Determination of the Ideal Gas Law Constant (R) Using Hydrogen Gas Generation

Law Constant: Determination of the Ideal Gas Law Constant (R)

Fundamental Concepts

The experiment focuses on determining the ideal gas law constant, R, by generating hydrogen gas through a chemical reaction and measuring its properties. This process applies the Ideal Gas Law and concepts of gas collection over water, partial pressures, and stoichiometry.

• Ideal Gas Law: Relates pressure (P), volume (V), temperature (T), and amount of gas (n) by the equation: $PV = nRT$

• Partial Pressure: When a gas is collected over water, the total pressure is the sum of the gas pressure and the vapor pressure of water: $P_{total} = P_{gas} + P_{H_2O}$

• Stoichiometry: The balanced chemical equation for the reaction is: $Mg(s) + 2HCl(aq) \rightarrow MgCl_2(aq) + H_2(g)$

Experimental Procedure

The experiment involves reacting a known mass of magnesium metal with excess hydrochloric acid to produce hydrogen gas, which is collected over water in a eudiometer tube. The volume of gas, temperature, and pressure are measured to calculate R.

1. Weigh a sample of magnesium metal and insert it into a eudiometer tube.

2. Add hydrochloric acid to the tube, causing the reaction to produce hydrogen gas.

3. Collect the gas over water and measure the volume of gas produced.

4. Record the temperature of the water (assumed to be the same as the gas).

5. Measure the atmospheric pressure and the height difference between water levels inside and outside the tube to correct for pressure.

6. Account for the vapor pressure of water at the measured temperature.

Data Collection and Calculations

Key data and calculations are required to determine the value of R:

• Mass of magnesium used (e.g., 0.090 g)

• Volume of hydrogen gas collected (e.g., 90.0 mL or 0.090 L)

• Atmospheric pressure (e.g., 29.80 in Hg, converted to mm Hg or atm)

• Temperature of gas (e.g., 24°C, converted to Kelvin)

• Vapor pressure of water at the measured temperature (e.g., 22.3 mm Hg at 24°C)

• Height difference between water levels (converted to mm Hg for pressure correction)

Sample Calculations

• Convert atmospheric pressure to mm Hg: $29.80\ \text{in Hg} \times 25.4\ \text{mm/in} = 757.92\ \text{mm Hg}$

• Correct for water level difference: $\text{Pressure difference} = \frac{\text{height in mm}}{13.6}$ (since the density of mercury is 13.6 times that of water)

• Calculate pressure of dry hydrogen gas: $P_{H_2} = P_{atm} - P_{H_2O} - P_{diff}$

• Convert all pressures to atmospheres: $1\ \text{atm} = 760\ \text{mm Hg}$

• Calculate moles of hydrogen gas produced: $n = \frac{\text{mass of Mg}}{\text{molar mass of Mg}}$

• Calculate R using the ideal gas law: $R = \frac{P V}{n T}$

• Calculate percent error: $\%\ \text{error} = \frac{|R_{\text{experimental}} - R_{\text{theoretical}}|}{R_{\text{theoretical}}} \times 100$

Example Data Table

Measurement	Value
Mass of Mg	0.090 g
Volume of H2 gas	90.0 mL (0.090 L)
Atmospheric Pressure	757.92 mm Hg
Vapor Pressure of Water	22.3 mm Hg
Temperature	24°C (297 K)
Corrected Pressure of H2	0.9627 atm
Moles of H2	0.0037 mol
Experimental R	0.0787 L·atm/(mol·K)
Theoretical R	0.0821 L·atm/(mol·K)
Percent Error	4%

Common Sources of Error and Error Analysis

• Gas escape: If hydrogen gas escapes, the calculated moles will be too low, making R too high.

• Neglecting water level difference: Not correcting for water level difference can cause errors in pressure calculation, affecting R.

• Incorrect vapor pressure: Not accounting for water vapor pressure will overestimate the pressure of hydrogen gas.

• Measurement inaccuracies: Errors in measuring mass, volume, or temperature will propagate into the calculation of R.

Practice Problems and Applications

1. What gas is generated when magnesium reacts with hydrochloric acid? Answer: Hydrogen gas (H2).

2. Write the balanced chemical equation for the reaction. Answer: $Mg(s) + 2HCl(aq) \rightarrow MgCl_2(aq) + H_2(g)$

3. Calculate the pressure difference between two gases separated by liquid water, given the height and densities. Example: $P_{H_2O} = 28.0\ \text{cm} \times \frac{1.00\ \text{g/mL}}{13.6\ \text{g/mL}} \times 10 = 20.6\ \text{mm Hg}$

4. Calculate percent error if the experimental R is 0.0801 L·atm/(mol·K) and the theoretical R is 0.08206 L·atm/(mol·K): $\%\ \text{error} = \frac{0.0801 - 0.08206}{0.08206} \times 100 = -2.39\%$

Summary Table: Key Equations and Constants

Equation/Constant	Description
$PV = nRT$	Ideal Gas Law
$P_{total} = P_{gas} + P_{H_2O}$	Dalton's Law of Partial Pressures
$R = 0.0821$ L·atm/(mol·K)	Theoretical value of the gas constant
$n = \frac{\text{mass}}{\text{molar mass}}$	Moles calculation
$T(K) = T(°C) + 273$	Temperature conversion

Additional info:

• Dalton's Law of Partial Pressures is essential when collecting gases over water, as the vapor pressure of water must be subtracted from the total pressure to obtain the pressure of the dry gas.

• Careful measurement and correction for all variables are crucial for accurate determination of R.