Molar heat capacity of \(\mathrm{CD}_{2} \mathrm{O}\) (deuterated form of formaldehyde) vapour at constant pressure is vapour 14 cal/K-mol. The entropy change associated with the cooling of \(3.2 \mathrm{~g}\) of \(\mathrm{CD}_{2} \mathrm{O}\) vapour from \(1000 \mathrm{~K}\) to \(900 \mathrm{~K}\) at constant pressure is (assume ideal gas behaviour for \(\mathrm{CD}_{2} \mathrm{O}\) ) \([\ln 0.9=-0.1]\) (a) \(+0.14 \mathrm{cal} / \mathrm{K}\) (b) \(-0.14 \mathrm{cal} / \mathrm{K}\) (c) \(-1.4 \mathrm{cal} / \mathrm{K}\) (d) \(+1.4 \mathrm{cal} / \mathrm{K}\)

Understanding molar heat capacity is integral to mastering concepts in thermodynamics and physical chemistry. It is defined as the amount of heat required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). Represented as \( C_p \), with the subscript \( p \) indicating that the measurement is done at constant pressure, it's a property that varies depending on the type of substance and the conditions of the environment.

The molar heat capacity of a gas at constant pressure is especially important because it can be used to calculate the entropy change of the gas when it undergoes a temperature change. For gases that exhibit ideal behavior, molar heat capacity values allow for straightforward calculations using the formula \( \Delta S = n * C_p * \(ln\frac{T2}{T1}\) \), where \(\Delta S\) is the entropy change, \(n\) is the number of moles, \(C_p\) is the molar heat capacity at constant pressure, and \(T1\) and \(T2\) are the initial and final temperatures, respectively.

It’s worth mentioning that the molar heat capacity of a substance can vary with temperature; however, in many classroom problems, this variation may be neglected for simplicity, assuming a constant value as is the case in the present problem we're discussing.

Entropy is a fundamental concept in thermodynamics often associated with the level of disorder or randomness in a system. Entropy change, denoted as \( \Delta S \), indicates how much the disorder of a system has changed during a process.

In the context of the given problem, entropy change is being calculated for the cooling of a gas. When a gas cools down, it typically becomes more ordered, and therefore one might expect a decrease in entropy. However, the relationship between heat transfer and entropy change is more nuanced. The entropy change of a system can be calculated using the formula mentioned in the solution, which takes into account the moles of substance, its molar heat capacity at constant pressure, and the logarithm of the ratio of the final to initial temperatures.

This thermodynamic calculation is widely applicable, from understanding the behavior of gases in engines to predicting the stability of chemical compounds. The precision in calculating entropy change is vital, as it can have significant implications in fields like environmental science, engineering, and even cosmology.

Ideal gas behavior is a cornerstone concept in chemistry and physics, describing a simplified model in which gas particles are considered to be point particles moving in random motion and experiencing no intermolecular forces. Despite being a simplification, the ideal gas model provides a useful approximation for the behavior of real gases at high temperatures and low pressures.

An ideal gas follows the ideal gas law, \( PV = nRT \), linking pressure (\(P\)), volume (\(V\)), number of moles (\(n\)), the ideal gas constant (\(R\)), and temperature (\(T\)). It’s essential to note that the ideal gas behavior assumes constant motion, random collisions, and energy conservation during collisions—conditions that are not always met in real-life applications.

In the context of the problem, assuming ideal gas behavior allows us to use straightforward formulae for calculating entropy change without worrying about the complexities of real gas interactions. This assumption simplifies the problem and is commonly used in educational settings to help students grasp the basic concepts before moving on to more complex and realistic scenarios.

The mole is a fundamental unit in chemistry, essential for quantifying the amount of a substance. One mole corresponds to Avogadro's number of particles, which is approximately \(6.022 \times 10^{23}\) entities. Calculating moles is a foundational skill in chemistry that allows for the conversion between mass, volume (for gases), and number of particles.

In the exercise we are considering, the first step requires converting the given mass of a substance to moles. This is done using the molar mass of the substance, which represents the mass of one mole of its entities. The molar mass is the sum of the atomic masses of all atoms in the molecule. For instance, in the compound \(CD_2O\), we sum the atomic masses of carbon, deuterium (twice, since there are two atoms), and oxygen to get the molar mass. Once we have the molar mass, we can calculate the number of moles by dividing the given mass by this molar mass value.

Being able to calculate moles is indispensable, not just for determining the entropy change in this particular problem but also for solving a wide range of problems in stoichiometry, thermodynamics, and reaction kinetics.