall). Another important form of energy is the energy associated with the molecules that make up the substance -- their kinetic energy of motion and the potential energy associated with inter-molecular forces. We cannot "see" this internal energy directly. But temperature is one measure of how much internal energy a system possesses.

One of the great accomplishments of nineteenth century science was to recognize

Safety Precaution: Be sure to use the insulated gloves when handling any heated objects.

1. Heat and Temperature Change

Start with about 300 ml of water in a 600 ml beaker. We will want to know the mass of the water, so first measure the mass of the empty beaker on the digital balance; then measure the mass of the beaker with water. Start the hotplate, set on high. Give it a minute or so to get up to its final temperature. Place the beaker on the hotplate, and start the clock to record the time. You don't have to turn the clock on and off. Just keep it going and record the time when you take a temperature reading. Record temperature (call it T) and time (call it t) every minute. (Before recording the temperature each time, stir the water gently.) Follow the temperature change from room temperature (about 25 degrees Celsius) up to between 75 and 80o C. Then remove the beaker and turn off the hotplate.

Later we will plot a graph with time on the vertical axis and temperature on the horizontal axis. Since heat enters the system at a constant rate, the vertical axis will represent the total heat that has entered the water at each reading.

2. Heat and Mass

Now we want to repeat the experiment for 150 ml and 450 ml of water. These quantities don't have to be exact, because, by weighing the water (as before), you will measure the mass precisely. [Before beginning to heat the water, make sure the beaker is cooled off, by running cold water in it a few times.] We will want to determine whether the amount of heat needed to increase the temperature of water a given amount is proportional to the mass of water.

3. Another Substance: Glycerine

We will do just one experiment, using a certain quantity of glycerine instead of water. The glycerine is already in a beaker, and we don't want to pour it out. So just weigh the beaker with the liquid in it, and assume the weight of the beaker is the same as the one you used for water. Heat the glycerine from room temperature to between 75 and 80o C. We will compare the heat needed to raise the temperature of glycerine with that required for water.

4. Analysis

Prepare a graph with time on the vertical axis and temperature on the horizontal axis. Choose the scales so that data for all three water experiments can be plotted on this one graph. First plot the points for the experiment with 300 ml of water. Use a ruler to draw the best straight line fit to the data. To the extent that the points are close to a straight line, this tells us that the amount of heat, Q, needed to change the temperature by an amount, DT, is proportional to DT. In symbols,

Q ~ DT,
where the symbol, ~, means "is proportional to".

Use the straight line (not the individual points), to find the time to raise the water temperature by 40o C.

Now plot the points for the 150 ml and 450 ml experiments. Use some method to make sure you can distinguish the points from the three different experiments. Again draw straight line fits to your data, and find the time to raise the temperature by 40o C.

Prepare a table showing the mass of water in the three experiments, and for each the time needed to raise the temperature by 40o C. Plot a separate graph of the data in this table, with time on the vertical axis and mass on the horizontal axis. Fit a straight line to these three points.

To the extent that this graph is linear, it shows that the heat needed to raise the temperature of water is proportional to mass. In symbols,

Q ~ m.

Equations (1) and (2) can be combined by writing,

Q ~ mDT.

Alternately, one can write this as an equation,

Q = cmDT,

where "c" is a constant. To say c is constant means that Eq. (4) determines the required heat for any mass, m, and temperature change, DT, in an experiment with water, since that's the only substance we have analyzed so far. With another substance, we expect a similar relation, but with a different constant, say, c'.

Now return to the glycerine data. First plot time vs. temperature for the glycerine data. (If it can be done clearly, do it on the same graph as the water data. Otherwise, use a separate graph.) Draw the best straight line fit to the data, and find the time for DT = 40o C. Add this to your table with the water data, including the measured mass of the glycerine. Also place this point on the graph which shows the DT = 40o time vs. mass, at the appropriate glycerine mass.

To compare the two substances, we would like to have experiments with the same mass. Since the glycerine mass may not match any of the three water values, we extend the line at the glycerine mass (see the Figure below) up to the line through the 3 water points. The ratio of these two coordinates, Yglyc/Ywater, measures the heat requirement for glycerine relative to water. It is the ratio of the constants used above at Equation (4), c'/c. The accepted value of this ratio is 0.6.

The discussion above avoids the question of defining a unit of heat. One way a heat unit is defined is as follows: One calorie is the heat needed to raise the temperature of one gram water by one degree. In this experiment we've assumed a fixed number of calories flow into the substance every minute, but we didn't determine how many calories that was. If we had measured calories instead of time units, Equation (1) would allow us to calculate c. As it is, we are able to calculate the ratio, c'/c, of glycerine compared to water. This is called the specific heat of glycerine. The specific heat of water, by definition, is 1.

It is worth noting that the specific heats of most substances are considerably less than 1. In other words, water is unusual in this respect. The large constant for water means that it takes a lot of heat to change the temperature of water. Water heats up less and less rapidly when it's in a hot environment (and correspondingly, cools off less in a cold environment), compared to other substances. This is why hot soup cools off more slowly than other hot food. It also explains why weather (temperature variations) is usually more moderate in places, like New York City, near large bodies of water.

Question: What flaws do you see in the assumption that the heat added to the water (or glycerine) is proportional to the time that the burner is on?