Problem 101 - Chemistry: An Atoms-Focused Approach

Question

A sample of oxygen is collected over water at \(25^{\circ} \mathrm{C}\) and 1.00 atm. If the total sample volume is 0.480 \(\mathrm{L}\), how many moles of \(\mathrm{O}_{2}\) are collected?

Step-by-Step Solution

Answer

Answer: Approximately 0.0195 moles of O₂ are present in the collected sample.

Step 1: Convert temperature to Kelvin

Given that the temperature is \(25^{\circ} \mathrm{C}\), we need to convert it to Kelvin:$$ T_{K} = T_{\mathrm{C}} + 273.15 $$So, the temperature in Kelvin is:$$ T_{K} = 25 + 273.15 = 298.15 \, \mathrm{K} $$

Step 2: Find the vapor pressure of water

We need to find the vapor pressure of water at \(25^{\circ} \mathrm{C}\) (298.15 K) to correct the pressure of the gas sample. It can be found in a reference table or online. For this problem, the vapor pressure of water at \(25^{\circ} \mathrm{C}\) is approximately 23.76 mmHg.

Step 3: Convert vapor pressure to atm

The given pressure is in atm, so we need to convert the vapor pressure of water (which is in mmHg) to atm using the conversion factor:$$ 1 \, \mathrm{atm} = 760 \, \mathrm{mmHg} $$Hence, the vapor pressure of water in atm:$$ P_{\mathrm{H_2O}} = \frac{23.76 \, \mathrm{mmHg}}{760 \, \mathrm{mmHg}} = 0.0313 \, \mathrm{atm} $$

Step 4: Subtract the vapor pressure from the total pressure

Now, subtract the vapor pressure of water from the total pressure to find the pressure of \(\mathrm{O}_{2}\):$$ P_{\mathrm{O_2}} = P_{\mathrm{total}} - P_{\mathrm{H_2O}} = 1.00 \, \mathrm{atm} - 0.0313 \, \mathrm{atm} = 0.9687 \, \mathrm{atm} $$

Step 5: Use the ideal gas law to find the number of moles

Use the ideal gas law equation, \(PV = nRT\), to find the number of moles (\(n\)) of \(\mathrm{O}_{2}\):$$ n = \frac{PV}{RT} $$We know the pressure \(P_{\mathrm{O_2}}\), the volume \(V\) is 0.480 L, \(R\) is the gas constant (\(0.0821 \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}}\)), and the temperature \(T_K\). Plug in the values:$$ n = \frac{0.9687 \, \mathrm{atm} \cdot 0.480 \, \mathrm{L}}{0.0821 \, \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}} \cdot 298.15 \, \mathrm{K}} \approx 0.0195 \, \mathrm{mol} $$So, there are approximately 0.0195 moles of \(\mathrm{O}_{2}\) in the collected sample.

Key Concepts

Vapor Pressure

Vapor pressure is an important concept when gases are collected over water. It refers to the pressure exerted by the vapor that is in equilibrium with its liquid form. At a given temperature, a liquid exerts a characteristic vapor pressure, which increases with temperature.

In the context of collecting gas over water, the vapor pressure of water needs to be taken into account because it contributes to the overall pressure measured. This is why, when determining the pressure of a collected gas sample, like oxygen, one must subtract the vapor pressure of water from the total pressure.

For example, at 25°C, the vapor pressure of water is about 23.76 mmHg. It's important to convert this value to atmospheres since most pressure measurements and calculations involving the Ideal Gas Law are done in atm:

• The conversion is performed using the relation 1 atm = 760 mmHg.
• This yields a vapor pressure of water of approximately 0.0313 atm.

Gas Collection Over Water

When gases are collected using the method of displacement of water, they are always mixed with water vapor. This setup makes it imperative to correct the gas pressure by accounting for water's vapor pressure. The total pressure in this setup is the sum of the partial pressures of the gas collected and the water vapor present.

This is why understanding how to separate these components is crucial. By subtracting the water vapor pressure from the total pressure, you obtain the pressure of the dry gas. In practice:

• Find the vapor pressure of water at the given temperature from a standard reference.
• Convert the vapor pressure of water to the same units as the total pressure.
• Subtract the water vapor pressure from the total pressure to find the pressure of the gas alone.

Temperature Conversion

In chemistry, temperature is often more conveniently measured in Kelvin, especially when dealing with gas laws. The Kelvin scale starts at absolute zero, the thought lowest thermal motion point possible in physics. To convert Celsius to Kelvin, you simply add 273.15.

For instance, in the problem where gas is collected at 25°C, it must be converted:

• Add 273.15 to the Celsius temperature to obtain the equivalent temperature in Kelvin.
• Thus, 25°C becomes 298.15 K.

Why is this important? The Ideal Gas Law, which establishes the relationship among pressure, volume, number of moles, and temperature, utilizes the Kelvin scale because it ensures that the calculations always involve positive numbers, and accurately reflect thermal motion.

Dalton's Law of Partial Pressures

Dalton's Law of Partial Pressures is a key principle in understanding gas mixtures. It 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. Each gas in a mixture behaves independently and exerts a pressure as if it were alone in the container.

In the context of gases collected over water, Dalton's Law is applied to find the pressure of the dry gas. This law supports the method of subtracting the vapor pressure of water from the total pressure measured to isolate the oxygen pressure:

• Measure the total pressure over the liquid.
• Determine the vapor pressure of water at the given temperature.
• Subtract the vapor pressure from the total pressure to discover the dry gas pressure.

Understanding this principle is vital for accurately gauging the properties of individual gases in a mixture, such as collected oxygen in this exercise.