Question

Show that for an ideal gas \(\bar{c}_{p}=\bar{c}_{v}+R_{u}\).

Short answer

Question: Prove that for an ideal gas, the molar heat capacity at constant pressure (cp) is equal to the molar heat capacity at constant volume (cv) plus the gas constant Ru. Answer: Using the definitions of molar heat capacities, the relationship between enthalpy and internal energy, the ideal gas law, and partial differentiation with respect to temperature, we have shown that $\bar{c}_{p}$ = $\bar{c}_{v} + Ru$, which is the desired relationship for an ideal gas.

Step-by-step solution

Understand the molar heat capacities

The molar heat capacities cp and cv are defined as the amount of energy required to raise the temperature of one mole of a substance by 1 Kelvin under constant pressure (cp) or constant volume (cv) conditions, respectively. Mathematically, we can express the molar heat capacities as: $$\bar{c}_{p} = \left(\frac{\partial{H}}{\partial{T}}\right)_{P}$$ and $$\bar{c}_{v} = \left(\frac{\partial{U}}{\partial{T}}\right)_{V}$$ where H is the enthalpy of the substance and U is the internal energy of the substance.

Use the Ideal Gas Law and relationships between internal energy, enthalpy, and heat capacities

We know that for an ideal gas, the enthalpy H and internal energy U are related through the equation: $$H = U + PV$$ The molar heat capacities cp and cv can also be related to enthalpy and internal energy through their respective definitions: $$\bar{c}_{p} = \left(\frac{\partial{H}}{\partial{T}}\right)_{P}$$ $$\bar{c}_{v} = \left(\frac{\partial{U}}{\partial{T}}\right)_{V}$$ Recalling that the ideal gas law is given by PV=nRuT, we have: $$P=nRu \frac{U}{T}$$

Differentiate the equations with respect to temperature

Now, differentiate the equation of enthalpy H with respect to temperature T, and keep the pressure P constant: $$\frac{\partial{H}}{\partial{T}}= \frac{\partial{U}}{\partial{T}}+\frac{\partial{(PV)}}{\partial{T}}$$ Since the molar heat capacities cp and cv are defined by the partial derivatives of H and U with respect to T, we can substitute them into the equation above: $$\bar{c}_{p} = \bar{c}_{v} + \frac{\partial{(PV)}}{\partial{T}}$$

Substitute the ideal gas law and simplify the equation

Now, substitute the expression for the pressure P we derived earlier from the ideal gas law into the equation: $$\bar{c}_{p} = \bar{c}_{v} + \frac{\partial{(nRuT)}}{\partial{T}}$$ Now, differentiate the right-hand side of the equation with respect to temperature T: $$\bar{c}_{p} = \bar{c}_{v} + nRu$$ Since we are working with molar heat capacities (per mole), we can divide both sides of the equation by n: $$\bar{c}_{p} = \bar{c}_{v} + Ru$$ Now, we have derived the relationship we set out to prove: $$\bar{c}_{p} = \bar{c}_{v} + Ru$$

Key concepts

Ideal Gas Law

The ideal gas law is a pivotal principle in understanding the behavior of gases under various conditions. It provides a relationship between the pressure (P), volume (V), temperature (T), and the number of moles (n) of an ideal gas.

The law is mathematically expressed as:
\[ PV = nRT \]
Where:

• \( P \) is the pressure of the gas,
• \( V \) is the volume of the gas,
• \( n \) is the number of moles of the gas,
• \( R \) is the ideal gas constant,
• \( T \) is the temperature of the gas in Kelvins.

When applying the ideal gas law in thermodynamic processes, it is assumed that the gas particles are point masses with no intermolecular forces of attraction or repulsion and that they move randomly. This simplification allows us to analyze and predict the changes in physical properties of gases with relative ease. It is essential to note that real gases deviate from ideal behavior at high pressures and low temperatures, where intermolecular forces become significant.

Enthalpy

Enthalpy, denoted as \( H \), is a measure of the total energy of a thermodynamic system and includes the internal energy of the system plus the energy required to sustain the system at constant pressure. It is an extensive property, meaning it depends on the amount of substance present.

Mathematically, enthalpy is given by:\[ H = U + PV \]
where:

• \( U \) is the internal energy,
• \( P \) is the pressure,
• \( V \) is the volume.

Enthalpy changes are often encountered in calculations involving heat transfer during chemical reactions, particularly at constant pressure. Because enthalpy is a state function, the change in enthalpy (\( delta H \)) between two states does not depend on the path taken by the system, only on the initial and final states. This characteristic makes enthalpy a very useful concept in thermodynamics and chemistry.

Internal Energy

Internal energy, designated as \( U \), encompasses the energy stored within a system, which includes both the kinetic energy of particles in motion and the potential energy resulting from intermolecular forces.

Within the perspective of thermodynamics, changes in internal energy can occur due to heat transfer and work done by or on the system. The first law of thermodynamics, which is essentially a statement of the conservation of energy, relates the change in internal energy to heat added to the system and work done by the system:\[ delta U = q - w \]

• \( q \) is the heat transferred to the system,
• \( w \) is the work performed by the system.

In an isochoric process, where the volume is held constant, no work is done and the change in internal energy is directly proportional to the heat added or removed. Understanding internal energy is essential to grasp other concepts, such as enthalpy and the molar heat capacities.

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, temperature, and the statistical behavior of systems with a large number of particles. It lays the groundwork for analyzing energy conversions and the directionality of physical and chemical processes.

Core principles of thermodynamics include:

The Zeroth Law

It establishes the concept of thermal equilibrium and temperature.

The First Law

It is the law of conservation of energy, stating that energy cannot be created or destroyed, only transformed.

The Second Law

It introduces the concept of entropy, suggesting that the total entropy of an isolated system can never decrease over time.

The Third Law

It states that as the temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant minimum.
Thermodynamics plays a critical role in various fields, from classical mechanics to modern engineering, and even in cosmology and biology. The four laws provide a comprehensive framework that underpins nearly all processes in nature.