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Understanding how much energy is required to change the temperature of a substance is key in thermodynamics. Specific heat capacity is the quantity of heat required to change the temperature of one mole of a substance by one degree Kelvin. Each substance has its own specific heat capacity, influenced by its structure and bonding.
The heat capacity at constant volume for a substance is denoted as \( C_v \). The formula \( q = mc\Delta T \) allows us to calculate the energy needed to change temperature, where \( q \) is the heat added, \( m \) is the number of moles, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.
In our exercise, this formula helps calculate the energy for both diatomic air and xenon, illustrating differences based on their molecular nature.

Diatomic gases, like oxygen and nitrogen in air, consist of two atoms per molecule. These gases exhibit specific heat capacities that differ from monoatomic gases.
Under constant volume conditions, the specific heat capacity \( C_v \) for a diatomic gas is given by \( C_v = \frac{5}{2} R \). This accounts for both translational and rotational movements of the molecules. Diatomic gases have more degrees of freedom compared to monoatomic gases, allowing them to store more heat energy, hence a larger \( C_v \).
In our calculation, we used this value for air, considering it as a diatomic gas, to determine that 249.39 Joules of heat is needed to increase its temperature.

Monoatomic gases, like xenon, are composed of single atoms. These gases have a simplified structure, influencing their specific heat capacity.
The heat capacity at constant volume for a monoatomic gas is \( C_v = \frac{3}{2} R \). This lower value arises due to fewer degrees of freedom; monoatomic gases only possess translational motion as opposed to rotational or vibrational.
For xenon, this principle demonstrated why it requires less energy—149.64 Joules—to achieve the same temperature increase as diatomic air, reflecting its simpler atomic nature.

The universal gas constant \( R \) is a crucial element in thermodynamic equations. It represents the molar energy scale for gases and connects the microscopic world of molecules to macroscopic properties like volume and pressure. In energy units, \( R \) is valued at 8.314 Joules per mole per Kelvin.
This constant is integrated into various gas equations, including those for specific heat calculations \( C_v = \frac{f}{2} R \), where \( f \) is the degrees of freedom.
By using \( R \) in our heat calculations, we ensure accuracy when determining how much energy is needed to change the temperature of different gases. The uniformity offered by \( R \) allows us to compare different gas behaviors under similar conditions.