Problem 54 - University Physics with Modern Physics

Question

Evaporation of sweat is an important mechanism for temperature control in some warm-blooded animals. (a) What mass of water must evaporate from the skin of a 70.0-kg man to cool his body 1.00 \(\mathrm{C}^{\circ} ?\) The heat of vaporization of water at body temperature \(\left(37^{\circ} \mathrm{C}\right)\) is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .\) The specific heat of a typical human body is 3480 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) (see Exercise 17.31\() .\) (b) What volume of water must the man drink to replenish the evaporated water? Compare to the volume of a soft-drink can \(\left(355 \mathrm{cm}^{3}\right)\).

Step-by-Step Solution

Answer

Around 0.101 kg of water must evaporate; this is about 101 cm³, or 1/3 of a soft-drink can volume.

Step 1: Determine Heat Needed to Cool the Body

To calculate the heat needed to cool the body, we use the equation for heat transfer: \( Q = mc\Delta T \), where \( m \) is the mass, \( c \) is the specific heat, and \( \Delta T \) is the temperature change. Here, \( m = 70 \text{ kg} \), \( c = 3480 \text{ J/(kg}\cdot\text{K)} \), and \( \Delta T = 1.00 \underline{\phantom{xxx}} ^{\circ} \text{C} \). Thus, \( Q = 70 \times 3480 \times 1 = 243,600 \text{ J} \).

Step 2: Calculate Mass of Water Needed to Evaporate

Since vaporization requires energy, we use the equation \( Q = m_v L_v \), where \( m_v \) is the mass of water that must evaporate, and \( L_v \) is the latent heat of vaporization. From part (a), \( Q = 243,600 \text{ J} \) and \( L_v = 2.42 \times 10^6 \text{ J/kg} \). Solving for \( m_v \), we find \( m_v = \frac{243,600}{2.42 \times 10^6} \approx 0.101 \text{ kg} \).

Step 3: Determine Volume of Water Evaporated

To find the volume of the evaporated water, use the density of water \( \rho = 1000 \text{ kg/m}^3 \). The volume \( V \) is given by \( V = \frac{m_v}{\rho} \). So, \( V = \frac{0.101 \text{ kg}}{1000 \text{ kg/m}^3} = 0.000101 \text{ m}^3 \).

Step 4: Convert Volume to Compare with a Soft-Drink Can

Convert \( 0.000101 \text{ m}^3 \) to \( \text{cm}^3 \), knowing that \( 1 \text{ m}^3 = 10^6 \text{ cm}^3 \). Thus, \( 0.000101 \text{ m}^3 = 101 \text{ cm}^3 \). Now compare this to a typical soft-drink can volume of \( 355 \text{ cm}^3 \). It takes roughly 1/3 of a soft drink can to replenish the evaporated water.

Key Concepts

Evaporation

Evaporation is a fascinating natural process where a liquid turns into a gas. This phenomenon occurs when molecules at the surface of a liquid gain enough energy to break free from the liquid and enter the air. Think of a puddle on a sunny day slowly disappearing as water molecules transition to water vapor in the air.
Evaporation plays a crucial role in thermoregulation in warm-blooded animals. When we sweat, the water from our skin evaporates into the air, taking heat with it. This process helps cool our bodies.

• Evaporation occurs at the surface of the liquid.
• Molecules need energy to break intermolecular bonds.
• Evaporation can happen at any temperature.

This passive cooling mechanism is what keeps us comfortable on hot days.

Latent Heat of Vaporization

The latent heat of vaporization is the energy required to transform a unit mass of a liquid into a gas at a constant temperature. For water, this is a considerable energy amount, which is why boiling a pot of water takes some time.
At body temperature, the latent heat of vaporization for water is given as \(2.42 \times 10^6 \text{ J/kg}\). This value hints at how much energy is required to turn water from liquid sweat into vapor to cool our bodies.

• Latent heat "hides" energy during phase changes without raising temperature.
• High latent heat means effective cooling—excellent for thermoregulation.

In the human body, evaporating sweat utilizes this large energy transfer to effectively remove heat.

Specific Heat Capacity

Specific heat capacity refers to the amount of heat energy required to raise the temperature of one kilogram of a substance by one degree Celsius (or one Kelvin).
For the human body, specific heat capacity is estimated at \(3480 \text{ J/(kg} \cdot \text{K)}\). This indicates how much heat is needed to change body temperature, which tends to resist quick changes.

• High specific heat means more energy is needed for temperature change.
• Helps maintain stable internal body temperatures.
• Ensures body warming and cooling are controlled processes.

This concept is essential for understanding temperature regulation in biological entities.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. It can happen in several ways, including conduction, convection, and radiation.
In our exercise problem, heat transfer is crucial for calculating how much sweat must evaporate to cool the body. The formula \(Q = mc\Delta T\) helps us figure out the heat energy transferred as the temperature changes.

• Conduction transfers heat through direct contact.
• Convection involves heat moving through fluids (liquids and gases).
• Radiation transmits heat through electromagnetic waves.

Understanding heat transfer is vital for predicting temperature changes and strategizing ways to maintain or alter them efficiently.