Question

Two flasks, each with a volume of \(1.00 \mathrm{L},\) contain \(\mathrm{O}_{2}\) gas with a pressure of \(380 \mathrm{mm}\) Hg. Flask \(\mathrm{A}\) 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?

Short answer

Flask B contains more \( O_2 \) molecules.

Step-by-step solution

Understanding the 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 gas constant, and \( T \) is the temperature in Kelvin. We need to use the same equation to compare the number of moles, as they relate to the number of molecules, in each flask.

Convert Temperatures to Kelvin

Temperatures must be in Kelvin for the ideal gas law. Convert the temperatures from Celsius to Kelvin using the formula \( T(K) = T(°C) + 273.15 \):\[ T_A = 25 + 273.15 = 298.15 \, K \]\[ T_B = 0 + 273.15 = 273.15 \, K \]

Compare Pressure, Volume, and Temperature

The volume \( V \) is the same for both flasks at \( 1.00 \, L \), and pressure \( P \) is the same for both flasks at \( 380 \, mm \, Hg \). Hence, the only differing factor that affects the number of moles \( n \) is temperature \( T \).

Using the Ideal Gas Law to Compare Moles

Since \( PV = nRT \), and both flasks have the same \( P \) and \( V \), we focus on \( n \) being inversely proportional to \( T \). Therefore,\[ n_A = \frac{P \times V}{R \times T_A} \]\[ n_B = \frac{P \times V}{R \times T_B} \]Since \( T_A > T_B \), \( n_A < n_B \).

Conclusion on Molecules

Since the number of moles \( n \) is directly proportional to the number of molecules, flask B contains more \( O_2 \) molecules because it is at a lower temperature.

Key concepts

Kelvin Temperature Conversion

When dealing with gases and the ideal gas law, temperature must be measured in Kelvin. This is crucial because Kelvin is an absolute temperature scale with no negative values, ensuring that our calculations remain consistent and meaningful. To convert from Celsius to Kelvin, simply add 273.15 to the Celsius temperature.

In our exercise, Flask A is at 25°C, which converts to 298.15 K, and Flask B is at 0°C, converting to 273.15 K. This conversion helps us accurately use the ideal gas law. Remember:

• K = °C + 273.15

Using Kelvin allows us to directly relate temperature to other gas properties like pressure and volume.

Moles and Molecules

The concept of moles is fundamental in chemistry as it measures the amount of substance. One mole contains Avogadro's number (6.022 × 10^23) of molecules. In our exercise, to find out how many oxygen molecules are in each flask, we first need to determine the number of moles of oxygen gas using the ideal gas law:

\(PV = nRT\) helps us relate pressure, volume, and temperature to moles. Once moles ( n ) are known, we can easily find the number of molecules by multiplying by Avogadro's number.

This equation tells us that if temperature changes while pressure and volume are constant, the number of moles will also change accordingly. Hence, more moles mean more molecules, as seen in Flask B.

Pressure and Volume Relationship

The relationship between pressure and volume is a key part of understanding gas behavior under the ideal gas law. At constant temperature and moles ( n ), Boyle's law tells us that pressure and volume are inversely proportional: increasing volume decreases pressure and vice versa.

However, in our exercise, both flasks have constant volume ( 1.00 ext{ L} ) and pressure ( 380 ext{ mm Hg} ), so we see the effect of temperature on the number of moles.

With both pressure and volume being the same, only the temperature affects how many moles are present in each flask, demonstrating how temperature changes can directly impact the number of molecules.