One mole of oxygen is heated from \(0^{\circ} \mathrm{C}\), at constant pressure, till its volume increased by \(10 \%\). The specific heat of oxygen, under these conditions, is \(0.22 \mathrm{cal} / \mathrm{g}-\mathrm{K}\). The amount of heat required is (a) \(32 \times 0.22 \times 27.3 \times 4.2 \mathrm{~J}\) (b) \(16 \times 0.22 \times 27.3 \times 4.2 \mathrm{~J}\) (c) \(\frac{32 \times 0.22 \times 27.3}{4.2} \mathrm{~J}\) (d) \(\frac{16 \times 0.22 \times 27.3}{4.2} \mathrm{~J}\)

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of our specific heat calculation problem, it's essential to understand how heat energy can cause changes in temperature and volume of gases at constant pressure.

In thermodynamics, one of the most crucial principles is the conservation of energy, which implies that energy cannot be created or destroyed, only transferred from one form to another. When we heat oxygen, we are transferring energy to it, resulting in a change in temperature and volume as observed in the exercise.

Moreover, the first law of thermodynamics, which is essentially the law of energy conservation, plays a significant role in understanding how the heat added to the system, in this case, oxygen, converts to internal energy causing a temperature rise.

Heat capacity, or thermal capacity, is a physical property that represents the amount of heat energy required to raise the temperature of a substance by a certain amount. The specific heat capacity is the heat capacity per unit mass, which in our exercise is given as 0.22 cal/g-K for oxygen.

The higher the specific heat capacity, the more heat is needed to change the temperature of a substance. Calculating the amount of heat required using the specific heat involves the formula: \( Q = mc\Delta T \), where \( Q \) is the heat in joules, \( m \) is the mass in grams, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature in Kelvin.

Understanding heat capacity is crucial in physical chemistry as it influences various processes, including phase changes, reaction kinetics, and thermal stability of substances.

The ideal gas law is a fundamental equation in physical chemistry that describes the behavior of ideal gases. It is expressed as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

In the given problem, we use the ideal gas law implicitly by assuming that a 10% increase in volume at constant pressure indicates a 10% increase in temperature. This assumption is based on the fact that for an ideal gas at constant pressure, the volume is directly proportional to the temperature (Charles's Law).

The ideal gas law provides a way to relate the physical properties of a gas under ideal conditions, which are usually closely met by gases like oxygen at standard temperatures and pressures.

Solving physical chemistry problems requires a strong grasp of concepts such as thermodynamics, the ideal gas law, and heat capacity. Problems often involve calculations that reveal how these concepts interact in real-world scenarios.

For instance, the given exercise involves calculating the amount of heat required to cause a specific temperature change in oxygen. To solve such problems, clear step-by-step approaches are crucial. One must identify the correct formulas, understand the relationships between variables, and ensure units are consistent throughout the calculation.

Students tackling physical chemistry problems should take care to follow a logical problem-solving process, double-check conversions (e.g., from calories to joules), and compare their calculations against the provided options to verify their accuracy.