# How to Calculate the Heat Gained by the Calorimeter ••• SubstanceP/iStock/GettyImages

At some point in your life, you have probably wondered what a calorie is after looking at a nutrition-information label for a given food. Other than something many people like to see lower numbers associated with when they scan such labels, what is a calorie?

And how do "calories" add mass to living systems, if this is in fact what happens? And how can you be sure the number of calories listed for a given item — be this value reassuring or depressing — has been accurately determined?

Heat is one of many properties of the ambient world that you can probably describe well in a few of your own well-chosen words, but it has a more focused meaning in the physical sciences. The calorie is a measure of heat, as is the joule (J) and the British thermal unit (btu). The study of heat exchange is a branch of physical science known as calorimetry, which in turn relies on devices called calorimeters.

Intuitively, you might find it odd that chilled or frozen foods like ice cream and cheesecake can pack a lot of what is supposedly heat into a small serving. Also, if calories somehow translate to heat, shouldn't foods that supply more of it actually lead to weight loss rather than added body mass?

These are good questions, and after you "burn" through the rest of this article, you'll have these answers and much more to take to your next calorimetry lab or sports-nutrition discussion.

## What Is Heat in Physics?

Heat can be thought of chiefly as thermal energy. Like other forms of energy, it has units of joules (or the equivalent in non-SI units). Heat is an elusive quantity in that it is difficult to measure directly. Instead, changes in temperature under controlled experimental conditions can be used to determine whether a system has gained or lost heat.

The fact that heat is treated as energy means that keeping track of it is a mathematically straightforward exercise, even if experiments sometimes make it difficult to establish conditions in which no heat energy escapes and eludes measurement. But because of fundamental realities such as the law of conservation of energy, heat tabulation is fairly simple in principle.

Materials have different levels of resistance to changing temperatures when a given amount of heat is added to a fixed amount of that substance. That is, if you took 1 kilogram of substance A and 1 kilogram of substance B and added the same amount of heat to each, with no heat permitted to leave either system, the temperature of A might increase by only one-fifth as much as the temperature of substance B does.

This would mean that substance A has a specific heat five times that of substance A, a concept to be explored in detail below.

## Units of Heat and the "Calorie"

The "calorie" listed on nutrition labels is in fact a kilocalorie, or kcal. So in reality, a typical can of sugared soda has about 120,000 calories, expressed by convention as a calorie in everyday communication.

• Calor is the Latin word for, appropriately enough, heat.

The calorie is equivalent to about 4.184 J, meaning that the kcal treated as a calorie on food labels is equal to 4,184 J or 4.184 kJ. The rate of energy expenditure (joules per second) in physical science is called power, and the SI unit is the watt (W), equal to 1 J/s. One kcal is therefore a sufficient amount of energy to power a system humming along at 0.35 to 0.4 kW (350 J/s) for about 12 seconds:

P = E/t, so t = E/P = 4.186 kJ/(0.35 kJ/s) = 12.0.

• A trained endurance athlete such as a bicyclist or runner is capable of maintaining such a power output over extended periods. In theory, then, a 100-"calorie" (100-kcal) energy drink could keep an Olympic road cyclist or marathon runner going for about 100 times 12 seconds, or 20 minutes. Because the human system is not nearly 100 percent mechanically efficient, it actually requires more than 300 kcal to operate at close to full aerobic capacity for this long.

The calorie is defined as the amount of heat required to increase the temperature of 1 gram of water by 1 degree Celsius. One problem with this is that there is a slight variation of the c of water with temperature across the range of temperatures at which H2O is a liquid. The "specific" in "specific heat" refers not only to specific materials but to a specific temperature.

• The specific heats of most materials are given at 20

°C or 25 °C.

## Heat Capacity and Specific Heat Defined

Technically, the terms "heat capacity" and "specific heat capacity" mean different things, even though you may see these used interchangeably in less-rigorous sources.

Heat capacity, when originally coined, referred simply to the amount of heat required to warm an entire object (which may be made of multiple materials) by a given amount. Specific heat capacity refers to the amount of heat needed to raise the temperature of 1 gram of a specific material by 1 degree Celsius or Kelvin (°C or K).

• While the Celsius and Kelvin temperature scales are not the same, they are different by a fixed amount, as °C + 273 = K where K cannot be negative. This means that a given numerical change in temperature in one scale produces the same magnitude of change in the other, unlike the case with Fahrenheit-Celsius interconversions.

Rather than shorten "specific heat capacity" to "heat capacity," instead use the term specific heat, as is the convention in reputable sources.

## What Is Calorimetry?

The purpose of a calorimeter is to capture the heat released in some process, such as an exothermic chemical reaction, that would otherwise be lost to the environment. When the temperature change of the system and the mass and specific heat of the calorimeter assembly are known, the amount of heat put into the system by the process can be determined. Examples are provided in a subsequent section.

A calorimeter can be built from a number of different materials, with the condition that they be insulating (i.e., not permissive of heat transfer; the term is also used in electromagnetism to refer to resistance against electrical charge transfer).

One common version can be made from a Styrofoam cup and a well-fitting lid. In this coffee-cup calorimeter, water is usually used as a solvent, and a thermometer and (if needed) stirring stick are fitted snugly through small holes in the lid of the cup.

## The Calorimetry Formula

The change in heat of a closed system (positive by definition in the case of a calorimeter) is given by the product of the mass of the system, the heat capacity of the calorimeter and the change in temperature of the system:

Q = mC∆T

Where:

• Q = heat evolved (equal to heat absorbed − heat released) in joules (J)
• m = mass in kilograms (kg)
• c = specific heat capacity in J/kg⋅°C (or J/kg⋅K)
• ∆T = temperature change in °C (or K)

The heat that is liberated from whatever exothermic (heat-releasing) chemical reaction occurs in the calorimeter would ordinarily disperse into the environment. This is a loss chalked up to a change in a thermodynamic quantity known as enthalpy that describes both the internal energy of the system and changes in the system's pressure-volume relationship. This heat is instead trapped between the solvent and the lid of the cup.

Earlier, the idea of conservation of energy was introduced. Because the heat entering the calorimeter must equal the heat liberated by the system within the calorimeter consisting of the reactants and products themselves, the sign of the heat change for this system is negative and has the same magnitude as the heat gained by the calorimeter.

The above and related statements assume that only no heat or negligible amounts of heat escape from the calorimeter. Heat moves from warmer to cooler areas when insulation is not present, so without proper insulation, heat will leave the calorimeter assembly for the ambient environment unless the environmental temperature is warmer than that of the calorimeter.

## Some Common Specific Heat Capacities

The following chart includes the specific heat in J/kg⋅°C of some commonly encountered elements and compounds.

• H2O, ice: 2.108
• H2O, water: 4.184
• H2O, water vapor: 2.062
• Methanol: 2.531
• Ethanol: 2.438
• Benzene: 1.745
• Carbon, graphite: 0.709
• Carbon, diamond: 0.509
• Aluminum: 0.897
• Iron: 0.449
• Copper: 0.385
• Gold: 0.129
• Mercury: 0.140

• Table salt (NaCl): 0.864

• Quartz: 0.742
• Calcite: 0.915

Note that water has an unusually large heat capacity. It is perhaps counterintuitive that a gram of water will warm by less than one-tenth as much as a gram of water given the same amount of added heat, but this is important to life around the planet.

Water makes up about three-fourths of your body, making you able to tolerate major swings in environmental temperature. More broadly, the oceans act as heat reservoirs to help stabilize temperatures worldwide.

## The Heat Capacity of a Calorimeter

Now you are ready for some calculations involving calorimeters.

Example 1: First, take the simple case of a gram of sodium hydroxide (NaOH) being dissolved in 50 mL of water at 25 °C. Take the heat capacity of water at this temperature to be 4.184 J/kg⋅°C and consider the 50 mL of water to have a mass of 50 grams, or 0.05 kg. If the temperature of the solution increases to 30.32 °C, how much heat is gained by the calorimeter?

You have Q = mc∆T = (0.05 kg)(4.184 kJ/kg⋅°C)(30.32 − 5.32 °C)

= 1.113 kJ or 1,113 J.

Example 2: Now consider the case of a home solar energy storage unit, a device becoming more popular over time. Assume this device uses 400 L of water for storing thermal energy.

On a clear summer day, the initial temperature of the water is 23.0 °C. During the course of the day, the temperature of the water rises to 39.0 °C as it circulates through the "water wall" of the unit. How much energy has been stored in the water?

Again, assume the mass of water is 400 kg, that is, that the density of water can be considered to be exactly 1.0 within this temperature range (this is a simplification).

The equation of interest this time is:

Q = mc∆T = (400 kg)(4.184 kJ/kg⋅°C)(39 °C − 23 °C)

= 26,778 J = 26.78 kJ.

This is enough energy to power a 1.5-kW space heater for about 17 seconds:

(26.78 kJ)(kW/(kJ/s)/(1.5 kW) = 17.85 s

Most likely, the homeowners have a different use planned for it if they live in a solar house.

## Calorimetry Calculator

You can use online calculators that allow you to convert easily between units of specific heat, including unusual but not completely extinct units such as Btu/lbmoF.

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