Question

A 1-g sample of a certain fuel is burned in a bomb calorimeter that contains 2 kg of water in the presence of \(100 \mathrm{g}\) of air in the reaction chamber. If the water temperature rises by \(2.5^{\circ} \mathrm{C}\) when equilibrium is established, determine the heating value of the fuel in \(\mathrm{kJ} / \mathrm{kg}\).

Short answer

Question: Given the mass of water, the mass of air, the change in temperature, and the mass of fuel burned, determine the heating value of the fuel in kJ/kg. Given data: - Mass of water = 2000 kg - Mass of air = 100 kg - Change in temperature = 2.5 K - Mass of fuel burned = 0.001 kg - Specific heat capacity of water = 4185 J/(kg*K) - Specific heat capacity of air = 1005 J/(kg*K) Solution steps: 1. Calculate the heat absorbed by water and air (Q_water and Q_air, respectively) using the specific heat capacities and change in temperature. 2. Determine the total heat absorbed (Q_total = Q_water + Q_air). 3. Calculate the energy released by the fuel (E_fuel = Q_total). 4. Compute the energy per mass of the fuel sample (E_mass = E_fuel / mass of fuel). 5. Convert the energy per mass of the fuel sample to the heating value in kJ/kg (HV = E_mass/1000 kJ/kg).

Step-by-step solution

Calculate the heat absorbed by water and air when the fuel was burned

Let's calculate the heat absorbed by the water and air. The specific heat capacities of water and air are 4185 J/(kgଏC) and 1005 J/(kgଏC) respectively. Heat absorbed by water (\(Q_{water}\)) = mass of water * specific heat capacity of water * change in temperature \(Q_{water} = 2000 \times(4185 \frac{\text{J}}{\text{kg} \times \text{K}}) \times (2.5 \text{K})\) Now, we will find the heat absorbed by the air (\(Q_{air}\)). Heat absorbed by air (\(Q_{air}\)) = mass of air * specific heat capacity of air * change in temperature \(Q_{air} = 100 \times(1005\frac{\text{J}}{\text{kg}\times \text{K}}) \times (2.5 \text{K})\) Now, we need to find the total heat absorbed (\(Q_{total}\)). \(Q_{total} = Q_{water} + Q_{air}\)

Determine the energy released by the fuel

Now that we've calculated the total heat absorbed by the water and air, we need to find the energy released by the fuel. Since the energy released by the fuel (\(E_{fuel}\)) should be equal to the heat absorbed by the water and air, we have: \(E_{fuel} = Q_{total}\)

Compute the energy per mass of the fuel sample

Now that we've found the energy released by the fuel, let's compute the energy per mass of the fuel sample. Energy per mass (\(E_{mass}\)) = \(\frac{E_{fuel}}{\text{mass of fuel}}\) \(E_{mass} = \frac{E_{fuel}}{0.001 \text{kg}}\)

Convert the energy per mass of the fuel sample to the heating value in kJ/kg

Finally, we will convert the energy per mass of the fuel sample to the heating value in kJ/kg. Heating value of the fuel (HV) = \( \frac{E_{mass}}{1000} \frac{\text{kJ}}{\text{kg}}\) Now, you can calculate the numerical values and obtain the heating value of the fuel in kJ/kg.