Problem 102 - Chemistry and Chemical Reactivity

Question

Two flasks, each with a volume of \(1.00 \mathrm{L},\) contain \(\mathrm{O}_{2}\) gas with a pressure of 380 mm Hg. Flask \(A\) is at \(25^{\circ} \mathrm{C}\) and flask \(\mathrm{B}\) is at \(0^{\circ} \mathrm{C}\). Which flask contains the greater number of \(\mathrm{O}_{2}\) molecules?

Step-by-Step Solution

Answer

Flask B contains more \( \text{O}_2 \) molecules.

Step 1: Understand Ideal Gas Law

The ideal gas law equation is \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin. This equation helps us calculate the number of moles given the other variables.

Step 2: Convert Temperatures to Kelvin

To use the ideal gas law, convert the temperatures from Celsius to Kelvin. \( T(K) = T(°C) + 273.15 \). Flask \( A \) at \( 25^{\circ}C \) has a temperature of \( 298.15 \) K, and Flask \( B \) at \( 0^{\circ}C \) has a temperature of \( 273.15 \) K.

Step 3: Apply the Ideal Gas Law

Since both flasks have the same \( P \), \( V \), and \( R \) values, the number of moles \( n \) depends on \( T \). According to the ideal gas law, \( n = \frac{PV}{RT} \). Thus, \( n \) is inversely proportional to \( T \).

Step 4: Compare Moles Based on Temperature

Since \( n \) is inversely proportional to \( T \), the flask with the lower temperature \( (T) \) will have more moles. Flask \( B \) at \( 273.15 \) K has a lower temperature than Flask \( A \).

Step 5: Conclusion

Flask \( B \) contains more \( \text{O}_2 \) molecules than Flask \( A \) because it is at a lower temperature, making \( n \) greater.

Key Concepts

Understanding Temperature Conversion

When working with the ideal gas law, an important first step is converting temperatures from Celsius to Kelvin. This conversion is crucial because the Kelvin scale is an absolute temperature scale used in scientific equations. The conversion is quite simple: just add 273.15 to the Celsius temperature, as Kelvin starts from absolute zero.

For instance, in the problem at hand, Flask A is at 25°C, which converts to 298.15 K, and Flask B is at 0°C, which converts to 273.15 K. Remembering this conversion is essential for accurate calculations when using the ideal gas law.

This straightforward conversion allows us to apply the ideal gas law correctly, ensuring that our calculations reflect the true thermodynamic behavior of gases.

Mole Calculations and the Ideal Gas Law

The ideal gas law is pivotal when it comes to calculating moles in a gas sample. The equation is given by \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.

In this equation, when you know \( P \), \( V \), and \( T \), and use a constant \( R \), you can solve for \( n \) (the number of moles), using the formula \( n = \frac{PV}{RT} \). This relationship shows that the number of moles is directly influenced by temperature and inversely proportional to it.

This means for the same pressure and volume, a gas at a lower temperature will have more moles. Therefore, when you calculate \( n \), pay attention to how a change in temperature affects it, as seen in the provided problem where Flask B, with a lower temperature, has more moles than Flask A.

Pressure-Volume Relationship in Gases

Though the pressure and volume in this exercise remain constant for both flasks, understanding their relationship is critical in gas calculations. Generally, the relationship between pressure and volume is illustrated through Boyle's Law, which states that for a fixed amount of gas at constant temperature, the pressure is inversely proportional to its volume: \( P \propto \frac{1}{V} \).

In scenarios where either pressure or volume changes and other conditions like temperature or moles remain constant, this understanding helps predict how a gas will behave. However, in this particular exercise, since both flasks have the same volume of 1.00 L and the same pressure of 380 mm Hg, comparisons on a PV basis can focus on temperature effects instead.

To master gaseous calculations, appreciating how pressure and volume interact is vital, but remember: for realistic scenarios such as in this problem, temperature must be factored in to get the full understanding of gas behavior.