Question

You have four gas samples: 1\. 1.0 L of \(\mathrm{H}_{2}\) at STP 2\. 1.0 L of Ar at STP 3\. \(1.0 \mathrm{L}\) of \(\mathrm{H}_{2}\) at \(27^{\circ} \mathrm{C}\) and \(760 \mathrm{mm} \mathrm{Hg}\) 4\. 1.0 L of He at \(0^{\circ} \mathrm{C}\) and \(900 \mathrm{mm} \mathrm{Hg}\) (a) Which sample has the largest number of gas particles (atoms or molecules)? (b) Which sample contains the smallest number of particles? (c) Which sample represents the largest mass?

Short answer

(a) He sample; (b) \(\mathrm{H}_2\) at 27°C; (c) Ar sample.

Step-by-step solution

Understanding STP Conditions

Standard Temperature and Pressure (STP) is defined as 0°C (273.15 K) and 1 atm (760 mm Hg). According to Avogadro's Law, 1 mole of any gas occupies 22.4 L at STP.

Calculate Moles for H2 at STP

For Sample 1, we have 1.0 L of \(\mathrm{H}_{2}\) at STP. The number of moles is calculated as: \[ \text{Moles of } \mathrm{H}_{2} = \frac{1.0 \, \text{L}}{22.4 \, \text{L/mol}} \approx 0.0446 \, \text{moles} \]

Calculate Moles for Ar at STP

For Sample 2, we have 1.0 L of Ar at STP. The number of moles is calculated as: \[ \text{Moles of Ar} = \frac{1.0 \, \text{L}}{22.4 \, \text{L/mol}} \approx 0.0446 \, \text{moles} \]

Calculate Moles for H2 at 27°C and 760 mm Hg

Convert 27°C to Kelvin: \(27 + 273.15 = 300.15\) K. Use the ideal gas law \(PV = nRT\) with \(R = 0.0821 \, \text{L atm/mol K}\) and pressure in atm (760 mm Hg = 1 atm):\[ n = \frac{(1.0 \times 1)}{(0.0821 \times 300.15)} \approx 0.0403 \text{ moles} \]

Calculate Moles for He at 0°C and 900 mm Hg

Convert pressure to atm (900 mm Hg = 1.184 atm). Use the ideal gas law: \[ n = \frac{(1.0 \times 1.184)}{(0.0821 \times 273.15)} \approx 0.0525 \text{ moles} \]

Identify Largest and Smallest Number of Particles

The sample with the largest number of moles has the largest number of particles. Sample 4 (He) has the largest number of moles (0.0525 moles) and, therefore, the largest number of particles. Sample 3 (H2 at 27°C) has the smallest number of moles (0.0403 moles) and, therefore, the smallest number of particles.

Calculate Mass of Each Sample

Use molar masses: \(\mathrm{H}_2 = 2.02\) g/mol, \(\mathrm{Ar} = 39.95\) g/mol, \(\mathrm{He} = 4.00\) g/mol.\[ \begin{align*} \text{Sample 1 (H}_2\text{):} & \ 0.0446 \times 2.02 \approx 0.0901 \text{ g} \ \text{Sample 2 (Ar):} & \ 0.0446 \times 39.95 \approx 1.784 \text{ g} \ \text{Sample 3 (H}_2\text{):} & \ 0.0403 \times 2.02 \approx 0.0814 \text{ g} \ \text{Sample 4 (He):} & \ 0.0525 \times 4.00 \approx 0.210 \text{ g} \end{align*} \] Sample 2 (Ar) has the largest mass.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of gases. It is expressed as \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin. This law allows for the calculation of any one of the variables if the other three are known. In the context of the exercise, the Ideal Gas Law is crucial for determining the number of moles of gas present when conditions are not at standard temperature and pressure (STP). For example, for Sample 3 in the exercise, the temperature is 27°C, which needs to be converted to Kelvin (300.15 K) before using it in the equation. Similarly, pressure needs to be in atmospheres (1 atm = 760 mm Hg) for accurate calculations. By substituting these values into the Ideal Gas Law formula, we can determine the moles of gas, which reflects the number of particles.

Avogadro's Law

Avogadro's Law states that equal volumes of gases, at the same temperature and pressure, contain an equal number of particles, typically molecules or atoms. This law is a key principle when comparing samples at Standard Temperature and Pressure (STP). According to Avogadro, 1 mole of any gas at STP occupies 22.4 liters. This is crucial for the exercise because Samples 1 and 2 are both 1.0 L at STP. Therefore, they have approximately the same number of moles, calculated as \( \frac{1.0 \, \text{L}}{22.4 \, \text{L/mol}} = 0.0446 \, \text{moles} \). Even though these samples have different chemical identities (\( \mathrm{H}_{2} \) for Sample 1 and \( \mathrm{Ar} \) for Sample 2), their molecular count remains similar due to Avogadro's Law.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a set of conditions used for measuring gases, where the temperature is 0°C (273.15 K) and the pressure is 1 atm (760 mm Hg). STP provides a reference point that allows scientists to compare the behavior of gases under standardized conditions. In the given exercise, Samples 1 and 2 are at STP, which simplifies calculations using Avogadro's constant 22.4 L/mol. Understanding STP is vital because it directly influences how we measure gaseous substances. When dealing with gases not at STP, as in Samples 3 and 4, adjustments using the Ideal Gas Law are required to find equivalent conditions. This adaptability helps in determining moles from volume and pressure, aiding in the comprehensive understanding of the number of particles and mass present.

Moles Calculation

Moles Calculation is essential to quantify the amount of a substance. The mole correlates the number of particles to a measurable amount of mass based on Avogadro's number \( 6.022 \times 10^{23} \) particles per mole. In the exercise, moles are calculated to determine which gas sample has the largest number of particles. For instance, for Sample 4 (He), we use the Ideal Gas Law to calculate the moles, finding that it has 0.0525 moles. In contrast, Sample 3 (H\(_2\) at different conditions) calculates to 0.0403 moles. This calculation allows for direct comparison, revealing that Sample 4 has the most particles. The relationship between the moles and mass is made clear through molar masses, where multiplying the moles by the molar mass gives the mass in grams. Such calculations help determine which gas sample has the largest mass, with Sample 2 (Ar) having the greatest mass due to its higher molar mass.