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Chapter 17: Problem 36

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掳C) you need in the bath so that its temperature remains at 0掳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.

Short Answer

Expert verified
You need a minimum of 3.65 kg of ice.

Step by step solution

01

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,\[ Q = 70.0 \times 3480 \times (37.0 - 32.0) \]\[ Q = 70.0 \times 3480 \times 5 \]\[ Q = 1,218,000 \, \text{J} \]
02

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 \( m \):\[ 1,218,000 = m \times 334,000 \]\[ m = \frac{1,218,000}{334,000} \]\[ m \approx 3.65 \, \text{kg} \]
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳C.In the exercise, the specific heat capacity of the human body is given as 3480 J/kg掳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掳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.

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Most popular questions from this chapter

CP A person of mass 70.0 kg is sitting in the bathtub. The bathtub is 190.0 cm by 80.0 cm; before the person got in, the water was 24.0 cm deep. The water is at 37.0\(^\circ\)C. Suppose that the water were to cool down spontaneously to form ice at 0.0\(^\circ\)C, and that all the energy released was used to launch the hapless bather vertically into the air. How high would the bather go? (As you will see in Chapter 20, this event is allowed by energy conservation but is prohibited by the second law of thermodynamics.)

BIO Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 K. What would Celsius and Fahrenheit thermometers read for this temperature? (b) Elevated body temperature. During very vigorous exercise, the body鈥檚 temperature can go as high as 40\(^\circ\)C. What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about 7 C\(^\circ\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at 4.0\(^\circ\)C lasts safely for about 3 weeks, whereas blood stored at -160\(^\circ\)C lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body鈥檚 temperature is above 105\(^\circ\)F for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.

A technician measures the specific heat of an unidentified liquid by immersing an electrical resistor in it. Electrical energy is converted to heat transferred to the liquid for 120 s at a constant rate of 65.0 W. The mass of the liquid is 0.780 kg, and its temperature increases from 18.55\(^\circ\)C to 22.54\(^\circ\)C. (a) Find the average specific heat of the liquid in this temperature range. Assume that negligible heat is transferred to the container that holds the liquid and that no heat is lost to the surroundings. (b) Suppose that in this experiment heat transfer from the liquid to the container or surroundings cannot be ignored. Is the result calculated in part (a) an overestimate or an underestimate of the average specific heat? Explain.

A metal sphere with radius 3.20 cm is suspended in a large metal box with interior walls that are maintained at 30.0\(^\circ\)C. A small electric heater is embedded in the sphere. Heat energy must be supplied to the sphere at the rate of 0.660 J/s to maintain the sphere at a constant temperature of 41.0\(^\circ\)C. (a) What is the emissivity of the metal sphere? (b) What power input to the sphere is required to maintain it at 82.0\(^\circ\)C? What is the ratio of the power required for 82.0\(^\circ\)C to the power required for 41.0\(^\circ\)C? How does this ratio compare with 2\(^4\)? Explain.

In very cold weather a significant mechanism for heat loss by the human body is energy expended in warming the air taken into the lungs with each breath. (a) On a cold winter day when the temperature is -20\(^\circ\)C, what amount of heat is needed to warm to body temperature (37\(^\circ\)C) the 0.50 L of air exchanged with each breath? Assume that the specific heat of air is 1020 J / kg \(\cdot\) K and that 1.0 L of air has mass \(1.3 \times 10{^-}{^3} kg\). (b) How much heat is lost per hour if the respiration rate is 20 breaths per minute?
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