Problem 12 - Chemistry An Atoms First Approach

Question

If you have any two gases in different containers that are the same size at the same pressure and same temperature, what is true about the moles of each gas? Why is this true?

Step-by-Step Solution

Answer

Under the given conditions where both gases have the same pressure, temperature, and equal volume containers, the moles of both gases are equal (i.e., they have the same number of moles), according to the Ideal Gas Law. This is true due to the direct proportionality of pressure, volume, and temperature to the moles of an ideal gas.

Step 1: Ideal Gas Law Equation

The Ideal Gas Law is a mathematical equation relating the volume, temperature, pressure, and amount of substance (in moles) for an ideal gas. The Ideal Gas Law equation is as follows: \(PV = nRT\) 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 gas constant (8.314 J/(mol K)), - T is the temperature of the gas in Kelvin.

Step 2: Apply the given conditions

In this exercise, we know that both gases are at the same pressure, same temperature, and in containers of the same size (equal volume). Let's denote the conditions on gas 1 as \(P_1\), \(V_1\), \(n_1\), \(T_1\) and the conditions on gas 2 as \(P_2\), \(V_2\), \(n_2\), \(T_2\). Given that \(P_1 = P_2\), \(V_1 = V_2\), and \(T_1 = T_2\), let's apply the Ideal Gas Law to both gases: Gas 1: \(P_1V_1 = n_1RT_1\) Gas 2: \(P_2V_2 = n_2RT_2\)

Step 3: Comparing the moles of both gases

Since the pressure, volume, and temperature of both gases are equal, we can now compare the moles of both gases. Divide the equation for gas 1 by the equation for gas 2: \(\frac{P_1V_1}{P_2V_2} = \frac{n_1RT_1}{n_2RT_2}\) Since \(P_1 = P_2\), \(V_1 = V_2\), and \(T_1 = T_2\), this simplifies to: \(\frac{n_1}{n_2} = 1\) Which means: \(n_1 = n_2\)

Step 4: Conclusion

Under the given conditions where both gases have the same pressure, temperature, and equal volume containers, the moles of both gases are equal (i.e., they have the same number of moles), according to the Ideal Gas Law. This is true due to the direct proportionality of pressure, volume, and temperature to the moles of an ideal gas.