Problem 88 - Chemistry and Chemical Reactivity

Question

You drink \(350 \mathrm{mL}\) of diet soda that is at a temperature of \(5^{\circ} \mathrm{C}\) (a) How much energy will your body expend to raise the temperature of this liquid to body temperature \(\left(37^{\circ} \mathrm{C}\right) ?\) Assume that the density and specific heat capacity of diet soda are the same as for water. (b) Compare the value in part (a) with the caloric content of the beverage. (The label says that it has a caloric content of 1 Calorie.) What is the net energy change in your body resulting from drinking this beverage? (1 Calorie = \(1000 \mathrm{kcal}=4184 \mathrm{J} .)\) (c) Carry out a comparison similar to that in part (b) for a nondiet beverage whose label indicates a caloric content of 240 Calories.

Step-by-Step Solution

Answer

To heat soda to body temperature costs 46,928 J. Diet soda nets -42,744 J; non-diet nets +957,232 J.

Step 1: Identify Known Values

First, note the known information: - Volume of diet soda: 350 mL - Initial temperature of soda: 5°C - Final (body) temperature: 37°C - Density of water: 1 g/mL (equating mL to grams) - Specific heat capacity of water: 4.18 J/g°C.

Step 2: Calculate Mass of Soda

Since the density of soda is the same as water, the mass of 350 mL of soda is 350 grams (since density = 1 g/mL).

Step 3: Calculate Energy to Heat Soda (a)

Use the formula for heat energy:\[ Q = mc\Delta T \]Where:\( m = 350 \) g (mass of diet soda), \( c = 4.18 \) J/g°C (specific heat capacity), \( \Delta T = 37°C - 5°C = 32°C \) (change in temperature).Substitute to find \( Q \):\[ Q = 350 \times 4.18 \times 32 = 46,928 \text{ Joules} \].

Step 4: Compare Energy with Caloric Content (b)

1 dietary Calorie = 4184 Joules. The caloric content of diet soda is 1 Calorie or 4184 Joules. Energy to heat soda: 46,928 Joules. Net energy = Energy from soda - Energy expended = 4184 - 46,928 = -42,744 Joules.

Step 5: Calculate Net Energy for Non-diet Beverage (c)

For non-diet soda with 240 Calories:Convert Calories to Joules: \( 240 \times 4184 = 1,004,160 \text{ Joules} \).Net energy = Energy from non-diet soda - Energy expended = 1,004,160 - 46,928 = 957,232 Joules.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a property of a material that tells us how much energy is needed to raise the temperature of one gram of the substance by one degree Celsius. It is a measure of how well a substance can absorb and store heat. In equations, it is commonly denoted as \( c \). For water, the specific heat capacity is \( 4.18 \text{ J/g°C} \). This means that to heat one gram of water by one degree Celsius, 4.18 Joules of energy is required.

Understanding specific heat capacity is crucial in calorimetry, which involves measuring the energy changes in physical and chemical processes. In this exercise, the diet soda's specific heat capacity is considered the same as that of water, allowing us to use the known value to calculate the energy needed to change the soda's temperature. This demonstrates how specific heat capacity directly influences the amount of energy required for temperature changes.

Energy Expended

Energy expended, in this context, refers to the amount of energy needed to raise the temperature of the diet soda from \( 5^{\circ} \text{C} \) to body temperature \( 37^{\circ} \text{C} \).

To find this energy (\( Q \)), we use the formula:- \[ Q = mc\Delta T \]

Where:- \( m \) is the mass of the soda, \( 350 \text{g} \)- \( c \) is the specific heat capacity, \( 4.18 \text{ J/g°C} \)- \( \Delta T \) is the temperature change, \( 32^{\circ} \text{C} \) (\( 37^{\circ} - 5^{\circ}\))Substituting into the equation gives us \( 46,928 \text{ Joules} \). This signifies the amount of energy derived from the body's resources to warm the drink to body temperature. Such calculations help in understanding how our bodies metabolize different substances energetically.

Temperature Change

Temperature change is a fundamental part of this exercise. The temperature of the diet soda changes from its initial temperature to match the body's temperature, which is \( 37^{\circ} \text{C} \).

The actual temperature change, \( \Delta T \), can be found as:- \( \Delta T = T_{\text{final}} - T_{\text{initial}} \)- In this case, \( \Delta T = 37ºC - 5ºC = 32ºC \).The temperature change plays a crucial role in calculating energy expenditure. A larger temperature change would require more energy. Moreover, the energy requirement does not depend on the initial and final temperatures directly but rather on how significantly these temperatures differ. Each degree of temperature change involves a set amount of energy defined by the specific heat capacity.

Caloric Content

Caloric content is an important measure when evaluating the energy supplied by food and drink compared to the energy expended. It is often measured in Calories (with a capital C), where 1 Calorie equals 4,184 Joules.

In this problem, the diet soda has a caloric content of 1 Calorie, translating to 4,184 Joules of energy. When compared to the energy the body expends to warm the drink – 46,928 Joules – the net energy change is negative. This suggests that more energy is expended than consumed, an important consideration in diet and metabolism.

For the non-diet drink with 240 Calories, the total caloric content is 1,004,160 Joules. Our calculations reveal a positive net energy, as significantly more energy is taken in than expended. Understanding caloric content helps make informed dietary choices, highlighting the concept of energy balance in nutrition.