Measurement of Heat by Temperature

Author: John Hutchinson

We can define in a variety of ways a temperature scale which permits quantitative measurement of "how hot" an object is. Such scales are typically based on the expansion and contraction of materials, particularly of liquid mercury, or on variation of resistance in wires or thermocouples. Using such scales, we can easily show that heating an object causes its temperature to rise.

It is important, however, to distinguish between heat and temperature. These two concepts are not one and the same. To illustrate the difference, we begin by measuring the temperature rise produced by a given amount of heat, focusing on the temperature rise in 1000 g of water produced by burning 1.0 g of methane gas. We discover by performing this experiment repeatedly that the temperature of this quantity of water always rises by exactly 13.3 °C. Therefore, the same quantity of heat must always be produced by reaction of this quantity of methane.

If we burn 1.0 g of methane to heat 500 g of water instead, we observe a temperature rise of 26.6 °C. If we burn 1.0g of methane to heat 1000 g of iron, we observe a temperature rise of 123 °C. Therefore, the temperature rise observed is a function of the quantity of material heated as well as the nature of the material heated. Consequently, 13.3 °C is not an appropriate measure of this quantity of heat, since we cannot say that the burning of 1.0 g of methane "produces 13.3 °C of heat". Such a statement is clearly revealed to be nonsense, so the concepts of temperature and heat must be kept distinct.

Our observations do reveal that we can relate the temperature rise produced in a substance to a fixed quantity of heat, provided that we specify the type and amount of the substance. Therefore, we define a property for each substance, called the heat capacity, which relates the temperature rise to the quantity of heat absorbed. We define q to be the quantity of heat, and ΔT to be the temperature rise produced by this heat. The heat capacity C is defined by

q=CΔT

[1]

This equation, however, is only a definition and does not help us calculate either q or C, since we know neither one.

Next, however, we observe that we can also elevate the temperature of a substance mechanically, that is, by doing work on it. As simple examples, we can warm water by stirring it, or warm metal by rubbing or scraping it. (As an historical note, these observations were crucial in establishing that heat is equivalent to work in its effect on matter, demonstrating that heat is therefore a form of energy.) Although it is difficult to do, we can measure the amount of work required to elevate the temperature of 1g of water by 1 °C. We find that the amount of work required is invariably equal to 4.184 J. Consequently, adding 4.184 J of energy to 1 g of water must elevate the energy of the water molecules by an amount measured by 1 °C. By conservation of energy, the energy of the water molecules does not depend on how that energy was acquired. Therefore, the increase in energy measured by a 1 °C temperature increase is the same regardless of whether the water was heated or stirred. As such, 4.184 J must also be the amount of energy added to the water molecules when they are heated by 1 °C rather than stirred. We have therefore effectively measured the heat q required to elevate the temperature of 1g of water by 1 °C. Referring back to equation 1, we now can calculate that the heat capacity of 1g of water must be 4.184 J/°C. The heat capacity per gram of a substance is referred to as the specific heat of the substance, usually indicated by the symbol cs. The specific heat of water is 4.184 J/°C.

Determining the heat capacity (or specific heat) of water is an extremely important measurement for two reasons. First, from the heat capacity of water we can determine the heat capacity of any other substance very simply. Imagine taking a hot 5.0 g iron weight at 100 °C and placing it in 10.0 g of water at 25 °C. We know from experience that the iron bar will be cooled and the water will be heated until both have achieved the same temperature. This is an easy experiment to perform, and we find that the final temperature of the iron and water is 28.8 °C. Clearly, the temperature of the water has been raised by 3.8 °C. From equation 1 and the specific heat of water, we can calculate that the water must have absorbed an amount of heat q = (10.0 g)*(4.184J/g°C)*(3.8 °C) = 159 J. By conservation of energy, this must be the amount of heat lost by the 1g iron weight, whose temperature was lowered by 71.2 °C. Again referring to equation 1, we can calculate the specific heat of the iron bar to be c_s = -159J/(-71.2°C)/(5.0g) = 0.45 J/g°C. Following this procedure, we can easily produce extensive tables of heat capacities for many substances.

Second, and perhaps more importantly for our purposes, we can use the known specific heat of water to measure the heat released in any chemical reaction. To analyze a previous example, we observed that the combustion of 1.0g of methane gas released sufficient heat to increase the temperature of 1000g of water by 13.3°C. The heat capacity of 1000 g of water must be (1000g)*(4.184J/g°C) = 4184 J/°C. Therefore, by equation 1, elevating the temperature of 1000g of water by 13.3°C must require 55,650 J = 55.65 kJ of heat. Therefore, burning 1.0g of methane gas produces exactly 55.65 kJ of heat.

The method of measuring reaction energies by capturing the heat evolved in a water bath and measuring the temperature rise produced in that water bath is called calorimetry. This method is dependent on the equivalence of heat and work as transfers of energy, and on the law of conservation of energy. Following this procedure, we can straightforwardly measure the heat released or absorbed in any easily performed chemical reaction. For reactions which are difficult to initiate or which occur only under restricted conditions or which are exceedingly slow, we will require alternative methods.