Question

Show how Boyle's gas law can be derived from the ideal gas law.

Short answer

To derive Boyle's gas law from the ideal gas law, start with the formula: \(PV = nRT\). Next, set the temperature constant and remove it by denoting the constant as \(k\), where \(k = nRT\). Rewrite the ideal gas law formula as \(PV = k\). This equation demonstrates Boyle's gas law, showing that the product of pressure and volume remains constant when the temperature is constant.

Step-by-step solution

Write down the ideal gas law formula

The ideal gas law is represented by the formula: \( PV = nRT \) 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

Set the temperature constant, and remove it from the equation

To derive Boyle's law, we need to consider the relationship between pressure and volume at a constant temperature. This means that the temperature (\(T\)) and the number of moles (\(n\)) in the system should remain constant as well. Therefore, we can treat the product of the number of moles, the gas constant, and the temperature (\(nRT\)) as a constant. Let's denote this constant as \(k\): \( k = nRT \) Now, we can rewrite the ideal gas law formula as: \( PV = k \)

Derive Boyle's gas law

Boyle's gas law states that the product of the pressure and volume of a gas is constant as long as the temperature remains constant. Now that we have the simplified equation, we can see that this is indeed the case: \( PV = k \) This equation demonstrates Boyle's gas law, as it shows that the product of pressure and volume remains constant when temperature remains constant.