Problem 29 - Chemistry The Molecular Science

Question

Which of these gas samples contains the largest number of molecules and which contains the smallest? (a) \(1.0 \mathrm{~L} \mathrm{H}_{2}\) at \(\mathrm{STP}\) (b) \(1.0 \mathrm{~L} \mathrm{~N}_{2}\) at \(\mathrm{STP}\) (c) \(1.0 \mathrm{~L} \mathrm{H}_{2}\) at \(27{ }^{\circ} \mathrm{C}\) and \(760 . \mathrm{mmHg}\) (d) \(1.0 \mathrm{~L} \mathrm{CO}_{2}\) at \(0{ }^{\circ} \mathrm{C}\) and \(800 . \mathrm{mmHg}\)

Step-by-Step Solution

Answer

The \( \mathrm{CO}_2 \) sample has the largest number of molecules; \( \mathrm{H}_2 \) at 27°C and 760 mmHg has the smallest.

Step 1: Recall the ideal gas law

The ideal gas law is given by the formula \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is temperature in Kelvin.

Step 2: Understand the conditions at STP

Standard Temperature and Pressure (STP) is defined as 0°C (273.15 K) and 1 atm (760 mmHg). Under these conditions, 1 mole of an ideal gas occupies 22.4 L.

Step 3: Calculate moles for gases at STP

Both 1.0 L of \( \mathrm{H}_2 \) at STP and 1.0 L of \( \mathrm{N}_2 \) at STP have the same conditions, so they each contain \( \frac{1.0}{22.4} = 0.04464 \) moles.

Step 4: Convert non-STP conditions to STP equivalents

For 1.0 L \( \mathrm{H}_2 \) at 27°C (300.15 K) and 760 mmHg: Convert temperature to Kelvin and use \( n = \frac{PV}{RT} \) with \( R = 0.0821 \ { \rm L \, atm / K \, mol} \) and \( P = 1 \ { \rm atm} \). Therefore, \( n = \frac{1.0 \times 1}{0.0821\times300.15} = 0.0403 \) moles.

Step 5: Calculate moles for conditioned \( CO_2 \)

For 1.0 L \( \mathrm{CO}_2 \) at 0°C and 800 mmHg: Convert the pressure to atm (800 mmHg = 1.0526 atm) and use the ideal gas law: \( n = \frac{1.0526 \times 1}{0.0821 \times 273.15} = 0.0463 \) moles.

Step 6: Compare moles to determine the number of molecules

Moles are directly proportional to the number of molecules (Avogadro's number \(6.022 \times 10^{23}\)). Thus, more moles mean more molecules. Comparing the moles calculated: 0.0463 (\( \mathrm{CO}_2 \)) > 0.04464 (\( \mathrm{H}_2 \) at STP and \( \mathrm{N}_2 \) at STP) > 0.0403 (\( \mathrm{H}_2 \) at 27°C).

Step 7: Final conclusion

The \( \mathrm{CO}_2 \) sample has the largest number of molecules, while \( \mathrm{H}_2 \) at 27°C and 760 mmHg contains the smallest number of molecules.

Key Concepts

Standard Temperature and Pressure

When talking about gases in chemistry, Standard Temperature and Pressure (STP) is a very important concept. It provides a common reference point so scientists can easily compare results and make calculations.

**What does STP mean exactly?**
To simplify, STP conditions are defined as 0°C (or 273.15 Kelvin, more scientifically) and 1 atm of pressure, which equals 760 mmHg.

Under these conditions, one mole of any ideal gas occupies 22.4 liters of volume. This is a very useful number because it allows comparisons and calculations involving different gases without having to account for varying conditions like temperature or pressure.

You can think of STP as the universal language for gases! Whether it's hydrogen, nitrogen, or carbon dioxide, knowing that 1 mole will fill 22.4 L helps chemists work out how many molecules are present in a sample under these conditions.

Moles and Molecules

In chemistry, moles and molecules are like cups and drops of water. A mole is a counting unit, just like a dozen or a pair, but much larger. One mole equals Avogadro's number, which is approximately 6.022 x 10^23 molecules or atoms.

**Why use moles?**
Moles let chemists count large numbers of small entities, like molecules, in a practical way. Without a unit like the mole, it would be daunting to count every single molecule in a sample.

**Moles are linked to volume through STP:**
- At STP conditions, 1 mole of an ideal gas occupies 22.4 liters.
- This helps determine how many moles and therefore molecules are in a given volume of gas.

Using the ideal gas law (\(PV = nRT\)), where \(n\) is the number of moles, you can calculate how many molecules a sample contains just by knowing its volume and the conditions it's kept in. This calculation is crucial for comparing different gases like we did when we determined which sample had the most molecules.

Gas Laws in Chemistry

The behavior of gases can be described using the ideal gas law, a fundamental equation in chemistry. This is expressed as \(PV = nRT\), where:

• \(P\) stands for pressure
• \(V\) is the volume
• \(n\) is the number of moles
• \(R\) is the ideal gas constant
• \(T\) is the temperature in Kelvin

This law helps predict how changes in temperature, volume, and pressure can affect a gas. For example, if you increase the temperature of a gas while keeping its volume constant, the pressure will also increase.

**Understanding Real-world Conditions:**
In chemistry, it is important to understand how the variables in \(PV = nRT\) interact.

- **Pressure and Volume:** As per Boyle's Law, if the volume decreases, the pressure increases, given a constant temperature.
- **Volume and Temperature:** Charles's Law tells us that if you increase temperature, the volume increases if pressure is constant.
- **Pressure and Temperature:** Gay-Lussac’s Law states that pressure increases with temperature if volume is constant.

These interconnections help us understand not just laboratory settings but also natural phenomena like weather and even breathing.