Question

Assume that for a domestic hot water supply, \(160 \mathrm{~kg}\) of water per day must be heated from \(10^{\circ} \mathrm{C}\) to \(60^{\circ} \mathrm{C}\) and gaseous fuel propane, \(\mathrm{C}_{3} \mathrm{H}_{8}\), is used for this purpose. What volume of propane gas at STP would have to be used for heating domestic water, with efficiency of \(40 \%\) ? Heat of combustion of propane is \(-500 \mathrm{kcal} / \mathrm{mol}\) and specific heat capacity of water is \(1.0 \mathrm{cal} / \mathrm{K}-\mathrm{g}\). (a) \(896 \mathrm{~L}\) (b) \(908 \mathrm{~L}\) (c) \(896 \mathrm{~m}^{3}\) (d) \(908 \mathrm{~m}^{3}\)

Short answer

The volume of propane gas at STP required is 896 L.

Step-by-step solution

Calculate the heat required to raise the temperature of water

First, determine the amount of heat needed to raise the temperature of water from 10°C to 60°C using the formula: \( Q = m \cdot c \cdot \Delta T \), where \( m = 160 \text{ kg} \), \( c = 1.0 \text{ cal/K-g} \), and \( \Delta T = (60-10) \text{°C} = 50 \text{°C} \). Convert \( m \) from kilograms to grams by multiplying by 1000 (since 1 kg = 1000 g). Apply the formula to find \( Q \).

Calculate the heat provided by the combustion of propane

Using the efficiency of the heating system, calculate the actual amount of heat provided when burning propane. The amount of heat released per mole of propane at 100% efficiency is given as 500 kcal/mol. If the efficiency is 40%, the actual heat provided per mole is \( 500 \times 0.40 \).

Determine the moles of propane required

Divide the total heat required to heat the water by the actual heat provided by one mole of propane to find the number of moles of propane needed: \( n = \frac{Q}{\text{heat per mole of propane at given efficiency}} \).

Calculate the volume of propane at STP

At STP conditions, one mole of any gas occupies 22.4 L. Use this information and the number of moles calculated in the previous step to determine the volume of propane gas required: \( V = n \times 22.4 \text{ L/mol} \).

Select the correct answer

Based on the calculations, compare the volume calculated with the answers provided and select the option that is mathematically correct.

Key concepts

Thermochemistry

Thermochemistry is a branch of physical chemistry that deals with the energy changes associated with chemical and physical processes. It's pivotal in calculating how heat is either absorbed or released during these reactions. A fundamental concept in thermochemistry is the heat of combustion, which is the amount of heat released when a substance undergoes complete combustion with oxygen under standard conditions.

To understand thermochemistry in the context of our exercise, consider the combustion of propane to heat water. Energy is transferred from the chemical bonds in propane to the water, increasing its temperature. The efficiency of this energy transfer is vital as it determines how much propane is needed to achieve the desired heating effect. Thermochemical calculations are essential for a range of applications, from designing energy-efficient heating systems to understanding the environmental impact of fuel usage.

Molar Enthalpy

Molar enthalpy, also known as molar heat of combustion, refers to the amount of heat released when one mole of a substance combusts completely. It is usually measured in kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol). In the given exercise, the molar enthalpy of propane is provided as -500 kcal/mol, which means that 500 kcal of energy are released in the form of heat when one mole of propane burns.

This 'negative' sign indicates the exothermic nature of the reaction – energy is released to the surroundings. The term also signifies that energy is a product of the reaction, rather than a reactant. Molar enthalpy helps us calculate the total heat exchanged in a reaction and, subsequently, the amount of fuel needed when considering the efficiency of the process. It's relevant to energy management, cost calculation, and environmental concerns in various industrial processes.

Specific Heat Capacity

Specific heat capacity is a property of a substance that indicates the amount of heat required to raise the temperature of one gram of the substance by one degree Celsius. It is expressed in units of energy per mass per temperature, such as cal/g°C or J/g°C. Water, with its high specific heat capacity of 1.0 cal/g°C, can absorb a lot of heat before increasing in temperature, which makes it an excellent medium for thermal storage and transfer.

In our exercise scenario, understanding the specific heat capacity is crucial to determine how much heat energy is needed to raise the temperature of the water from one temperature to another. It allows us to calculate the thermal energy change involved in heating water and designing heating systems in applications ranging from domestic to industrial settings. Accurate measurements of specific heat capacity are vital for precision in thermal engineering and energy conservation strategies.