Question

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

Short answer

Answer: The volume expansion coefficient for an ideal gas is 𝛽=1/T.

Step-by-step solution

Define the volume expansion coefficient

The volume expansion coefficient, denoted by \(\beta\), is a property of materials that describes how its volume changes with temperature, keeping pressure constant. It is mathematically defined as: $$ \beta = \frac{1}{V}\frac{\partial V}{\partial T} $$ Where \(V\) is the volume and \(T\) is the absolute temperature.

Identify the ideal gas law

The ideal gas law relates the pressure (\(P\)), volume (\(V\)), number of moles (\(n\)), and temperature (\(T\)) of an ideal gas with a constant \(R\). The relation can be expressed as: $$ PV=nRT $$ We will use this equation to derive the volume expansion coefficient for an ideal gas.

Solve the ideal gas law for the volume

In order to use the ideal gas law to express the volume of the ideal gas as a function of the temperature, we will first isolate the volume \(V\) in the equation \(PV=nRT\). To do this, we can divide both sides of the equation by \(P\): $$ V = \frac{nRT}{P} $$

Take the partial derivative of volume with respect to temperature

Now that we have the volume as a function of temperature, we can take the partial derivative of \(V\) with respect to \(T\), considering the pressure \(P\), number of moles \(n\), and gas constant \(R\) as constants: $$ \frac{\partial V}{\partial T}=\frac{nR}{P}\frac{\partial (T)}{\partial T} $$ Since the partial derivative of \(T\) with respect to \(T\) is \(1\), our expression for the partial derivative of volume with respect to temperature becomes: $$ \frac{\partial V}{\partial T}=\frac{nR}{P} $$

Calculate the volume expansion coefficient for an ideal gas

Now that we have both the volume as a function of temperature and the partial derivative of volume with respect to temperature, we can use the definition of the volume expansion coefficient from Step 1 and substitute our expressions: $$ \beta = \frac{1}{V}\frac{\partial V}{\partial T} = \frac{1}{\frac{nRT}{P}} \cdot \frac{nR}{P} $$ Simplify the expression by cancelling out the terms: $$ \beta = \frac{1}{\frac{nRT}{P}} \cdot \frac{nR}{P} = \frac{1}{T} $$ Hence, we have shown that for an ideal gas, the volume expansion coefficient is equal to \(\beta=1/T\).