Problem 33 - College Physics

Question

\(\cdot\) Treatment for a stroke. 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 capacity 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 required.

Step 1: Determine Temperature Change

To calculate the amount of ice needed, first find the temperature change the patient's body undergoes. The patient's initial temperature is 37.0°C, and their final temperature is 32.0°C. Therefore, the temperature change is given by:\[ \Delta T = 37.0^{\circ} \mathrm{C} - 32.0^{\circ} \mathrm{C} = 5.0^{\circ} \mathrm{C} \]

Step 2: Calculate Heat Loss from Body

Next, calculate the amount of heat the patient's body loses as it cools down. Use the formula:\[ Q = mc\Delta T \]where:- \( m = 70.0 \, \mathrm{kg} \) is the mass of the patient,- \( c = 3480 \, \mathrm{J}/(\mathrm{kg} \cdot \mathrm{C}^{\circ}) \) is the specific heat capacity,- \( \Delta T = 5.0^{\circ} \mathrm{C} \) is the temperature change.Substitute the known values into the equation:\[ Q = 70.0 \, \mathrm{kg} \times 3480 \, \mathrm{J}/(\mathrm{kg} \cdot \mathrm{C}^{\circ}) \times 5.0^{\circ} \mathrm{C} \]\[ Q = 1218000 \, \mathrm{J} \]So, the body loses 1,218,000 joules of heat.

Step 3: Calculate Mass of Ice Required

To find the mass of ice needed to absorb this amount of energy, use the formula for the heat absorbed by ice melting:\[ Q = m_f L_f \]where:- \( m_f \) is the mass of ice required,- \( L_f = 334,000 \, \mathrm{J/kg} \) is the latent heat of fusion for ice.Since the heat lost by the body is equal to the heat absorbed by the melting ice:\[ 1218000 \, \mathrm{J} = m_f \times 334,000 \, \mathrm{J/kg} \]Solve for \( m_f \):\[ m_f = \frac{1218000 \, \mathrm{J}}{334,000 \, \mathrm{J/kg}} \approx 3.65 \, \mathrm{kg} \]Thus, a minimum of 3.65 kg of ice is required to maintain the bath at 0°C while cooling the patient.

Key Concepts

Specific Heat Capacity

Specific heat capacity is a crucial concept in thermodynamics, especially when dealing with changes in temperature. It tells us how much heat energy is needed to change the temperature of a substance by a certain amount. For the human body, the specific heat capacity is noted as 3480 J/(kg°C). This means that to increase or decrease 1 kg of body mass by 1°C, 3480 joules of energy is necessary.
This characteristic allows us to understand how much energy it takes to cool or warm the human body during medical treatments such as in the exercise where a patient's body is cooled down. Knowing the specific heat capacity allows us to calculate the amount of energy loss or gain accurately, ensuring effective temperature regulation.

Latent Heat of Fusion

Latent heat of fusion is another vital parameter in thermodynamic calculations, particularly when a substance changes state from solid to liquid without changing its temperature. This is the hidden or 'latent' energy required to melt a solid without altering its temperature. In this context, ice (at 0°C) absorbs heat and melts, a process governed by its latent heat of fusion. For ice, this value is 334,000 J/kg, meaning that 334,000 joules are required to melt 1 kg of ice. Understanding latent heat is key to determining how much ice is necessary to absorb heat and help regulate temperature in treatments like ice baths. By calculating the energy absorbed by melting ice, we ensure the patient's body temperature reaches the desired level effectively.

Temperature Change

Temperature change is a fundamental aspect in understanding how heat energy affects the body or any material. It refers to the difference in initial and final temperatures. In these exercises, it assists in quantifying how much heat energy changes occur. For example, if a stroke patient is initially at 37°C and lowers to 32°C, the temperature change is the difference of 5°C. This simple subtraction helps us track how the total energy transfer occurs, based on the patient's body mass and its specific heat capacity. Understanding the relationship between temperature changes and energy requirements is essential for precise controlled adjustments, as seen in therapeutic scenarios like cooling therapies.

Heat Transfer

Heat transfer is the movement of thermal energy from one object to another, driven by temperature differences. This pivotal thermodynamics concept helps us understand how energy changes between different states and materials. When placing a patient in an ice bath, heat transfer takes place from the warmer human body to the colder water and ice. The amount of heat transferred can be calculated and is manifested as the energy loss from the patient's body. This concept helps us determine how much ice is required by equating the heat lost to the body with the heat gained by the ice, either through temperature change or melting. Mastering heat transfer principles allows us to perform accurate calculations for how solutions like cooling a patient's body to prevent further damage post-stroke work effectively.