Question

Show that the volume expansion coefficient of an ideal gas is \(\beta=1 / T\), where \(T\) is the absolute temperature.

Short answer

Question: Show that the volume expansion coefficient of an ideal gas is inversely proportional to the absolute temperature. Answer: The volume expansion coefficient (β) of an ideal gas is inversely proportional to the absolute temperature, which means β = 1 / T.

Step-by-step solution

Understand the volume expansion coefficient and ideal gas law

The volume expansion coefficient (β) of a substance is a measure of how the volume of the substance changes with temperature, at a constant pressure. Mathematically, it is defined as: \(\beta = \frac{1}{V} \frac{\partial V}{\partial T}\Big|_P\) where V is the volume, T is the absolute temperature, and P is the pressure. The ideal gas law relates the pressure (P), volume (V), and temperature (T) of an ideal gas: \(PV = nRT\) where n is the number of moles of the gas and R is the ideal gas constant.

Express the volume as a function of temperature and pressure

From the ideal gas law, we can express the volume (V) as a function of temperature (T) and pressure (P): \(V = \frac{nRT}{P}\)

Calculate the partial derivative of the volume with respect to the temperature

Now, we will calculate the partial derivative of the volume with respect to the temperature (keeping the pressure constant): \(\frac{\partial V}{\partial T}\Big|_P = \frac{\partial}{\partial T} \left(\frac{nRT}{P}\right)\Big|_P = \frac{nR}{P}\) Here, we treat n, R, and P as constants.

Calculate the volume expansion coefficient

Finally, we will plug our expression for the partial derivative of the volume with respect to the temperature back into the equation for the volume expansion coefficient: \(\beta = \frac{1}{V} \frac{\partial V}{\partial T}\Big|_P = \frac{1}{\frac{nRT}{P}} \frac{nR}{P}\) Simplifying, we get: \(\beta = \frac{1}{T}\) Therefore, the volume expansion coefficient of an ideal gas is inversely proportional to the absolute temperature, β = 1 / T.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics, which describes how gases behave under various conditions. It connects pressure \(P\), volume \(V\), and temperature \(T\) in a simple way. The law is expressed as:\[PV = nRT\]where:

• \(n\) represents the number of moles of the gas.
• \(R\) is the ideal gas constant, valued at 8.314 J/(mol·K).

This equation implies that if you know any three properties of a gas, you can calculate the fourth. The relationship shows that as temperature increases, volume increases if pressure remains constant. It forms the basis for understanding gas behaviors and is widely applicable in various science fields. This can be particularly helpful when dealing with changes in conditions such as temperature and pressure.

Volume Change with Temperature

Volume change with temperature is often observed in gases and is measured by the volume expansion coefficient \(\beta\). This coefficient indicates how much the volume of a substance changes with a temperature shift while under constant pressure. Mathematically, it's expressed as:\[\beta = \frac{1}{V} \frac{\partial V}{\partial T}\Big|_P\]The intuition here is simple: if you heat a gas, it will generally expand or increase in volume if the pressure is the same. For ideal gases, as derived in the exercise, \(\beta = \frac{1}{T}\), indicating that the expansion is inversely proportional to temperature. This insight is critical for applications like designing systems and processes where temperature fluctuations are a concern.

Partial Derivative

Partial derivatives play a vital role in understanding how a function changes as one specific variable changes, while others are held constant. It's a common tool in calculus used to explore multi-variable functions. In the context of thermodynamics, it helps us determine the rate of change of variables like volume and temperature.For instance, in calculating the volume expansion coefficient of an ideal gas, the partial derivative of volume with respect to temperature is:\[\frac{\partial V}{\partial T}\Big|_P = \frac{nR}{P}\]This step is crucial because it shows how the volume changes as temperature changes, specifically when pressure is held steady. Understanding partial derivatives helps in analyzing complex systems where multiple factors interact.

Thermodynamics

Thermodynamics is the branch of physics that deals with heat, work, and energy. It examines how energy is transferred and transformed within systems. The fundamental laws of thermodynamics provide the framework for understanding these processes. Key principles include:

• Energy can neither be created nor destroyed (First Law of Thermodynamics).
• The entropy of a system tends to increase (Second Law of Thermodynamics).

With regards to gas laws, thermodynamics helps explain how gases expand and contract with changes in temperature and pressure. The concept of the volume expansion coefficient, as discussed in the exercise, is a perfect example of applying thermodynamic principles to predict how an ideal gas will behave under specific conditions. Understanding these relationships is vital in fields ranging from engineering to natural sciences.