(III) The specific heat per mole of potassium at low temperatures is given by C_V = aT + bT³, where a = 2.08 mJ/mol · K² and b = 2.57 mJ/mol· K⁴ . Determine (by integration) the entropy change of 0.15 mol of potassium when its temperature is lowered from 5.0 K to 1.0 K.

Verified step by step guidance

1. Identify the given values and the formula to use. Here, the specific heat per mole of potassium, $C_V$, is given by $C_V = aT + bT^3$, where $a = 2.08 \, \text{mJ/mol} \cdot \text{K}^2$ and $b = 2.57 \, \text{mJ/mol} \cdot \text{K}^4$. The amount of potassium is 0.15 mol, and the temperature change is from 5.0 K to 1.0 K.
2. Convert the units of $a$ and $b$ from mJ/mol·K² and mJ/mol·K⁴ to J/mol·K² and J/mol·K⁴ respectively, since the standard unit for energy in the International System of Units (SI) is joules (J).
3. Use the formula for entropy change $\Delta S$ for a process at constant volume, which can be calculated by integrating the expression for $C_V/T$ with respect to temperature from the initial temperature $T_i$ to the final temperature $T_f$: $\Delta S = \int_{T_i}^{T_f} \frac{C_V}{T} \, dT$.
4. Substitute the expression for $C_V$ into the integral and simplify: $\Delta S = \int_{5.0 \, \text{K}}^{1.0 \, \text{K}} \frac{aT + bT^3}{T} \, dT = \int_{5.0 \, \text{K}}^{1.0 \, \text{K}} (a + bT^2) \, dT$.
5. Evaluate the integral from 5.0 K to 1.0 K, and multiply the result by the number of moles (0.15 mol) to find the total entropy change for the potassium.

Key Concepts

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a substance by one degree Celsius per mole. In this context, the specific heat per mole of potassium is expressed as a function of temperature, indicating how the heat capacity varies with temperature, which is crucial for calculating changes in thermal energy.

Entropy

Entropy is a measure of the disorder or randomness in a system, often associated with the amount of energy unavailable for doing work. In thermodynamics, the change in entropy can be calculated from the heat transfer divided by the temperature, reflecting how energy disperses in a system as it undergoes a temperature change.

Integration in Thermodynamics

Integration is a mathematical technique used to calculate quantities that accumulate over a range, such as heat capacity over a temperature interval. In this problem, integrating the specific heat capacity function from the initial to the final temperature allows us to determine the total heat transfer, which is essential for calculating the change in entropy for the potassium sample.