 One accomplishment of early thermal science was the development of the concept of heat capacity and the related idea of specific heat. These investigations showed that temperature and quantity of heat, while often confused previously, are actually not the same phenomenon. This became clear in Fahrenheit's mixing experiments. He observed temperature changes when different substances were mixed. For example, a volume V1 of water at temperature T1, mixed with a volume V2 of mercury at temperature T2, produces a mixture with volume V and temperature T. The final volume is, to a good approximation, equal to the sum of the two initial volumes, but the temperature is not a weighted sum of the initial temperatures with volumes as weighting factors. Because mercury is so much denser than water, you could argue that the volumes are not useful measures of the relative amounts of material. Instead, you might use masses as weighting factors, but this also gives incorrect results for the final temperature. The correct weighting factor for an object is called its heat capacity. Suppose Q calories of heat are added to a body, and we observe a temperature change delta T. Recall that one calorie is the heat required to raise the temperature of one gram of water by one degree Celsius. Then the body's heat capacity, denoted by capital C, is the ratio of Q to delta T, with units of calories per degree Celsius. When two bodies are combined and reach thermal equilibrium, the resulting heat capacity is the sum of the individual heat capacities, and the equilibrium temperature is the weighted sum of the individual temperatures with the heat capacities as weighting factors. If we know one of the heat capacities, and we measure the three temperatures, then we can use this last equation to solve for the unknown heat capacity. Suppose the first body is water with heat capacity C1 and temperature T1. The heat capacity of water is, by definition, equal to its mass in grams. Since the calorie is, by definition, the heat required to raise one gram of water by one degree. If the water is the hotter of the two bodies, then in cooling to the equilibrium temperature T, it will give up a heat Q equals C1 times its temperature change, T1 minus T. This heat will be absorbed by the second body, causing it to increase in temperature by T minus T2 degrees. Therefore, the heat capacity of the second body is the ratio of Q to this temperature change. The heat capacity of an object depends both on the material it's made of and its size. The specific heat capacity, or specific heat, characterizes the material independent of the object's size. A body with mass M and heat capacity C has specific heat, denoted by lowercase C, equal to the heat capacity divided by the mass. The units are calories per gram degrees Celsius. For a particular object, the heat capacity is its mass times the specific heat of the material it's made of. Specific heat is an inherent property of a material. Values for a few substances are shown here. The specific heat of water, by definition, is one calorie per gram degrees Celsius. For Mercury, the value is about one-thirtieth of this. This illustrates the difference between temperature and quantity of heat. Given equal masses of water in Mercury, raising their temperatures by an equal amount takes about 30 times as much heat for the water than it does for the Mercury. We'll consider the reason for this difference in a future video. Let's do a bit of kitchen science to illustrate specific heat. We start with 202 grams of nails in a foam cup. In what follows, we assume the foam is a good enough insulator that we can neglect heat loss to the environment. The nails are at room temperature, which we measure as 19.6 degrees. We take some warm tap water at 29.3 degrees and pour 104 grams of it over our nails. Then we cover the mixture and let it come to thermal equilibrium. The final temperature is 27.3 degrees. The heat capacity of the nails is the heat they absorb divided by their temperature change. The absorbed heat equals the heat given up by the warm water. The heat lost by the water is its heat capacity, numerically equal to its 104 gram mass, times its temperature change, 29.3 minus 27.3. The nails temperature change is 27.3 minus 19.6. This gives a heat capacity of 27 calories per degree Celsius. Dividing this by the nails 202 gram mass, we find a specific heat of 0.134 calories per gram degree Celsius. So far, we've qualified heat using calories. The amount of heat needed to raise the temperature of 1 gram of water by 1 degree Celsius. The calorie is related to our mechanical concepts of energy and work by the so-called mechanical equivalent of heat. This was established experimentally by James Joule, after whom the unit of energy, the Joule, is named. He presented his experimental results in 1849. The concept of the mechanical equivalent of heat underlies the so-called first law of thermodynamics, which we'll consider in a future video. The idea that mechanical work, a force acting through a displacement, can be converted to heat is obvious, given the heating caused by friction. More subtle is the implication that heat can be converted into mechanical work. If heat and mechanical work are just two forms of energy, then converting between them should be simply a matter of identifying an appropriate physical mechanism. The working principle of Joule's apparatus is shown in this illustration from his publication. And in this illustration from an 1869 magazine article. The heart of the apparatus is a water filled chamber, through its top is a shaft with attached paddles. When the shaft turns, the paddles push against the water. Here's a photograph of the actual device. Outside the chamber, strings are wound around the shaft, and connected by pulleys to weights. Initially, a coupling connecting the interior and exterior parts of the shaft is disconnected, and the exterior shaft is turned with an attached crank. This raises the weights without turning the paddles. The energy stored in each is simply their weight times the height to which they are raised. Then the shaft coupling is connected, and the weights are allowed to fall. In doing so, they turn the paddles, which do work against the water. This transfers the known mechanical energy stored in the raised weights to the water. The transferred energy heats the water. A thermometer measures the temperature increase. Joule knew the heat capacity of the water plus the copper chamber mechanism, so he could convert the temperature increase into the number of calories of added heat. Comparing this number of calories to the known mechanical energy stored in the weights, he was able to determine the mechanical energy equivalent of heat energy. The modern value is one calorie of heat is equivalent to 4.184 joules of energy. To be precise, this is the value of the so-called thermochemical calorie. The heat capacity of water actually varies slightly with temperature. This means that if the calorie is defined as the energy required to raise the temperature of one gram of water by one degree Celsius, its value in joules will change slightly with water temperature. With the ability to measure temperature, an understanding of the relation between transfer of heat and temperature change, and the ability to quantify heat in terms of mechanical and other forms of energy, we are in a position in future videos to develop a rigorous theory of thermodynamics.