Question

The normal boiling point for methyl hydrazine \(\left(\mathrm{CH}_{3} \mathrm{~N}_{2} \mathrm{H}_{3}\right)\) is \(87^{\circ} \mathrm{C}\). It has a vapor pressure of \(37.0 \mathrm{~mm} \mathrm{Hg}\) at \(20^{\circ} \mathrm{C}\). What is the concentration (in \(\mathrm{g} / \mathrm{L}\) ) of methyl hydrazine if it saturates the air at \(25^{\circ} \mathrm{C}\) ?

Short answer

Answer: The concentration of methyl hydrazine in air at 25°C is 0.123 g/L.

Step-by-step solution

Use the Clausius-Clapeyron equation to find the vapor pressure at the desired temperature.

We'll be using the following form of the Clausius-Clapeyron equation: \(\ln \frac{P_{2}}{P_{1}}=-\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)\) where: \(P_{1}=37.0 \text{ mm Hg}\) (vapor pressure at \(T_{1}=20^{\circ} \mathrm{C}\)) \(T_{1}=20+273.15=293.15\mathrm{K}\) \(P_{2}\) is the vapor pressure at \(T_{2}=25^{\circ} \mathrm{C}\) \(T_{2}=25+273.15=298.15\mathrm{K}\) \(\Delta H_{vap}\) is the enthalpy of vaporization, which is constant for a given substance \(R=8.314 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}\) (universal gas constant) To find the enthalpy of vaporization, we first need to find out the molar mass of methyl hydrazine: \(\mathrm{CH}_{3} \mathrm{~N}_{2} \mathrm{H}_{3}\) Molar mass: \(12.01 \mathrm{g/mol} (\mathrm{C})+3(1.008 \mathrm{g/mol})(\mathrm{H})+2(14.007 \mathrm{g/mol})(\mathrm{N})=45.08 \mathrm{g/mol}\) Since methyl hydrazine boils at \(87^{\circ} \mathrm{C}\), we can assume that \(\Delta H_{vap} \approx 40.7 \text{ kJ/mol}\) which is the typical value for substances with a similar boiling point. Now we can use the Clausius-Clapeyron equation to find \(P_{2}\): \(\ln \frac{P_{2}}{37.0}=-\frac{40700}{8.314}\left(\frac{1}{298.15}-\frac{1}{293.15}\right)\) Solving for \(P_{2}\): \(P_{2}=37.0 \cdot e^{-\frac{40700}{8.314}\left(\frac{1}{298.15}-\frac{1}{293.15}\right)}=50.72 \mathrm{~mm} \mathrm{Hg}\)

Use the Ideal Gas Law to find the concentration of methyl hydrazine in g/L

Using the ideal gas law: \(PV=nRT\), we can find the concentration of methyl hydrazine. First, convert the vapor pressure from mm Hg to atm: \(P_{2}=\frac{50.72 \mathrm{~mm} \mathrm{Hg}}{760 \mathrm{~mm} \mathrm{Hg/atm}}=0.0668 \mathrm{~atm}\) Now, we will use the ideal gas law to find the number of moles of methyl hydrazine in one liter of air (\(25^{\circ} \mathrm{C}\)): \(0.0668 \mathrm{V}=n(0.0821 \cdot 298.15)\) Solving for n, assuming V = 1L: \(n=\frac{0.0668}{0.0821 \cdot 298.15}=2.72 \times 10^{-3} \text{mol}\) Finally, we will convert the number of moles to the concentration in g/L using the molar mass: Concentration = \(n \cdot \text{molar mass} =2.72 \times 10^{-3} \text{mol/L} \cdot 45.08\text{g/mol}=0.123 \mathrm{g} / \mathrm{L}\) Thus, the concentration of methyl hydrazine if it saturates the air at \(25^{\circ} \mathrm{C}\) is \(0.123 \mathrm{g} / \mathrm{L}\).

Key concepts

Vapor Pressure

Vapor pressure is a key concept in understanding phase changes in liquids. It refers to the pressure exerted by the vapor of a liquid in equilibrium with its liquid phase at a given temperature. As the temperature increases, the kinetic energy of molecules in a liquid increases, which leads to more molecules escaping into the vapor phase. This results in an increase in vapor pressure.

Vapor pressure depends on the nature of the liquid and the temperature. For instance, a liquid with strong intermolecular forces will generally have a lower vapor pressure since the particles are not as free to enter the vapor phase. In the exercise, the vapor pressure of methyl hydrazine at a specific temperature was needed to further compute its concentration in air. We saw that at 20°C, the vapor pressure is 37.0 mm Hg. Finding the vapor pressure at 25°C using the Clausius-Clapeyron equation helps us determine how much of the liquid will evaporate into the air and potentially impact the concentration calculations.

Enthalpy of Vaporization

The enthalpy of vaporization ( $ Delta H_{vap} ) is the energy required to convert one mole of a liquid into its vapor phase at a constant temperature. It is an intrinsic property of each substance and reflects how much energy is needed to overcome intermolecular forces during the phase change. For substances with higher boiling points, the enthalpy of vaporization tends to be higher as well, since stronger forces must be overcome to vaporize the substance.

In this task, we assumed that the enthalpy of vaporization of methyl hydrazine was approximately 40.7 kJ/mol. This assumption helped us calculate the change in vapor pressure from one temperature to another using the Clausius-Clapeyron equation. This process shows the importance of $ Delta H_{vap} in predicting how a substance behaves under temperature changes.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the four basic properties of gases: pressure, volume, number of moles, and temperature, expressed by the equation \( PV = nRT \). Here, \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.This law assumes that gas molecules occupy no volume and have no intermolecular forces - an approximation that simplifies calculations but holds true mostly for ideal gases under a range of conditions.

In our solution, we used the Ideal Gas Law to calculate the concentration of methyl hydrazine in air, assuming the air volume was 1 liter and the vapor was behaving ideally. By converting the calculated vapor pressure to atmospheres, we solved it with the Ideal Gas Law to determine the number of moles per liter. Converting moles to grams using the molar mass gave us the concentration in grams per liter ($mathrm{g} / mathrm{L}), answering our original question.