Question

Starting with the Gibbs equation \(d h=T d s+v d P\) obtain the expression for the change in enthalpy of an ideal gas having constant specific heats during the isentropic pro\(\operatorname{cess} P v^{k}=\) constant.

Short answer

Question: Derive an expression for the change in enthalpy of an ideal gas having constant specific heats during an isentropic process. Answer: The expression for the change in enthalpy of an ideal gas having constant specific heats during an isentropic process is given as: $$C_v(T_2 - T_1) = nR \ln \left(\frac{v_1}{v_2}\right)^k$$

Step-by-step solution

Simplify the Gibbs Equation for Isentropic Process

Since the process is isentropic, the change in entropy is zero (\(ds=0\)). So, the Gibbs equation becomes: $$dh = v dP$$

Replace Specific Volume Using the Ideal Gas Equation

Using the ideal gas equation \(PV = nRT\), we can express the specific volume (v) as: $$v = \frac{nRT}{P}$$ Substitute this expression for v in the Gibbs equation: $$dh = \frac{nRT}{P} dP$$

Integrate Both Sides

Integrate both sides with respect to pressure (P) to obtain the change in enthalpy: $$\int_{h_1}^{h_2} dh = nR \int_{P_1}^{P_2} \frac{1}{P} dP$$ Upon integrating, we get: $$(h_2 - h_1) = nR \ln \frac{P_2}{P_1}$$

Express Pressure and Enthalpy Using Specific Heats and Temperature

Recall that for an ideal gas with constant specific heats, we have: $$h_2 - h_1 = C_p(T_2 - T_1)$$ $$v_2 - v_1 = \frac{1}{nR}(P_2 - P_1)$$ Further, for an isentropic process, we have: $$\frac{P_2}{P_1}=\left(\frac{v_1}{v_2}\right)^k$$ Since the change in enthalpy is the main focus, we can use these equations and the conclusion from Step 3 to find an expression for the change in enthalpy for an isentropic process.

Derive the Final Expression for Change in Enthalpy

From Steps 3 and 4, we have: $$C_p(T_2 - T_1) = nR \ln \frac{P_2}{P_1}$$ Replace \(P_2\) in terms of \(P_1\) using the isentropic process equation: $$C_p(T_2 - T_1) = nR \ln \frac{P_1 (v_1 / v_2)^k}{P_1}$$ Simplify the equation: $$C_p(T_2 - T_1) = nR \ln \left(\frac{v_1}{v_2}\right)^k$$ Divide both sides by \(C_p\) and multiply by \(C_v\) to get: $$C_v(T_2 - T_1) = nR \ln \left(\frac{v_1}{v_2}\right)^{C_p / C_v}$$ Since \(k = \frac{C_p}{C_v}\), the final expression for change in enthalpy is: $$C_v(T_2 - T_1) = nR \ln \left(\frac{v_1}{v_2}\right)^k$$ This is the expression for the change in enthalpy of an ideal gas having constant specific heats during an isentropic process.

Key concepts

Gibbs Equation

The Gibbs equation is a fundamental relationship in thermodynamics that expresses the differential form of enthalpy (\(dh\)) as a function of temperature (\(T\)), entropy change (\(ds\)), and volume change (\(vdP\)). For an isentropic process, where entropy does not change (\(ds=0\)), the Gibbs equation simplifies to \(dh = vdP\).

To better understand this equation, let's visualize the concept of enthalpy as the total heat content of a system. It's a measure of the energy that is transferred in or out as heat during a process at constant pressure. The Gibbs equation helps us track this energy during minute changes in the system's state. In an ideal gas with constant specific heats, we can further refine the equation using other thermodynamic relationships to express enthalpy change in terms of temperature, pressure, volume, and specific heats.

Isentropic Process

An isentropic process is a thermodynamic process in which the entropy of the system remains constant. It's an idealization often used in engineering because it represents a reversible and adiabatic process—no heat is exchanged with the surroundings, and no energy is lost due to friction or other dissipative effects.

In the context of the exercise, the isentropic condition simplifies our calculations because we can set the entropy change to zero. This assumption leads us to an expression that links changes in pressure and volume during an isentropic compression or expansion of an ideal gas, often referred to as \(Pv^k =\text{constant}\), where \(k\) is the specific heat ratio.

Specific Heats

Specific heats are properties of substances that describe how much heat energy is required to raise the temperature of a unit mass of the substance by one degree. There are two specific heats commonly discussed for gases: specific heat at constant pressure (\(C_p\)) and specific heat at constant volume (\(C_v\)).

These values dictate how a gas will respond to heating under different conditions. For an ideal gas, the ratio of the specific heats (\(k = C_p/C_v\)) is a dimensionless number, known as the heat capacity ratio, which is used in various thermodynamic equations, including those for isentropic processes. Understanding specific heats is crucial when dealing with processes involving temperature and heat transfer in thermodynamics.

Ideal Gas Law

The ideal gas law is a cornerstone equation in thermodynamics and physical chemistry, represented by \(PV = nRT\), where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. It is an approximation that describes the behavior of an ideal gas.

However, real gases behave like ideal gases under many conditions, making this law incredibly useful for making calculations in science and engineering. It allows us to derive the specific volume (\(v\)) and to link pressure, temperature, and volume in a straightforward way. By relating pressure to temperature and volume, the ideal gas law can be used to find changes in these properties during thermodynamic processes, like the isentropic process we discussed earlier.