Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 17: Temperature and Heat

Problem 36

Chapter 17, Problem 36

One suggested treatment for a person who has suffered a stroke is immersion in an ice-water bath at 0°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°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 C°, and recall that normal body temperature is 37.0°C.

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Determine the amount of heat that needs to be removed from the patient's body to lower the temperature from 37.0°C to 32.0°C. Use the formula for heat transfer: \( Q = mc\Delta T \), where \( m \) is the mass of the patient, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

Calculate the change in temperature \( \Delta T \) as \( 37.0°C - 32.0°C = 5.0°C \).

Substitute the known values into the heat transfer formula: \( Q = (70.0 \text{ kg})(3480 \text{ J/kg°C})(5.0°C) \). This will give you the total heat \( Q \) that needs to be removed from the patient's body.

To find the minimum amount of ice required, use the formula for the heat absorbed by the ice: \( Q = mL_f \), where \( m \) is the mass of the ice and \( L_f \) is the latent heat of fusion for ice, which is 334,000 J/kg.

Set the heat removed from the body equal to the heat absorbed by the ice: \( (70.0 \text{ kg})(3480 \text{ J/kg°C})(5.0°C) = m(334,000 \text{ J/kg}) \). Solve for \( m \) to find the minimum mass of ice needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It is crucial in this problem to calculate the heat loss needed to lower the patient's body temperature from 37.0°C to 32.0°C. The specific heat capacity of the human body is given as 3480 J/kg°C, which will be used to determine the total heat energy that must be removed.

Heat Transfer

Heat transfer is the process of thermal energy moving from a hotter object to a cooler one. In this scenario, heat will transfer from the patient's body to the ice-water bath. Understanding this concept is essential to calculate how much ice is needed to absorb the heat lost by the patient, ensuring the bath remains at 0°C.

Latent Heat of Fusion

The latent heat of fusion is the amount of heat required to change a unit mass of a solid into a liquid without a temperature change. For ice, this is the energy needed to melt it at 0°C. This concept is vital to determine how much ice must melt to absorb the heat from the patient, ensuring the bath's temperature remains constant at 0°C.

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