Question

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

Short answer

To derive Charles's Law from the Ideal Gas Law, start with the Ideal Gas Law formula \(PV = nRT\). For two different temperatures, \(T_1\) and \(T_2\), with constant pressure \(P_1\) and constant number of moles \(n_1\), write the equations: \(P_1V_1 = n_1RT_1\) and \(P_1V_2 = n_1RT_2\). Divide the two equations to eliminate the constants, leading to \(\frac{V_2}{V_1} = \frac{T_2}{T_1}\), which represents Charles's Law, showing that the volume of a gas is directly proportional to its temperature when the pressure and number of moles are constant.

Step-by-step solution

Write down the Ideal Gas Law formula

The Ideal Gas Law is given by the equation: \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature in Kelvin.

Hold the pressure and number of moles constant

Charles's Law states that the volume of a gas is directly proportional to its temperature when the pressure and number of moles are held constant. So, we need to make sure that the pressure (P) and the number of moles (n) remain constant for our derivation. We can denote this constant pressure as \(P_1\) and constant number of moles as \(n_1\).

Write the Ideal Gas Law formula for two different temperatures

To show the relationship between volume and temperature, we need to compare the volumes and temperatures under the initial pressure and the constant pressure. Thus, we write the Ideal Gas Law for two different temperatures, \(T_1\) and \(T_2\), using the same pressure and number of moles: - For the initial temperature, \(T_1\) and initial volume, \(V_1\): \(P_1V_1 = n_1RT_1\) - For the final temperature, \(T_2\) and final volume, \(V_2\): \(P_1V_2 = n_1RT_2\)

Divide the two equations

To isolate the relationship between volume and temperature, divide the equation for final temperature by the equation for initial temperature to eliminate the constants \(P_1\), \(n_1\), and\(R\): \(\frac{P_1V_2}{P_1V_1} = \frac{n_1RT_2}{n_1RT_1}\)

Simplify the equation

Now, cancel out the common terms on both sides of the equation: \(\frac{V_2}{V_1} = \frac{T_2}{T_1}\)

Conclude the derivation

We have reached the equation that represents Charles's Law: \(\frac{V_2}{V_1} = \frac{T_2}{T_1}\), which tells us that the volume of a gas is directly proportional to its temperature, when the pressure and number of moles remain constant.