Question

The label on a soft drink states that 12 fl. oz \((355 \mathrm{~g})\) provides \(150 \mathrm{kcal}\). The drink is cooled to \(10.0^{\circ} \mathrm{C}\) before it is consumed. It then reaches body temperature of \(37^{\circ} \mathrm{C} .\) Find the net energy content of the drink. (Hint: You can treat the soft drink as being identical to water in terms of heat capacity.)

Short answer

Answer: The net energy content of the soft drink is approximately 595,758.7 J.

Step-by-step solution

Determine the energy content of the drink in Joules

First, let's convert the given energy content from kilocalories to Joules. We know that 1 kcal is equal to 4184 J, so we have: $$ 150 ~\text{kcal} \times \frac{4184 ~\text{J}}{1 ~\text{kcal}} = 627600 ~\text{J} $$

Calculate the energy required to heat the drink

To find the energy required to raise the temperature of the drink from 10°C to 37°C, we will use the specific heat formula: $$ Q = mc\Delta T $$ where \(Q\) is the energy needed, \(m\) is the mass of the soft drink, \(c\) is the specific heat of water, and \(\Delta T\) is the change in temperature. Since we assume that the heat capacity of the soft drink is similar to that of water, we can use the specific heat of water: \(c = 4.18 \frac{\mathrm{J}}{\mathrm{g \cdot K}}\). The mass of the drink, \(m\), is given as 355 g, and the change in temperature, \(\Delta T\), is equal to \((37 - 10)^\circ \mathrm{C} = 27^\circ \mathrm{C}\). Plugging these values into the formula, we get: $$ Q = (355 ~\text{g}) \times (4.18 ~\frac{\text{J}}{\text{g}\cdot \text{K}}) \times (27 ~\text{K}) = 4.18 \times 355 \times 27 ~\text{J} = 31841.3 ~\text{J} $$

Find the net energy content of the drink

Now, we subtract the energy required to heat the drink from the initial energy content to find the net energy content: $$ \text{Net Energy Content} = 627600 ~\text{J} - 31841.3 ~\text{J} = 595758.7 ~\text{J} $$ The net energy content of the soft drink is approximately \({595758.7}\mathrm{~J}\).

Key concepts

Specific Heat

Specific heat is a property of a substance that measures the energy required to raise the temperature of one gram of the substance by one degree Celsius (or one Kelvin). It is denoted by the symbol \(c\), and its units are typically expressed in joules per gram per degree Celsius \(\frac{J}{g\cdot^\circ C}\) or joules per gram per Kelvin \(\frac{J}{g\cdot K}\). For water, which is often used as a standard due to its high specific heat, the value is approximately \(4.18 \frac{J}{g\cdot K}\).

Understanding specific heat is crucial because it explains why different substances respond differently to heating. For example, metals, with their low specific heat values, warm up and cool down quickly, while water, with its higher specific heat, requires more energy to change its temperature.

Temperature Change

Temperature change, denoted by the symbol \(\Delta T\), represents the difference between the final and initial temperatures of a substance when it is heated or cooled. Measured in degrees Celsius \(^\circ C\) or Kelvin \(K\), temperature change is a critical factor in calculations involving heat transfer. It is an intrinsic part of the heat equation \(Q = mc\Delta T\), linking how much heat energy \(Q\) is necessary to change the temperature of a given mass \(m\) of a substance by a specific amount.

For the soft drink from the exercise, the temperature change from being cooled to consumption is from \(10.0^\circ C\) to \(37^\circ C\), which is a \(\Delta T\) of \(27^\circ C\) or \(27 K\), since the size of the degree is the same in both scales.

Heat Capacity

Heat capacity is the amount of heat energy required to raise the temperature of a given quantity of a substance by one degree. Unlike specific heat, which is an intensive property referring to the heat required for one gram of a substance, heat capacity is an extensive property; it depends on the amount of the substance present. Heat capacity is generally denoted by the symbol \(C\), and its unit of measurement is joules per degree Celsius \(J/{}^\circ C\) or joules per Kelvin \(J/K\).

The concept of heat capacity becomes important in practical applications involving thermal energy storage and temperature regulation. Specifically, in the case of the soft drink in the exercise, we calculated how much energy it takes to raise its entire mass to a higher temperature, reflecting its overall heat capacity.

Calorie to Joule Conversion

Converting calories to joules is a necessary step in many thermodynamics calculations, as different systems and regions use different units of energy. One calorie (cal) is traditionally defined as the amount of energy required to raise the temperature of one gram of water by one degree Celsius. However, for scientific purposes, specially nutritional labeling, the kilocalorie (kcal) is commonly used, which is equivalent to 1000 standard calories. One kilocalorie is equal to 4184 joules (J), making it a straightforward conversion: \(1 kcal = 4184 J\).

In dietary contexts, the term 'calorie' often refers to a kilocalorie, adding to some confusion. For our exercise involving the soft drink, the energy content listed on the label is given in kilocalories, which we converted to joules to calculate the net energy content when taking into account the temperature change of the drink.