Problem 41 - Physics

Question

Blood can carry excess energy from the interior to the surface of the body, where the energy is dispersed in a number of ways. While a person is exercising, \(0.6 \mathrm{~kg}\) of blood flows to the surface of the body and releases \(2000 \mathrm{~J}\) of energy. The blood arriving at the surface has the temperature of the body interior, \(37.0^{\circ} \mathrm{C}\). Assuming that blood has the same specific heat capacity as water, determine the temperature of the blood that leaves the surface and returns to the interior.

Step-by-Step Solution

Answer

The temperature of the blood returning to the interior is approximately \(36.2^{\circ} \text{C}\).

Step 1: Identify Known Values

The mass of the blood, \(m\), is given as \(0.6 \text{ kg}\). The energy released, \(Q\), is \(2000 \text{ J}\). The initial temperature of the blood is \(37.0 ^{\circ} \text{C}\). The specific heat capacity of water (and therefore blood) is \(c = 4186 \text{ J/kg}^\circ \text{C}\).

Step 2: Understand Formula Context

The formula to determine the change in temperature when energy is transferred is \(Q = mc\Delta T\), where \(\Delta T = T_{\text{final}} - T_{\text{initial}}\). We need to find \(T_{\text{final}}\).

Step 3: Rearrange Formula for \(T_{\text{final}}\)

From \(Q = mc(T_{\text{final}} - T_{\text{initial}})\), solve for \(T_{\text{final}}\): \[ T_{\text{final}} = \frac{Q}{mc} + T_{\text{initial}} \]

Step 4: Plug in the Values

Insert the known values into the rearranged formula: \[ T_{\text{final}} = \frac{-2000 \text{ J}}{0.6 \text{ kg} \times 4186 \text{ J/kg}^\circ \text{C}} + 37.0 ^{\circ} \text{C} \]

Step 5: Calculate \(T_{\text{final}}\)

Calculate \(T_{\text{final}}\): \[ T_{\text{final}} = \frac{-2000}{2511.6} + 37.0 \approx -0.795 + 37.0 = 36.205^{\circ} \text{C} \]

Key Concepts

Specific Heat Capacity

Specific heat capacity is like the bookkeeper of heat. It tells us how much heat energy a substance can store. When it comes to substances like water, it means how much energy is required to raise the temperature of 1 kilogram of the substance by 1°C. For water, this is \[4186 \text{ J/kg}^\circ \text{C}\]. This value is quite high, indicating that water can absorb a lot of heat without a significant rise in temperature. That's why we often compare other substances to water when discussing specific heat capacity.

• It's a material-specific property, meaning different substances have different capacities to store heat.
• The specific heat capacity of blood is assumed to be the same as that of water in many calculations, especially when dealing with the human body.
• This helps in calculating how much heat energy is needed to change the temperature of the blood as it flows through the body.

Energy Transfer

Energy transfer is a process where energy moves from one object or substance to another. In thermodynamics, energy can be transferred in the form of heat. When blood flows to the surface of the body, it carries with it the energy generated by metabolic processes inside the body. Here, the blood releases about 2000 joules of energy.

• Energy transfer occurs to maintain equilibrium in the body, moving heat from warmer to cooler areas.
• It's a natural process helping prevent overheating during exercise by dissipating excess heat through the skin.
• The amount of energy transferred directly affects the temperature change experienced by the blood.

It's important to understand how this transfer aids in temperature regulation and how calculations can help us determine changes in temperature. In this case, energy is leaving the blood, thereby lowering its temperature as it returns to the body's interior.

Temperature Change

Temperature change is the variation in temperature, resulting from the gain or loss of energy. When we talk about this concept in terms of blood and energy transfer, we're looking at how the loss of energy affects the blood's temperature.
We use the formula \[Q = mc\Delta T\]where:

• \(Q\) is the energy absorbed or released (in joules),
• \(m\) is the mass of the substance (in kilograms),
• \(c\) is the specific heat capacity, and
• \(\Delta T = T_{\text{final}} - T_{\text{initial}}\) is the temperature change.

Using this formula, we can calculate the final temperature of blood as it returns back to the interior. The negative sign of \(Q\) indicates energy is being lost, rather than gained. Solving this, we find that the blood's final temperature is slightly lower after releasing energy.