Question

If the specific heat \(c_{\mathrm{v}}\) of a gas obeying the van der Waals equation is given at a particular pressure, \(p^{\prime}\), by \(c_{v}=A+B T\), where \(A\) and \(B\) are constants, develop an expression for the change in specific entropy between any two states 1 and 2: \(\left[s\left(T_{2}, p_{2}\right)-s\left(T_{1}, p_{1}\right)\right]\)

Short answer

The change in specific entropy is: \[ s(T_2, p_2) - s(T_1, p_1) = A \ln{\frac{T_2}{T_1}} + B(T_2 - T_1) + R \ln{\frac{V_2-b}{V_1-b}} - \frac{a}{T}\left(\frac{1}{V_2} - \frac{1}{V_1}\right) \]

Step-by-step solution

Write the Differential Form of Entropy

The differential form of entropy change for a gas is given by the first law of thermodynamics: \[ ds = \frac{dq}{T} = \frac{dU + pdV}{T} \]

Relate Internal Energy and Specific Heat

Using the given specific heat expression, relate the change in internal energy (dU) to temperature (T): \[ dU = c_v dT = (A + BT) dT \]

Substitute into Entropy Differential

Substitute the expression for \(dU\) from Step 2 into the entropy differential from Step 1: \[ ds = \frac{(A + BT) dT + pdV}{T} \]

Use the Van der Waals Equation

Apply the van der Waals equation \[ p = \frac{RT}{V - b} - \frac{a}{V^2} \]. Substitute this expression for \(p\) in Step 3: \[ ds = \frac{(A + BT) dT}{T} + \left( \frac{RT}{V-b} - \frac{a}{V^2} \right)\frac{dV}{T} \]

Separate and Integrate Terms

Separate the terms involving \(dT\) and \(dV\): \[ ds = Ad\left(\ln{T}\right) + BdT + R\frac{dV}{V-b} - \frac{a}{T}d\left(\frac{1}{V}\right) \]Integrate each term from state 1 to state 2: \[ \int_{s_1}^{s_2} ds = A \int_{T_1}^{T_2} \frac{dT}{T} + B \int_{T_1}^{T_2} dT + R \int_{V_1}^{V_2} \frac{dV}{V-b} - a \int_{V_1}^{V_2} \frac{dV}{TV^2} \]

Solve the Integrals

Calculate the integrals: \[ s(T_2, p_2) - s(T_1, p_1) = A \ln{\frac{T_2}{T_1}} + B(T_2 - T_1) + R \ln{\frac{V_2-b}{V_1-b}} - \frac{a}{T}\left(\frac{1}{V_2} - \frac{1}{V_1}\right) \]

Key concepts

headline of the respective core concept

The van der Waals equation improves upon the ideal gas law by accounting for the finite size of gas molecules and the intermolecular forces between them. This provides a more accurate description of real gases, especially at high pressures and low temperatures.
The equation is written as follows:
\[ p = \frac{RT}{V - b} - \frac{a}{V^2} \]
Here:

• p is the pressure
• V is the volume
• R is the universal gas constant
• T is the temperature
• a and b are constants specific to each gas.

The term \( b \) corrects for the volume occupied by the gas molecules themselves, while \( a \) adjusts for the attractive forces between them. Because of these corrections, the van der Waals equation can predict real gas behavior more accurately than the ideal gas law, especially under non-ideal conditions.

headline of the respective core concept

Specific heat is the amount of heat energy required to raise the temperature of a substance per unit mass. It is denoted by c, and it varies depending on whether the process is at constant volume (c_v) or constant pressure (c_p).
For gases, c_v is used when the volume remains constant, and it's related to changes in internal energy. In this exercise, the specific heat at constant volume, c_v, is given by:
\[ c_{v} = A + B T \]
where A and B are constants, and T is the temperature.
To use specific heat in thermodynamic calculations, remember these points:

• Units of Specific Heat: It's often expressed in J/(kg·K).
• Temperature Dependence: For real materials, specific heat can vary with temperature, hence the inclusion of the BT term.
• Role in Energy Calculations: Specific heat helps calculate changes in internal energy and entropy.

Thus, specific heat is crucial for understanding how energy is stored and transferred in a substance.

headline of the respective core concept

Thermodynamics is the branch of physics that deals with heat and temperature, and their relation to energy and work. It describes how thermal energy is converted to and from other forms of energy and affects matter.
Here are key principles in thermodynamics:

• The First Law of Thermodynamics: This is the law of energy conservation. It states that the total energy of an isolated system is constant. Mathematically, it's described as:
  \[ \text{dU} = \text{dq} - \text{dw} \]
  where dU is the change in internal energy, dq is the heat added to the system, and dw is the work done by the system.
• The Second Law of Thermodynamics: This law states that the total entropy of an isolated system can never decrease over time. Entropy measures disorder; for instance, heat naturally flows from hot to cold objects.
• Entropy (s): Entropy is a measure of disorder or randomness in a system. A key focus in thermodynamics is understanding how entropy changes, as in the provided exercise which calculates the specific entropy change
  \[ s(T_2, p_2) - s(T_1, p_1) \] as a function of temperature and pressure using the van der Waals equation and specific heat.
  Understanding these laws and concepts helps us predict how energy changes within a system, and is critical for solving numerous practical problems in engineering, chemistry, and physics.