One mole of gas obeys Van der Waals equation of state. If...

One mole of gas obeys Van der Waals equation of state. If its molar internal energy is given by $u=c T-a / V$ (in which $V$ is the molar volume, $a$ is one of the constants in the equation of state, and $c$ is a constant), calculate the molar heat capacities $C_v$ and $C_p$.

Key Concepts

Molar Heat Capacities

Molar heat capacities are measures of the amount of heat required to raise the temperature of one mole of a substance by one degree under specified conditions. In thermodynamics, C? (at constant pressure) and C? (at constant volume) characterize how energy is stored in a system when it is heated at constant pressure or constant volume, respectively. These capacities are fundamental in understanding energy transfer, phase changes, and the response of a system to thermal inputs.

Internal Energy as a State Function

Internal energy is a thermodynamic property that quantifies the total energy contained within a system, which can depend on variables such as temperature and volume. When the internal energy is given as a function of temperature and volume (as in u = cT – a/V), it encapsulates both the kinetic energy of particles and potential energy due to intermolecular interactions, highlighting that energy changes can occur not only from temperature variations but also from changes in volume.

Enthalpy and Its Relationship to Internal Energy

Enthalpy is defined as the sum of a system's internal energy and the product of its pressure and volume (H = U + pV). This concept is crucial when dealing with processes at constant pressure because changes in enthalpy directly measure the heat exchanged. Understanding the relationship between enthalpy and internal energy is vital for deriving heat capacities, especially C?, which involves differentiating the enthalpy with respect to temperature at constant pressure.

Equations of State – Van der Waals Equation

Equations of state describe the relationship among pressure, volume, and temperature for a system. The Van der Waals equation is a modification of the ideal gas law that accounts for the finite size of molecules and the intermolecular forces. This equation gives a more realistic description of the behavior of real gases, and it plays a key role when analyzing how non-ideal effects influence thermodynamic properties like internal energy and heat capacities.

Thermodynamic Differentiation and Partial Derivatives

The use of partial derivatives is essential in thermodynamics for expressing how state functions change with respect to one variable while holding others constant. In calculating heat capacities, one needs to differentiate the internal energy and enthalpy with respect to temperature, applying the rules of differentiation to functions that depend on more than one variable. This mathematical tool allows us to connect microscopic energy expressions to macroscopic measurable quantities like C? and C?.

Please give Ace some feedback