The molar specific heat at constant pressure for a certain gas is given by \(C_{p}=a+b T+c T^{2},\) where \(a=33.6 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\), \(b=2.93 \times 10^{-3} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{2},\) and \(c=2.13 \times 10^{-5} \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}^{3} .\) Find the entropy change when 2 moles of this gas are heated from \(20^{\circ} \mathrm{C}\) to \(200^{\circ} \mathrm{C}\).

Molar specific heat is an essential concept in thermodynamics that refers to the amount of heat required to raise the temperature of one mole of a substance by one degree Kelvin (or Celsius). It's usually denoted by the symbol C, with subscripts p or v to indicate whether the measurement is taken at constant pressure or constant volume, respectively.

The given formula C_p = a + bT + cT² represents the variation of molar specific heat with temperature for the subject gas. The constants a, b, and c characterize the gas's unique thermal properties. Understanding how these constants function in the formula can help us better comprehend how the molar specific heat of a gas changes as its temperature increases.

This variable nature of C_p has significant implications when calculating thermal processes like the one in our exercise, where we determine entropy changes.

Thermodynamics is the study of heat, energy, and work, and how they interact with each other. It is a fundamental theory for various branches of science and engineering, explaining how thermal energy is converted to other energy forms and vice versa.

In our exercise, we focus on one of the central concepts of thermodynamics: entropy. Entropy is a measure of the disorder or randomness of a system and is significant because it describes the feasibility of a thermodynamic process. The Second Law of Thermodynamics tells us that entropy within a closed system tends to increase over time.

The finding of entropy change involves the calculation of heat exchange and understanding how it affects the disorder of the system. In this case, the heat exchange is characterized by the molar specific heat that changes with temperature.

Integral calculus is a mathematical tool frequently used in physics to calculate quantities that accumulate over time or space. It allows us to solve problems involving rates of change when the rates are expressed as functions.

In our entropy change calculation, integral calculus is indispensable because it helps us determine the total change when a variable specific heat is considered. As the function C_p varies with temperature, integrating it with respect to temperature allows us to accumulate the infinitesimal changes over the range from the initial to the final temperatures.

Applying Integral Calculus

By integrating the given function of C_p over the temperature limits, we are essentially summing up all the little bits of heat absorbed by the gas as its temperature raises incrementally, which then allows us to calculate the overall entropy change.

The Kelvin scale is an absolute thermodynamic temperature scale used widely in the physical sciences. Unlike Celsius or Fahrenheit, Kelvin begins at absolute zero, - the point where no more thermal energy can be extracted from a substance, which theoretically means no molecular motion.

In practical scenarios such as our exercise, converting Celsius to Kelvin is straightforward: we add 273.15 to the Celsius temperature. This conversion is critical because all thermodynamic calculations must be carried out in Kelvin to maintain consistency with the formulas' derivation, which assumes an absolute scale.

Importance in Calculations

When calculating thermodynamic properties like entropy, using the Kelvin scale ensures that we don't run into negative temperatures, which would be nonsensical in the context of the laws of thermodynamics. This also avoids the potential for mathematical errors that could arise if using a relative temperature scale like Celsius or Fahrenheit.