[From Adkins] Below 100 K the specific heat capacity of...

[From Adkins] Below $100 \mathrm{~K}$ the specific heat capacity of diamond varies as the cube of temperature: $c_p=a T^3$. A small diamond of mass $100 \mathrm{mg}$ is cooled to $77 \mathrm{~K}$ by immersion in liquid nitrogen, and then dropped into a bath of liquid helium at its boiling point of $4.2 \mathrm{~K}$ at atmospheric pressure. In cooling the diamond, some of the helium is boiled off. The gas is collected and found to occupy a volume $2.48 \times 10^{-5} \mathrm{~m}^3$ at $0^{\circ} \mathrm{C}$ and 1 atmosphere pressure. What is the value of $a$ in the formula for the specific heat capacity of diamond? [The latent heat of vaporization of helium at $1 \mathrm{~atm}$ is $21 \mathrm{~kJ} / \mathrm{kg}$.]

Key Concepts

Conservation of Energy in Thermal Processes

The principle of conservation of energy states that energy in an isolated system is conserved. In thermal processes, this means that the heat lost by a cooling body must be equal to the heat gained by its surroundings (or vice versa). This concept is fundamental in solving calorimetry problems where energy transfer between components, including phase changes, needs to be accounted for accurately.

Ideal Gas Law for Gas Calculations

The ideal gas law provides a relationship between the pressure, volume, temperature, and amount of gas. This concept is essential for calculating the amount (or mass) of a gas present when its volume and certain standard conditions (such as temperature and pressure) are provided, thereby linking macroscopic measurements to microscopic particle properties.

Temperature Dependent Specific Heat Capacity

This concept involves materials whose ability to store heat varies with temperature, such that the specific heat is not constant but depends on temperature (in this case, as the cube of temperature). Understanding this behavior is key in low-temperature physics and material science, where the thermal properties of solids deviate from classical predictions and need to be modeled properly using temperature dependent expressions.

Integration in Variable Heat Capacity Analysis

When the specific heat capacity depends on temperature, the total heat exchanged during a temperature change is determined by integrating the temperature-dependent specific heat over the range of temperatures. This approach is fundamental in thermodynamics for evaluating heat energy in systems where the heat capacity is not constant, allowing for the accurate calculation of energy changes.

Latent Heat of Vaporization

The latent heat of vaporization is the amount of energy required to change a substance from its liquid phase to its gaseous phase without a change in temperature. It is a crucial concept in understanding phase transitions and energy balance in systems where vaporization plays a role, serving as a bridge between thermal energy and mass change during phase change processes.

Transcript

Please give Ace some feedback