Question

Develop expressions for the volume expansivity \(\beta\) and the isothermal compressibility \(\kappa\) for (a) an ideal gas. (b) a gas whose equation of state is \(p(v-b)=R T\). (c) a gas obeying the van der Waals equation.

Short answer

The short answer must be shorter than three sentences.

Step-by-step solution

Understanding Volume Expansivity and Isothermal Compressibility

Volume expansivity, \(\beta\), is defined as \(\beta = \frac{1}{V} \(\frac{\text{d}V}{\text{d}T}\)|_p\), and isothermal compressibility, \(\kappa\), is defined as \(\kappa = -\frac{1}{V} \(\frac{\text{d}V}{\text{d}p}\)|_T\).

Key concepts

Volume Expansivity

Volume expansivity, also known as the thermal expansion coefficient, is a measure of how much a material expands upon heating. It is denoted by \( \beta \) and defined by the formula:\( \beta = \frac{1}{V} \frac{\text{d}V}{\text{d}T} \bigg|_p \), where:

• \( V \) is the volume
• \( T \) is the temperature
• \( p \) is the pressure

This equation tells us the rate of change of volume with respect to temperature while keeping the pressure constant. It helps in understanding how substances react to changes in temperature, especially gases. The higher the volume expansivity, the more sensitive the volume is to temperature changes. This is crucial in practical applications like designing engines and HVAC systems.

Isothermal Compressibility

Isothermal compressibility, symbolized as \( \, \kappa \, \) , measures the relative change in volume due to a change in pressure while holding temperature constant. It is given by:\( \, \kappa \ = -\frac{1}{V} \, \frac{\text{d}V}{\text{d}p} \, \bigg|_T \), where:

• \( V \) is the volume
• \( T \) is the temperature
• \( p \) is the pressure

The negative sign indicates that an increase in pressure typically causes a decrease in volume. Isothermal compressibility is essential in fields like material science and thermodynamics because it helps predict how a substance behaves under pressure. For instance, gases with high compressibility are easier to compress, making them useful in various industrial applications.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the relationship between pressure, volume, and temperature in ideal gases. The law is written as: \( PV = nRT \), where:

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

This equation assumes that the gas particles do not interact and occupy no volume. It provides a good approximation for the behavior of gases under many conditions. Although real gases deviate from this behavior at high pressures and low temperatures, the Ideal Gas Law is widely used because of its simplicity and predictive power.

Equation of State

In thermodynamics, an equation of state is a mathematical equation that describes the state of matter under a given set of physical conditions. The Ideal Gas Law is an example of an equation of state. Another example is the modified equation of state for a gas where \( (p(v-b)=RT) \). This equation adjusts for the volume occupied by gas molecules:

• \( v \) is the molar volume
• \( b \) is the volume correction factor
• \( p \) is the pressure
• \( R \) is the ideal gas constant
• \( T \) is the temperature

These equations of state are crucial because they allow us to predict how a given amount of substance behaves under varying conditions of pressure, volume, and temperature. They are invaluable in designing chemical processes and understanding natural phenomena.

Van der Waals Equation

The Van der Waals equation is an improvement over the Ideal Gas Law and accounts for the intermolecular forces and finite size of gas molecules. It is expressed as:\[ (P + a \frac{n^2}{V^2})(V - nb) = nRT \] where:

• \( P \) is the pressure
• \( V \) is the volume
• \( n \) is the number of moles of the gas
• \( R \) is the ideal gas constant
• \( T \) is the temperature
• \( a \) is a measure of the attraction between gas molecules
• \( b \) is the volume occupied by the gas molecules

The terms \( a \frac{n^2}{V^2} \) and \( nb \) adjust the pressure and volume, respectively, to account for the real behavior of gases. This equation provides a more accurate representation of gas behavior than the Ideal Gas Law, especially under high pressure and low temperature conditions. Understanding the Van der Waals equation is key to mastering more advanced concepts in thermodynamics and material science.