Problem 33 - College Physics

Question

One suggested treatment for a person who has suffered a stroke is to immerse the patient in an ice-water bath at \(0^{\circ} \mathrm{C}\) to lower the body temperature, which prevents damage to the brain. In one set of tests, patients were cooled until their internal temperature reached \(32.0^{\circ} \mathrm{C}\). To treat a \(70.0 \mathrm{~kg}\) patient, what is the minimum amount of ice (at \(0^{\circ} \mathrm{C}\) ) that you need in the bath so that its temperature remains at \(0^{\circ} \mathrm{C}\) ? The specific heat of the human body is \(3480 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right),\) and recall that normal body temperature is \(37.0^{\circ} \mathrm{C}.\)

Step-by-Step Solution

Answer

3.65 kg of ice is needed to lower the patient's temperature.

Step 1: Calculate Heat Lost by the Patient

To begin, we need to calculate the amount of heat the patient loses when cooled from their normal body temperature (\(37.0^{\circ} C\)) to the target temperature (\(32.0^{\circ} C\)).The heat lost, \( Q \), can be calculated using the formula:\[ Q = mc\Delta T \]where: - \( m = 70.0 \text{ kg} \) is the mass of the patient,- \( c = 3480 \text{ J/(kg°C)} \) is the specific heat capacity of the human body,- \( \Delta T = (32.0 - 37.0) \text{°C} = -5.0 \text{°C} \).Substituting these values, we find:\[ Q = 70.0 \times 3480 \times (-5.0) = -1,218,000 \text{ J} \]Since the heat lost by the patient will be gained by the ice, we take the absolute value: \( 1,218,000 \text{ J} \).

Step 2: Calculate the Mass of Ice Needed

Next, we need to find the mass of ice required to absorb this amount of heat, keeping in mind that the temperature remains at \(0^{\circ} C\). Ice absorbs heat without changing temperature when it melts, using its latent heat of fusion.The latent heat of fusion for ice is \( L = 334,000 \text{ J/kg} \).The mass of ice needed, \( m_{\text{ice}} \), can be found using:\[ Q = m_{\text{ice}} \times L \]Rearranging for \( m_{\text{ice}} \), we have:\[ m_{\text{ice}} = \frac{Q}{L} = \frac{1,218,000}{334,000} \approx 3.65 \text{ kg} \]Thus, approximately \(3.65\text{ kg}\) of ice is required.

Key Concepts

Specific Heat Capacity

The concept of specific heat capacity is fundamental in understanding how different materials respond to changes in temperature. It tells us how much heat is needed to change the temperature of a certain mass of a substance. Here's what you need to know:

• Specific heat capacity (\( c \)) is measured in joules per kilogram per degree Celsius (J/(kg°C)).
• For the human body, it's \( 3480 ext{ J/(kg°C)} \), indicating that it takes 3480 joules to raise the temperature of 1 kg of human body tissue by 1 °C.

This value plays a crucial role when determining the amount of heat the body loses or gains during temperature changes. In the exercise, cooling the body from 37 °C to 32 °C involves calculating the heat transfer using specific heat capacity, mass, and the temperature change. This helps establish how much the body's temperature can be altered without causing harm.

Latent Heat of Fusion

Latent heat of fusion is an important concept when a substance changes from solid to liquid, absorbing heat without changing its temperature. For water turning into ice or vice versa, this is highly applicable:

• Latent heat of fusion for ice is \( 334,000 ext{ J/kg} \).
• It indicates the amount of heat needed to melt 1 kg of ice without any change in temperature.

In the exercise, once the heat is removed from the patient, it is absorbed by the ice. The ice melts, requiring the latent heat of fusion. This calculation is essential to determine how much ice is needed to absorb the patient's heat and ensure the bath remains at freezing point. Thus, it facilitates decision-making about the mass of ice necessary in the therapeutic bath.

Temperature Change

Temperature change (\( \Delta T \)) involves the difference between the initial and final temperatures of a body. This change is pivotal in calculating heat transfer. Here's how it fits into our problem:

• The initial temperature of the patient is 37 °C.
• The final desired temperature is 32 °C.
• Thus, \( \Delta T \) = \( 32°C - 37°C = -5°C \)

The negative sign indicates cooling, meaning heat is being removed from the body. Understanding temperature change and how it relates to heat loss helps in determining the conditions under which the patient's body will reach the target temperature, ensuring the treatment’s effectiveness without causing discomfort or harm.

Heat Transfer

Heat transfer is the process through which heat moves from one body or substance to another. In this exercise, it's central for cooling the patient's temperature. Here’s how it functions:

• Heat energy always moves from a warmer substance to a cooler one.
• In the context of the exercise, heat is transferred from the warm body of the patient to the ice.
• The calculated heat loss of 1,218,000 J from the patient's body is absorbed by the ice.
• Through melting, the ice absorbs this energy, and it requires an adequate mass of ice to maintain the constant temperature of the bath.
Knowing how much heat needs to be transferred for a certain temperature change allows professionals to determine the exact conditions necessary for therapeutic practices like this. Heat transfer principles ensure that the change efficiently reaches the target internal body temperature.