Problem 64 - Chemistry and Chemical Reactivity

Question

Vapor pressures of \(\mathrm{NH}_{3}(\ell)\) at several temperatures are given in the table below. Use this information to calculate the enthalpy of vaporization of ammonia. $$\begin{array}{cc} \text { Temperature }\left(^{\circ} \mathrm{C}\right) & \text { Vapor Pressure }(\mathrm{atm}) \\ \hline-68.4 & 0.132 \\ -45.4 & 0.526 \\ -33.6 & 1.000 \\\ -18.7 & 2.00 \\ 4.7 & 5.00 \\ 25.7 & 10.00 \\ 50.1 & 20.00 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The enthalpy of vaporization of ammonia is approximately 30450 J/mol.

Step 1: Understanding the Clausis-Clapeyron Equation

The enthalpy of vaporization, \(\Delta H_{vap}\), can be calculated using the Clausius-Clapeyron equation: \[ \ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \] where \(P_1\) and \(P_2\) are vapor pressures at temperatures \(T_1\) and \(T_2\) respectively, and \(R\) is the universal gas constant \(8.314 \frac{J}{mol\cdot K}\).

Step 2: Selecting Data Points

From the table, choose two data points to use in the Clausius-Clapeyron equation. For example, let's select \(T_1 = -68.4^\circ C\) with \(P_1 = 0.132\) atm and \(T_2 = -33.6^\circ C\) with \(P_2 = 1.000\) atm.

Step 3: Convert Temperatures to Kelvin

Convert the temperatures from Celsius to Kelvin using the formula: \[ T(K) = T(°C) + 273.15 \] Hence, \(T_1 = -68.4 + 273.15 = 204.75\ K\) and \(T_2 = -33.6 + 273.15 = 239.55\ K\).

Step 4: Plug Values into the Clausius-Clapeyron Equation

Substitute the selected pressures and temperatures into the Clausius-Clapeyron equation: \[ \ln \left( \frac{1.000}{0.132} \right) = -\frac{\Delta H_{vap}}{8.314} \left( \frac{1}{239.55} - \frac{1}{204.75} \right) \] Solve for \(\Delta H_{vap}\).

Step 5: Solve for Enthalpy of Vaporization

Calculate the left side: \( \ln \left( \frac{1.000}{0.132} \right) = \ln(7.576) = 2.026 \). The right side is: \( \left( \frac{1}{239.55} - \frac{1}{204.75} \right) = -0.0005539\). So, \[ 2.026 = -\frac{\Delta H_{vap}}{8.314} \times -0.0005539 \] Re-arranging gives: \[ \Delta H_{vap} = 2.026 \times \frac{8.314}{0.0005539} \] \(\Delta H_{vap} \approx 30450\ J/mol \).

Key Concepts

Clausius-Clapeyron equation

The Clausius-Clapeyron equation is a powerful tool in thermodynamics that helps us understand the relationship between the vapor pressure of a substance and its temperature. It is particularly useful for determining the enthalpy of vaporization ( \(\Delta H_{vap}\)), which is the energy needed to convert a liquid into a gas at a constant pressure. This equation can be expressed as: \[ \ln \left( \frac{P_2}{P_1} \right) = -\frac{\Delta H_{vap}}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right) \]Where:

• \(P_1\) and \(P_2\) are the vapor pressures at temperatures \(T_1\) and \(T_2\) respectively.
• \(R\) is the universal gas constant, commonly \(8.314 \, \text{J/mol•K}\).

This formula is linear if you plot \(\ln(P)\) against \(1/T\). This makes it easier to calculate \(\Delta H_{vap}\) given vapor pressures at different temperatures. It's fundamental in predicting how a substance behaves as it changes state under different conditions.

vapor pressure

Vapor pressure is an essential concept in understanding phase changes. It refers to the pressure exerted by a vapor in equilibrium with its liquid or solid phase. Essentially, when a liquid is placed in a closed container, molecules at the surface escape into the vapor phase. As more molecules vaporize, they create pressure. There are a few key points to note about vapor pressure:

• Vapor pressure increases with temperature: As temperature rises, more molecules have sufficient energy to escape the liquid phase, increasing the vapor pressure.
• Dynamic equilibrium: When the liquid and vapor reach a state where the rate of evaporation equals the rate of condensation, the vapor pressure becomes constant.
• Unique property: Each substance has a specific vapor pressure at a given temperature, which is related to the strength of intermolecular forces.

Understanding vapor pressure is crucial in applications such as weather prediction, air conditioning, and even cooking, where knowing how a liquid will behave as it volatilizes can lead to better outcomes.

ammonia

Ammonia is a crucial compound with a formula \(\text{NH}_3\). It is a colorless gas with a distinct, pungent odor and is highly soluble in water. Ammonia is easily liquefied under pressure, making it ideal for various industrial applications, such as refrigeration and fertilizer production.Here are some key characteristics and uses of ammonia:

• Boiling Point: Ammonia boils at \(-33.34^\circ\text{C}\) or \(239.81 \, \text{K}\), which explains its high ease in transitioning to the vapor phase when heated.
• Applications: Besides fertilizers, ammonia is used in cleaning products and as a precursor for many nitrogen-containing compounds.
• Safety: Although useful, ammonia requires careful handling due to its corrosiveness and ability to form explosive mixtures with air.

Its properties of high vapor pressure and easy liquefaction make ammonia a versatile agent in both household and industrial settings. Understanding ammonia's behavior under different temperatures and pressures helps design systems that optimize its use without compromising safety.