A diatomic ideal gas initially at \(273 \mathrm{~K}\) is given 100 cal heat due to which system did \(210 \mathrm{~J}\) work. Molar heat capacity of the gas for the process is \((1 \mathrm{cal}=4.2 \mathrm{~J})\) (a) \(\frac{3}{2} R\) (b) \(\frac{5}{2} R\) (c) \(\frac{5}{4} R\) (d) \(5 R\)

The first law of thermodynamics is a principle that relates to energy conservation in thermodynamic processes. It essentially states that the energy within a closed system is conserved, meaning it can neither be created nor destroyed, but it can change forms. In mathematical terms, it's represented as \( \text{Δ}U = Q - W \), where \( \text{Δ}U \) is the change in internal energy of the system, \( Q \) is the heat added to the system, and \( W \) represents the work done by the system.

In the context of the given exercise, we apply this principle to determine the change in internal energy of a gas after heating and work done. To connect it with the textbook exercise, when 100 cal of heat is added to the diatomic ideal gas and it does 210 J of work, we must first convert the heat from calories to joules and then apply the first law to find the change in internal energy.

Understanding this law is fundamental in physical chemistry as it serves as the foundation for analyzing energy transfer within chemical reactions and transitions between states of matter.

Internal energy is the total of all forms of energy contained within a system, which includes kinetic and potential energies at the molecular level. In thermodynamics, we're often concerned with changes in internal energy (\( \text{Δ}U \)), rather than the absolute value, because it is easier to measure and more relevant to physical and chemical processes.

In the ideal gas we're examining, the change in internal energy can be illustrated through its temperature change, which is tied to molecular motion. The equation \( \text{Δ}U = n C_{\text{molar}} \text{Δ}T \) implies that this change in internal energy is directly proportional to the quantity of substance ('n' in moles) and its temperature change (\( \text{Δ}T \)), multiplied by the molar heat capacity (\( C_{\text{molar}} \)).

Hence, in our exercise, calculating the internal energy change is key for determining the molar heat capacity, which depicts the amount of energy required to raise the temperature of one mole of substance by one degree Kelvin.

The ideal gas law, represented as \( PV = nRT \), is a fundamental equation in chemistry linking the pressure (\( P \)), volume (\( V \)), number of moles (\( n \)), ideal gas constant (\( R \)), and temperature (\( T \)) of an ideal gas. It is derived based on the assumption that gas molecules move freely and don’t interact with each other, barring during perfectly elastic collisions.

To tie this into our textbook problem, while the ideal gas law itself isn't directly used in the solution, understanding the relationship between these variables helps grasp why knowing the number of moles and temperature change is essential when calculating molar heat capacity. The ideal behavior stipulated by this law simplifies the connection between heat transfer and its effect on a gas's properties, allowing us to infer that internal energy changes for an ideal gas are independent of pressure and volume.

A diatomic gas is composed of molecules that each consist of two atoms, such as oxygen (\( O_2 \)) or nitrogen (\( N_2 \)). These gases are significant because their behavior can be modeled to understand the translational, rotational, and vibrational movements of molecules, which all contribute to the total internal energy and heat capacity. In diatomic gases, rotational and vibrational modes are important considerations as they absorb and store heat energy.

The term 'molar heat capacity' implies the heat required to raise one mole of gas by one degree Kelvin. For diatomic gases at constant volume, the molar heat capacity (at low temperatures) can be approximated as \( \frac{5}{2} R \), where \( R \) is the ideal gas constant. This differs from monatomic gases, where the molar heat capacity is approximately \( \frac{3}{2} R \) due to lacking rotational and vibrational energy modes. In our exercise, recognizing that the gas is diatomic helps us to narrow down the correct option for molar heat capacity.