Problem 46 - University Physics with Modern Physics

Question

Before going in for his annual physical, a \(70.0-\mathrm{kg}\) man whose body temperature is \(37.0^{\circ} \mathrm{C}\) consumes an entire \(0.355-\mathrm{L}\) can of a soft drink (mostly water) at \(12.0^{\circ} \mathrm{C}\) . (a) What will his body temperature be after equilibrium is attained? Ignore any heating bythe man's metabolism. The specific heat of the man's body is 3480 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) . (b) Is the change in his body temperature great enough to be measured by a medical themometer?

Step-by-Step Solution

Answer

(a) The final temperature is approximately 36.8°C. (b) Yes, the change is measurable.

Step 1: Understand the Problem

We need to find the final equilibrium body temperature of the man after he consumes the cold drink. We will treat the man's body and the drink as a closed system where heat exchange occurs.

Step 2: Identify Known Values

The key values given are:- Mass of the man, \( m_1 = 70.0 \text{ kg} \)- Initial body temperature, \( T_{1i} = 37.0^{\circ} \text{C} \)- Volume of the drink, \( V_2 = 0.355 \text{ L} \) (which is approximately \(355 \text{ g}\) or \(0.355 \text{ kg}\) for water)- Initial temperature of the drink, \( T_{2i} = 12.0^{\circ} \text{C} \)- Specific heat of the body, \( c_1 = 3480 \text{ J/kg} \cdot \text{K} \)- Specific heat of water, \( c_2 = 4190 \text{ J/kg} \cdot \text{K} \)We need to find the final temperature \( T_f \) of the combined system at equilibrium.

Step 3: Set Up the Heat Exchange Equation

Assuming no heat loss to the environment, the heat lost by the man's body (hot system) will equal the heat gained by the drink (cold system):\[ m_1 \cdot c_1 \cdot (T_f - T_{1i}) + m_2 \cdot c_2 \cdot (T_f - T_{2i}) = 0 \]

Step 4: Substitute Known Values and Solve for Final Temperature

Substitute the known values into the heat exchange equation:\[ (70.0 \times 3480 \times (T_f - 37.0)) + (0.355 \times 4190 \times (T_f - 12.0)) = 0 \]Simplify the equation:\[ 243600 \times (T_f - 37.0) + 1489.45 \times (T_f - 12.0) = 0 \]Distribute and combine like terms:\[ 243600T_f - 9013200 + 1489.45T_f - 17873.4 = 0 \]Combine terms:\[ 245089.45T_f = 9031073.4 \]Finally, solve for \( T_f \):\[ T_f = \frac{9031073.4}{245089.45} \approx 36.8^{\circ} \text{C} \]

Step 5: Determine if the Change is Measurable

The initial body temperature was \( 37.0^{\circ} \text{C} \) and the final equilibrium temperature is approximately \( 36.8^{\circ} \text{C} \). This represents a change of \( 0.2^{\circ} \text{C} \). Medical thermometers typically measure changes of \( 0.1^{\circ} \text{C} \) or greater, so yes, this change is measurable.

Key Concepts

Heat Transfer

In the realm of thermodynamics, heat transfer is a fundamental concept that refers to the transfer of thermal energy between different bodies or systems due to a temperature difference. In our example, when the man drinks the colder soft drink, heat flows from his warmer body to the colder drink until both reach the same temperature. This is because heat always transfers from the hotter object to the cooler one to achieve thermal equilibrium.

Heat transfer can occur in several ways, but in this scenario, it happens primarily through:

• Conduction: Direct transfer of heat between the man's body and the drink across their boundary.

The rate at which heat is transferred depends on the temperature difference and the properties of the substances involved. Understanding this process is key to predicting how and when two systems will reach the same temperature.

Specific Heat

Specific heat is a property that describes how much heat energy is needed to change the temperature of a substance by a certain amount. Specifically, it is the heat required to raise 1 kilogram of a substance by 1 degree Celsius (or 1 Kelvin).

In the problem, we focus on two specific heat capacities:

• The specific heat of the man's body: 3480 J/kg·K
• The specific heat of water (the drink): 4190 J/kg·K

These values indicate that water requires more energy than the human body to change its temperature by the same amount. This is crucial when calculating how the heat exchange will balance out. The specific heat tells us how responsive a material is to heating; materials with high specific heats can absorb a lot of heat without much change in temperature.

Thermal Equilibrium

Thermal equilibrium occurs when two interacting systems reach the same temperature and the exchange of heat stops. At this point, there is no net heat flow between them. In our example, the man's body and the soft drink reach thermal equilibrium at a temperature lower than the initial body temperature but higher than the initial drink temperature.

After reaching equilibrium, the final temperature (calculated to be approximately 36.8°C) no longer changes without external influences. Understanding thermal equilibrium is crucial for predicting the final temperatures in systems where different temperature bodies interact. It helps in numerous practical applications, such as refrigeration, heating, and designing materials to maintain certain temperature conditions.

Temperature Change Measurement

Measuring the change in temperature is an essential aspect of understanding thermal dynamics and equilibrium processes. In this problem, we calculate that the body's temperature decreases from 37.0°C to approximately 36.8°C after consuming the drink.

To determine whether such a change is noteworthy, we consider the precision of common medical devices:

• Medical thermometers are sensitive to changes as small as 0.1°C.

So, the change of 0.2°C is detectably significant. This highlights the importance of accurate measurement tools in medical and scientific settings, allowing for precise monitoring and response to temperature fluctuations. Temperature change measurement provides insight into the effectiveness of heat transfer and thermal balance in systems.