Question

Calcium metal and water produce hydrogen gas. If the volume of wet gas collected over water is \(95.0 \mathrm{~mL}\) at \(20^{\circ} \mathrm{C}\) and \(758 \mathrm{~mm} \mathrm{Hg},\) what is the volume of dry gas at STP conditions? (The vapor pressure of water at \(20^{\circ} \mathrm{C}\) is \(17.5 \mathrm{~mm}\) Hg.)

Short answer

The volume of dry gas at STP is 85.7 mL.

Step-by-step solution

Understand the Objective

We need to find the volume of the dry hydrogen gas at Standard Temperature and Pressure (STP) from the volume of wet gas collected over water.

Use Dalton's Law

The total pressure of the wet gas includes the pressure of the dry gas and the vapor pressure of water. According to Dalton's Law, we calculate the pressure of the dry gas:\[P_{\text{dry}} = P_{\text{total}} - P_{\text{water}} = 758 \text{ mm Hg} - 17.5 \text{ mm Hg} = 740.5 \text{ mm Hg}\]

Convert to STP Conditions

Standard Temperature and Pressure (STP) is defined as 0°C (273 K) and 1 atm pressure. First, convert the pressure from mm Hg to atm:\[P_{\text{dry}} (\text{atm}) = \frac{740.5 \text{ mm Hg}}{760 \text{ mm Hg/atm}} = 0.974 \text{ atm}\]

Apply Combined Gas Law

Use the combined gas law to find the volume of the gas at STP:\[\frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2}\]where \(P_1 = 0.974 \text{ atm}\), \(V_1 = 95.0 \text{ mL}\), \(T_1 = 293 \text{ K}\), \(P_2 = 1 \text{ atm}\), \(T_2 = 273 \text{ K}\). Substituting the known values gives:\[(0.974 \text{ atm} \times 95.0 \text{ mL}) / 293 \text{ K} = (1 \text{ atm} \times V_2) / 273 \text{ K}\]Solve for \(V_2\):\[V_2 = \frac{0.974 \times 95.0 \times 273}{293} = 85.7 \text{ mL}\]

Calculate Final Volume

The volume of dry gas at STP is 85.7 mL.

Key concepts

Dalton's Law

Dalton's Law of Partial Pressures is a powerful tool used when you're dealing with gases, especially mixtures collected over water. This law states that the total pressure exerted by a mixture of non-reacting gases is equal to the sum of the partial pressures of individual gases. In simple terms, each gas in a mixture exerts pressure as if it were the only gas present in that space.
For instance, in our exercise, the total pressure measured included not only the pressure from hydrogen gas but also the vapor pressure from water since the gas was collected over water. By applying Dalton's Law, we subtract the water's vapor pressure from the total to find the pressure of just the dry hydrogen gas. Remember, water has its own vapor pressure, especially when not at 0°C. At 20°C, it is 17.5 mm Hg, and therefore needs to be subtracted from the total pressure to get the true pressure of the hydrogen gas. By doing this, we can isolate the pressure that only the hydrogen contributes to the total pressure.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) is a reference point used in chemistry to allow scientists to compare results. At STP, the temperature is a cool 0°C, or 273 Kelvin, and the pressure is exactly 1 atmosphere (atm).
This serves as a standard so that different experiments can be compared, because gas volumes fluctuate significantly with both temperature and pressure. If gas properties were reported under varying conditions, it would be hard to make accurate comparisons between different data sets.
In our problem, we are asked to convert the volume of hydrogen gas from the conditions it was initially collected in (20°C and 758 mm Hg) to STP conditions. This involves using the combined gas law, which helps convert the state of a gas when volume, pressure, and temperature change. Remember, at STP, conditions are standardized, which makes calculations, predictions, and comparisons relevant and consistent.

Gas Laws

Gas laws govern the theories and math operations that explain how gases behave. Among these, the Combined Gas Law interconnects Charles’s Law, Boyle’s Law, and Gay-Lussac's Law. It provides an understanding of how gases respond when the temperature, volume, and pressure change, but the number of molecules remains constant.
When using the Combined Gas Law, keep in mind the formula: \[\frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2}\]This relationship is invaluable for solving problems involving gas collected initially in one condition and then adjusted to another set of conditions (like STP). These calculations allow for precise changes calculation, showing exactly how a change in pressure or temperature affects the volume of gas.

• Remember, all temperatures should be converted to Kelvin (K = °C + 273), as this is the absolute scale used in gas laws.
• Pressures must be in consistent units, which often involves conversion, like from mm Hg to atm.
• Volume calculations should take into account these conditions, resulting in predictions and calculations that match real-world experiments.

Understanding gas laws is crucial because gases play a vital role in various fields, from meteorology to engineering. This knowledge is part of what makes scientists capable of predicting how gases will behave under diverse conditions.