Question

What quantity of energy is evolved (in joules) when 1.00 mol of liquid ammonia cools from \(-33.3^{\circ} \mathrm{C}\) (its boiling point) to \(-43.3^{\circ} \mathrm{C} ?\) (The specific heat capacity of liquid \(\mathrm{NH}_{3}\) is \(4.70 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\).) Compare this with the quantity of heat evolved by 1.00 mol of liquid water cooling by exactly \(10^{\circ} \mathrm{C} .\) Which evolves more heat per mole on cooling \(10^{\circ} \mathrm{C},\) liquid water or liquid ammonia? (The underlying reason for the difference in energy evolved is scientifically illuminating and interesting. You can learn more by searching the Internet for specific heat capacity and its dependence on molecular properties.)

Short answer

Ammonia evolves more heat than water per mole when cooled by \(10^{\circ} \mathrm{C}\).

Step-by-step solution

Converting Temperature from Celsius to Kelvin

The temperature change is the same in Celsius and Kelvin; however, to use the formula later, ensure temperatures are in Kelvin for consistency. 1. Initial temperature, \( T_1 = -33.3^{\circ} \mathrm{C} \). 2. Final temperature, \( T_2 = -43.3^{\circ} \mathrm{C} \).The temperature change is:\[ \Delta T = T_1 - T_2 = -33.3 - (-43.3) = 10.0^{\circ} \mathrm{C} = 10.0 \text{ K} \]

Calculating Mass of 1 mol of Liquid Ammonia

Molar mass of ammonia, \(\mathrm{NH_3}\) is given as approximately 17.03 g/mol.Thus, the mass of 1 mole of ammonia is:\[ m = 17.03 \text{ g} \]

Finding Energy Evolved for Ammonia Cooling

The formula for energy evolved or absorbed is given by:\[ q = m \cdot c \cdot \Delta T \]where:- \( m = 17.03 \text{ g} \) (mass of ammonia)- \( c = 4.70\, \mathrm{J/g \cdot K} \) (specific heat capacity of ammonia)- \( \Delta T = 10.0 \text{ K} \)Substitute these values to calculate the energy:\[ q_{\text{NH}_3} = 17.03 \times 4.70 \times 10.0 = 800.41 \text{ J} \]

Calculating Energy Evolved for Water Cooling

For liquid water,- Specific heat capacity \( c = 4.18 \mathrm{J/g \cdot K} \)- Molar mass of water is approximately 18.02 g/mol.Using the same formula, with:\[ q = m \cdot c \cdot \Delta T \]Where:- \( m = 18.02 \text{ g} \)- \( \Delta T = 10.0 \text{ K} \)Calculate the energy:\[ q_{\text{H}_2\text{O}} = 18.02 \times 4.18 \times 10.0 = 753.88 \text{ J} \]

Comparing Energy Evolved

To determine which substance evolves more energy:- Ammonia evolves \(800.41\, \text{J/mol}\).- Water evolves \(753.88\, \text{J/mol}\).Since \(800.41 > 753.88\), ammonia evolves more heat than water per mole when cooled by \(10^{\circ} \mathrm{C}\).

Key concepts

Energy Evolution

When discussing the concept of energy evolution, it's essential to understand what represents energy evolving in a system. In simple terms, energy evolution, in this context, refers to the release of thermal energy when a substance cools. As temperature falls, the internal energy decreases, resulting in the evolution of heat. For substances like ammonia and water, this means these substances release energy as they cool down.

The amount of energy released or evolved depends greatly on the substance’s specific heat capacity, given by the formula: \[ q = m \cdot c \cdot \Delta T \] where:

• \( q \) is the energy in joules,
• \( m \) is the mass in grams,
• \( c \) is the specific heat capacity,
• \( \Delta T \) is the temperature change in Kelvin.

In our exercise, ammonia, with its specific heat capacity of 4.70 J/g·K, releases 800.41 J when it cools. Comparatively, water, having a slightly lower specific heat capacity of 4.18 J/g·K, evolves 753.88 J for the same process, albeit slightly less than ammonia.

Temperature Change

Temperature change is critical in understanding energy evolution in thermodynamics. When you have a substance like ammonia cooling from one temperature to a lower one, the change in temperature \( \Delta T \) is what drives the energy evolution. In our exercise, both water and ammonia cool by exactly 10°C.

Despite the same temperature drop, the energy released differs due to their molecular properties. The change from Celsius to Kelvin for temperature calculations is straightforward, given that the padding for Kelvin (+273.15) is just scalar and does not affect differences. Hence, a fall in 10°C equals a fall of 10 K in energy calculations, keeping the math manageable without unnecessary conversions during calculations.

• Initial temperature: -33.3°C
• Final temperature: -43.3°C
• Temperature change, \( \Delta T \)= 10°C or 10 K

Molecular Properties

Molecular properties have a fascinating influence on specific heat capacity, which in turn affects how much energy is evolved during cooling. The heat capacity of a substance depends on its molecular structure and the bonds holding the atoms together. Generally, larger and more complex molecules or those with more hydrogen bonds like water tend to have different heat capacities compared to simpler molecules like ammonia.

With ammonia and water, both of which are relatively simple molecules, the disparity in energy release is largely due to the difference in molecular weights and the arrangement of atoms in the molecules. Ammonia, with a molecular weight of 17.03 g/mol, releases more energy upon cooling than water (18.02 g/mol) for the same temperature change. This suggests that despite ammonia’s slightly lower density and molecular weight, it has a higher specific heat capacity due to different intermolecular forces and structural properties, enabling it to release more energy when cooled by 10°C or 10 K.