Question

A sample of hydrogen is collected in a bottle over water. By carefully raising and lowering the bottle, the height of the water outside is adjusted so that it is just even with the water level inside (see figure) . When a sample of gas was collected the initial conditions were: volume \(=425 \mathrm{ml}\), pressure \(=753 \mathrm{~mm}\) and the temperature of the water (and thus, the gas also) \(=34^{\circ} \mathrm{C}\). Calculate the volume of the hydrogen if it were dry and at a pressure of \(760 \mathrm{~mm}\) and a temperature of \(0^{\circ} \mathrm{C}\) (STP)

Short answer

The final volume of the hydrogen gas if it were dry and at a pressure of 760 mm and a temperature of 0°C (STP) is approximately \(397.67\,\text{ml}\).

Step-by-step solution

Convert temperatures to Kelvin

To convert from Celsius to Kelvin, we simply add 273.15. Therefore, the initial temperature T₁ will be: T₁ = 34°C + 273.15 = 307.15 K The final temperature T₂ is: T₂ = 0°C + 273.15 = 273.15 K

Plug in the values into the combined gas law formula

Let's plug in the initial conditions V₁, P₁, T₁ and the final conditions P₂, T₂ into the combined gas law formula: \[ \frac{425\,\text{ml}}{307.15\,\text{K}} \frac{753\,\text{mm}}{760\,\text{mm}} = \frac{V_2}{273.15\,\text{K}} \]

Solve for V₂

Now we will solve the equation for the final volume (V₂) of the hydrogen gas at STP: \[ V_2 = \frac{425\,\text{ml}}{307.15\,\text{K}} \frac{753\,\text{mm}}{760\,\text{mm}} \times 273.15\,\text{K} \] V₂ ≈ \(397.67\,\text{ml}\) So, the final volume of the hydrogen gas if it were dry and at a pressure of 760 mm and a temperature of 0°C (STP) is approximately 397.67 ml.

Key concepts

Gas Collection Over Water

Collecting gases over water is a common laboratory technique used to measure the volume of gases generated in a reaction. When a gas is collected over water, it becomes saturated with water vapor. The pressure inside the collection container is the sum of the pressure of the gas being collected and the partial pressure of the water vapor at a given temperature. To determine the pressure of the dry gas, we must subtract the vapor pressure of water from the total pressure measured. This is an essential step because ignoring the vapor pressure of water can lead to inaccuracies in subsequent calculations involving the gas.

For students solving these types of problems, it's important to consult a table of water vapor pressures at different temperatures. When working through textbook solutions, make sure to clarify the concept of partial pressures, and highlight the need to account for water vapor pressure to ensure a comprehensive understanding of the gas collection over water technique.

Combined Gas Law Formula

The combined gas law connects the pressure, volume, and temperature of a gaseous sample when it undergoes a change in one or more of these conditions. The law is derived from Boyle's Law, Charles's Law, and Gay-Lussac's Law. The formula for the combined gas law is: \[\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\]where:

• \(P_1\) and \(P_2\) are the initial and final pressures,
• \(V_1\) and \(V_2\) are the initial and final volumes, and
• \(T_1\) and \(T_2\) are the initial and final temperatures in Kelvin.

This formula is invaluable for solving problems involving changes in the state of a gas, as it allows for predicting the final state when the initial conditions and one of the final state variables are known. Students can practise this law with various exercises to learn how to manipulate and apply it in different situations efficiently.

Conversion to Kelvin

Temperature measurements in gas law calculations must be in Kelvin to ensure accuracy and consistency. Kelvin is the SI unit for thermodynamic temperature and is used in scientific calculations because it is an absolute scale with zero Kelvin being the absolute zero - the point at which particles theoretically have minimal thermal motion.

To convert Celsius to Kelvin, you simply add 273.15 to the Celsius temperature. This step is crucial, as it avoids negative temperature values that aren’t workable in gas law equations. For students practicing problems that involve temperature changes, make it a habit to always convert to Kelvin first before plugging values into formulas. Understanding this temperature scale and its importance in gas laws is key to mastering related physics and chemistry problems.

STP Conditions

STP stands for Standard Temperature and Pressure, which are standard sets of conditions for the measurement of gas volume to allow proper comparison between different sets of data. The standard temperature is defined as 0°C, which is 273.15 K, and the standard pressure is 1 atmosphere, which is equivalent to 760 mm Hg, or 101.325 kPa.

Gas law problems often refer to STP when asking to calculate what the volume of a gas would be under these standardized conditions. It is essential for students to remember these conditions and to use them when solving for gas volumes at STP in exercises. Proper use of STP can significantly simplify calculations involving gases, as they provide a reference for comparing the behavior of gases under different conditions.