# Heat Energy and Temperature

In this module we need to be a little bit more precise about temperature and heat energy than we have been so far. Heat energy is usually measured in terms of calories. The calorie was originally defined as the amount of energy required to raise one gram of water one degree Celsius at a pressure of one atmosphere. This definition is not complete because the amount of energy required to raise one gram of water one degree Celsius varies with the original temperature of the water by as much as one percent. Since 1925 the calorie has been defined as 4.184 joules, the amount of energy required to raise the temperature of one gram of water from 14.5 degrees Celsius to 15.5 degrees Celsius. For our purposes here we can ignore the fact that the effect of one calorie of energy varies depending on the temperature of the water.

Newton's model of cooling can be thought of, more precisely, as involving two steps.

• Heat energy, measured in calories, flows across the boundary between two objects. The rate of heat flow is measured in calories per unit of time.

• The heat energy changes the temperature of each object. The change in temperature depends on the heat energy and the composition and mass of each object and is inversely proportional to the mass of each object. For example, if heat energy was flowing at the rate of A calories per hour into a container of water whose mass was m grams then the temperature of the water would rise at the rate of A / m degrees Celsius per hour.

The picture above shows a brick whose length is four centimeters. We mentally divide the brick into two unequal pieces. The lefthand piece has a length of one centimeter and the righthand piece has a length of three centimeters. Heat is flowing across the mental boundary between the two pieces from left to right at the rate of A calories per hour. As a result the average temperature of the lefthand piece is changing at the rate of -kA degrees Celsius per hour. The constant k depends on the composition of the brick and its cross-sectional area. The average temperature of the righthand piece is changing at the rate of kA / 3 degrees Celsius per hour. The three in the denominator comes from the fact that since the righthand piece is three times the length of the lefthand piece, its mass is three times as big.

Copyright c 1997 by Frank Wattenberg, Department of Mathematics, Montana State University, Bozeman, MT 59717