Problem 36 - University Physics with Modern Physics

Question

BIO Treatment for a Stroke. One suggested treatment for a person who has suffered a stroke is immersion in an ice-water bath at 0\(^\circ\)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\(^\circ\)C. To treat a 70.0-kg patient, what is the minimum amount of ice (at 0°C) you need in the bath so that its temperature remains at 0°C? The specific heat of the human body is 3480 J/kg \(\cdot\) C\(^\circ\), and recall that normal body temperature is 37.0\(^\circ\)C.

Step-by-Step Solution

Answer

You need a minimum of 3.65 kg of ice.

Step 1: Determine the Heat Loss of the Patient

First, calculate how much heat the patient loses to cool from 37.0\(^\circ\)C to 32.0\(^\circ\)C. Use the formula for heat loss: \( Q = mc\Delta T \), where \( m = 70.0 \, \text{kg} \), \( c = 3480 \, \text{J/kg}\cdot\text{C}\degree \), and \( \Delta T = 37.0 - 32.0 \). So,\[ Q = 70.0 \times 3480 \times (37.0 - 32.0) \]\[ Q = 70.0 \times 3480 \times 5 \]\[ Q = 1,218,000 \, \text{J} \]

Step 2: Calculate the Heat Required to Melt Ice

For the temperature of the bath to remain at 0\(^\circ\)C, the heat lost by the patient must be absorbed by melting ice. The heat required to melt ice is given by \( Q = mL \), where \( m \) is the mass of the ice and \( L = 334,000 \, \text{J/kg} \) is the latent heat of fusion. Set the heat lost by the patient equal to the heat absorbed by the ice and solve for \( m \):\[ 1,218,000 = m \times 334,000 \]\[ m = \frac{1,218,000}{334,000} \]\[ m \approx 3.65 \, \text{kg} \]

Step 3: Conclude

We found that at least 3.65 kg of ice is needed to keep the temperature of the ice-water bath at 0\(^\circ\)C.

Key Concepts

Heat Transfer

Heat transfer is the process by which thermal energy moves from one object or substance to another. This process can occur in three different ways: conduction, convection, and radiation.
Conduction is the transfer of heat between substances that are in direct contact with each other, like how the patient loses heat to the surrounding ice-water bath.
Convection occurs in fluids, where warmer areas of a liquid or gas rise to cooler areas, transferring heat.
Radiation involves heat transfer through electromagnetic waves, like the warmth you feel from the sun. In the context of the problem, heat transfer is crucial to understanding how the patient's body temperature is lowered as heat energy is transferred from the patient's body to the ice in the water bath.
The rate and amount of heat transfer depend on the temperature difference between the substances, the surface area in contact, and the thermal properties of the materials involved, such as their specific heat capacities.

Specific Heat Capacity

Specific heat capacity is a property of substances that tells us how much heat is required to change the temperature of a unit mass of the substance by one degree Celsius.
It's denoted by the symbol "c" and commonly measured in units of J/kg°C.In the exercise, the specific heat capacity of the human body is given as 3480 J/kg°C.
This means that it takes 3480 joules of energy to increase or decrease the temperature of one kilogram of the human body by one degree Celsius.When cooling the patient, you can calculate the heat needed to lower their body temperature using the formula:

• \( Q = mc\Delta T \), where:
  • \( Q \) is the heat energy transferred,
  • \( m \) is the mass of the patient,
  • \( c \) is the specific heat capacity,
  • \( \Delta T \) is the change in temperature.

Understanding specific heat capacity helps us appreciate how much energy is involved in temperature changes, which is vital for processes like medical treatments or designing heating/cooling systems.

Latent Heat of Fusion

Latent heat of fusion is the amount of heat needed to change a substance from a solid to a liquid at its melting point, without changing its temperature.
This energy helps break the bonds that hold the molecules together in a solid, such as ice. It is measured in J/kg.In our exercise, the latent heat of fusion for ice is used to calculate how much ice is required to absorb the heat lost by the patient.
The formula used here is:

• \( Q = mL \), where:
  • \( Q \) is the heat absorbed during the phase change,
  • \( m \) is the mass of the ice,
  • \( L \) is the latent heat of fusion.

Applying this formula allows us to find the amount of ice needed to ensure that the bath remains at 0°C while the patient is treated.
Understanding latent heat of fusion is important in fields like meteorology and food industry where phase transitions play a key role, as well as in medical processes involving temperature control.