To determine how long the cooking gas will last for a family of three, we first need to calculate the total energy available from the butane in the cylinder and then see how many days that energy can provide the required daily cooking energy.
Calculating the Total Energy from Butane
We know that the cylinder contains 14 kg of butane (C4H10). The first step is to convert this mass into moles, as the heat of combustion is given per mole.
Step 1: Convert Mass to Moles
The molar mass of butane (C4H10) can be calculated as follows:
- Carbon (C): 12.01 g/mol × 4 = 48.04 g/mol
- Hydrogen (H): 1.008 g/mol × 10 = 10.08 g/mol
- Total Molar Mass = 48.04 g/mol + 10.08 g/mol = 58.12 g/mol
Now, we can convert the mass of butane in the cylinder to moles:
Number of moles = Mass (g) / Molar Mass (g/mol) = 14,000 g / 58.12 g/mol ≈ 240.5 moles
Step 2: Calculate Total Energy Available
Using the heat of combustion (ΔHc) for butane, which is -2600 kJ/mol, we can find the total energy released when all the butane is burned:
Total energy = Number of moles × ΔHc = 240.5 moles × (-2600 kJ/mol) = -624,000 kJ
Since we are interested in the energy available for cooking, we can consider the absolute value:
Total energy available = 624,000 kJ
Daily Energy Requirement
The family requires 10,000 J of heat energy per day. To make calculations easier, we can convert this energy requirement into kilojoules:
Daily energy requirement = 10,000 J = 10 kJ
Determining the Duration of the Gas Supply
Now, we can find out how many days the gas will last by dividing the total energy available by the daily energy requirement:
Number of days = Total energy available / Daily energy requirement = 624,000 kJ / 10 kJ = 62,400 days
Final Thoughts
This means that the cooking gas in the cylinder would last an astonishing 62,400 days, which is approximately 171 years! However, this theoretical calculation assumes complete combustion and no energy losses, which is rarely the case in practical scenarios. Factors such as efficiency of the cooking appliance and heat losses to the environment would significantly reduce this duration in real-life situations.
