Problem 117 - University Physics with Modern Physics

Question

Careful measurements show that the specific heat of the solid phase depends on temperature (Fig. P17.117). How will the actual time needed for this cryoprotectant to come to equilibrium with the cold plate compare with the time predicted by using the values in the table? Assume that all values other than the specific heat (solid) are correct. The actual time (a) will be shorter; (b) will be longer; (c) will be the same; (d) depends on the density of the cryoprotectant.

Step-by-Step Solution

Answer

The actual time will (b) be longer if the specific heat increases with decreasing temperature.

Step 1: Understand the problem

We are asked to determine how the actual cooling time of a cryoprotectant will compare with the predicted time if the specific heat of the solid phase depends on temperature.

Step 2: Consider the effect of specific heat

Specific heat is a measure of how much energy is required to change the temperature of a substance. If specific heat varies with temperature, the energy required to reach equilibrium temperature may differ from predictions that assume a constant specific heat.

Step 3: Analyze the impact of a variable specific heat

If the specific heat increases with temperature, more energy is needed to lower the temperature, elongating the process compared to using a constant value. Conversely, if it decreases, less energy is required, potentially reducing the time.

Step 4: Relate to equilibrium time

The time to reach equilibrium is related to the energy exchanged, which depends on specific heat. Because the problem states the table's constant value might not be accurate, consider this variability in the specific heat to answer the effect on time.

Key Concepts

Temperature Dependence

Temperature dependence is a crucial factor in understanding how specific heat functions. The specific heat of a material indicates how much energy is required to increase the temperature of that material by one degree. For some materials, specific heat does not remain constant and can change as the temperature of the material changes.
This dependence means that as the temperature of a cryoprotectant varies, the specific heat may increase or decrease.

• If the specific heat increases at higher temperatures, more energy will be needed to continue raising the temperature, leading to longer cooling or heating times.
• Conversely, if the specific heat decreases, less energy will be needed, potentially shortening the time needed to change the temperature.

Understanding how specific heat changes with temperature can allow more accurate predictions of how quickly or slowly a material will reach a desired temperature.

Cooling Time

Cooling time refers to how long it takes for a substance to reach the desired thermal equilibrium with its surroundings. This is often influenced by the material's specific heat, which, as mentioned before, can depend on temperature.
In the case of a cryoprotectant, the cooling time is determined by how efficiently heat is exchanged between the cryoprotectant and the cold plate.

• An increase in specific heat implies that more energy is needed to lower temperatures, possibly resulting in a longer cooling time than predicted.
• If specific heat decreases with a drop in temperature, it might speed up the cooling process, shortening the cooling time.

When predicting the cooling time, considering the variable nature of specific heat is essential for accurate assessments.

Equilibrium Temperature

The equilibrium temperature is the final temperature reached when a substance and its surroundings reach thermal balance. For a substance like a cryoprotectant interacting with a cold plate, this means the point at which there is no further net energy exchange.
The process to reach this equilibrium is dependent on the material properties, such as specific heat, and factors in how specific heat varies with temperature.

• If a cryoprotectant has a specific heat that decreases with temperature, getting to equilibrium might occur more rapidly.
• In contrast, if the specific heat increases, the journey to equilibrium could take longer due to the need for more energy transfer.

The understanding of these properties helps engineers and scientists design processes to achieve desired temperature changes efficiently.

Cryoprotectant

A cryoprotectant is a substance used to protect biological tissue from freezing damage. This is especially relevant in processes like cryopreservation, where biological specimens are preserved at extremely low temperatures.
The specific heat of a cryoprotectant is important because it determines how the material will respond to changes in temperature.

• Understanding if its specific heat changes with temperature can help predict how quickly it can be cooled down safely without causing damage.
• It is crucial for ensuring that the delicate balance of freezing protective biological materials is maintained.

Considering these characteristics is vital for optimizing cryoprotectants and ensuring they perform as required to protect sensitive biological materials during temperature changes.