Problem 40 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

3.65 kg of ice is needed.

Step 1: Find the Heat Loss Required

First, calculate the total amount of heat that needs to be removed from the patient's body to lower its temperature from normal body temperature to 32.0°C. Use the formula for heat loss: \[ Q = mc\Delta T \]where:- \( m \) is the mass of the patient: 70.0 kg,- \( c \) is the specific heat capacity of the human body: 3480 J/kg⋅°C,- \( \Delta T \) is the change in temperature: \( 37.0^{\circ}C - 32.0^{\circ}C = 5.0^{\circ}C \).Substituting these values, we get:\[ Q = 70.0 \times 3480 \times 5.0 = 1,218,000 \text{ J} \]

Step 2: Determine the Ice Required

Next, calculate the mass of ice required to absorb this amount of heat. The energy absorbed by melting the ice must equal the heat loss from the patient. The latent heat of fusion of ice is 334,000 J/kg.Using the formula:\[ Q = m_l L_f \]where:- \( m_l \) is the mass of the ice needed,- \( L_f \) is the latent heat of fusion of ice: 334,000 J/kg,Rearranging gives:\[ m_l = \frac{Q}{L_f} = \frac{1,218,000}{334,000} \approx 3.65 \text{ kg} \]

Step 3: Conclusion

The minimum amount of ice needed to keep the bath at 0°C while cooling the patient to the necessary temperature is approximately 3.65 kg.

Key Concepts

Heat Transfer

In the world of thermodynamics, heat transfer plays a crucial role. It is the process by which thermal energy moves from one object to another. This can occur through several modes, such as conduction, convection, and radiation.
In biological systems, especially in medical scenarios like treating stroke patients, controlling heat transfer is vital. When patients are immersed in an ice-water bath, the primary mode of heat transfer is conduction. Here, the patient’s body, at a warmer temperature, transfers heat to the ice bath at a lower temperature.

• This heat transfer continues until the patient's body temperature decreases to meet treatment goals.
• During this process, the heat from the patient's body is used to melt ice, which in turn keeps the bath at a constant temperature of 0°C.

Understanding how heat is transferred is vital for effective medical treatments that involve temperature management. This knowledge ensures that the process is controlled and effective without any complications.

Specific Heat Capacity

Specific heat capacity is a fundamental concept in thermodynamics that measures how much heat a substance can store per unit mass for a temperature change of one degree Celsius. For the human body, it is approximately 3480 J/kg·°C.
This value is relatively high, indicating that our bodies can absorb or lose substantial amounts of heat energy with only a small change in temperature. In the treatment of the stroke mentioned, this concept explains why a significant amount of heat must be extracted from the body to lower its temperature by a few degrees.

• High specific heat means the body holds onto heat more before changing temperature, akin to a thermal reservoir.
• This property demands more energy to achieve the necessary temperature drop, hence why calculations are needed to determine the exact energy or heat to be removed.

The specific heat capacity is essential to ensure precise adjustments in body temperature, especially in critical medical procedures.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy required to change a substance from solid to liquid without changing its temperature. For ice, this value is 334,000 J/kg.
In the context of treating a stroke patient, this concept helps calculate the amount of ice required to absorb sufficient heat from the patient in the ice bath, ensuring the water remains at 0°C.

• Even as heat is absorbed by the ice, its temperature remains constant until it completely melts.
• This energy absorbed during melting is called latent heat because it doesn’t raise the temperature but contributes to a change of state from solid to liquid.

This process is crucial in medical applications where precise thermal management is necessary to prevent further health complications.

Thermoregulation

Thermoregulation is the body's ability to maintain its internal temperature within a healthy range despite external temperature changes. It is a complex system involving the brain, skin, and various physiological processes.
During the stroke treatment, the body’s thermoregulation is overridden by immersing it in an ice-water bath. This external method helps rapidly decrease the body's core temperature to prevent brain damage due to the stroke.

• Normally, the body strives to maintain a temperature around 37°C, but in emergencies, external cooling can help bring it down safely.
• The aim is to lower the temperature to a specific target (e.g., 32°C) swiftly and efficiently, using tools like ice baths that bypass natural regulatory mechanisms.

This controlled decrease in internal temperature is sometimes necessary to provide critical intervention in cases where natural processes could be detrimental.