One mole of a gas obeys the van der Waals equation of state: $$ \left(P+\frac{a}{v^{2}}\right)(v-b)=R T $$ and its molar internal energy is given by $$ u=c T-\frac{a}{v}, $$ where \(a, b, c\), and \(R\) are constants. Calculate the molar heat capacities \(c_{V}\) and \(c_{P}\).

The van der Waals equation is an adjustment to the ideal gas law, accounting for the volume occupied by gas molecules and the attractive forces between them. The equation is given by:

\[ \bigg(P + \frac{a}{v^2}\bigg)(v - b) = RT \]

Here, \( P \) is the pressure, \( v \) is the molar volume, \( R \) is the gas constant, and \( T \) is the temperature. The constants \( a \) and \( b \) account for the intermolecular forces and the volume occupied by gas molecules, respectively.

This equation is essential for real gases, especially under high pressure and low temperature conditions where deviations from ideal behavior are significant. It helps to explain why real gases don't follow the ideal gas law perfectly.

Molar internal energy refers to the total energy contained within one mole of a substance. It includes kinetic and potential energy due to molecular motion and intermolecular forces. For a gas obeying the van der Waals equation:

\[ u = cT - \frac{a}{v} \]

Here, \( u \) is the molar internal energy, \( c \) is a specific heat constant, \( T \) is the temperature, and \( \frac{a}{v} \) accounts for intermolecular forces. The term \( cT \) represents the kinetic energy, while \( -\frac{a}{v} \) accounts for potential energy due to attractions between molecules.

In essence, molar internal energy helps us understand how temperature and volume changes affect the energy within a gas.

Enthalpy is a thermodynamic quantity equivalent to the total heat content of a system. It is an essential concept in understanding heat transfer processes at constant pressure. Enthalpy \( H \) is given by:

\[ H = u + Pv \]

Using the van der Waals equation, we can replace \( P \) with:

\[ P = \frac{RT}{v - b} - \frac{a}{v^2} \]

Substituting this into the enthalpy equation:

\[ H = cT - \frac{a}{v} + \bigg(\frac{RT}{v - b} - \frac{a}{v^2}\bigg)v \]

Simplifying, we get:

\[ H = cT + \frac{RTv}{v - b} - \frac{ab}{v(v - b)} \]

For an ideal situation (large \( v \)), this reduces to \( H \thickapprox cT + RT \). This simplification helps in solving for molar heat capacities.

Partial derivatives are used to measure how a function changes with respect to one variable while keeping other variables constant. They are crucial in thermodynamics for finding molar heat capacities.

For example, molar heat capacity at constant volume \( c_{V} \) is found using the partial derivative of internal energy \( u \) with respect to temperature at constant volume \( V \):

\[ c_{V} = \bigg(\frac{\frac{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{\boldsymbol{}}}}}}}}{u}{T}}_}{\bigg)}_V \bigg) \]

Given \( u = cT - \frac{a}{v} \), the partial derivative gives:

\[ c_{V} = c \]

Similarly, the molar heat capacity at constant pressure \( c_{P} \) involves the partial derivative of enthalpy \( H \) with respect to temperature at constant pressure \( P \):

\[ c_{P} = \bigg(\frac{\frac{\boldsymbol{\boldsymbol{H}{T}}}{}}{\bigg)}_^{P}_{\bigg)} \]

For the simplified enthalpy expression \( H \thickapprox cT + RT \), the partial derivative gives:

\[ c_{P} = c + R \]

Thus, partial derivatives allow us to precisely find how heat capacities change with temperature, essential for accurate thermodynamic calculations.