Question 9.5: A calorimeter is to be used to compare the energy content of......
A calorimeter is to be used to compare the energy content of some fuels. In the calibration of the calorimeter, an electrical resistance heater supplies 100.0 J of heat and a temperature increase of 0.850°C is observed. Then 0.245 g of a particular fuel is burned in this same calorimeter, and the temperature increases by 5.23°C. Calculate the energy density of this fuel, which is the amount of energy liberated per gram of fuel burned.
Strategy The calibration step allows us to determine the calorimeter constant. Once this is known, the amount of heat evolved from the fuel can be determined by using Equation 9.6. Finally, we divide this heat by the mass of fuel that generated it to arrive at the requested energy density.
q = C_{\text{calorimeter}} × ΔT (9.6)
Step 1: Calibration
q=C_{\mathrm{calorimeter}}\times\Delta T
so
C_{\mathrm{calorimeter}}=q/\Delta T
= 100.0 J/0.850°C
C_{\mathrm{calorimeter}}=118\,J/^{\circ}\mathrm{C}
Step 2: Determination of heat evolved by fuel
q_{\mathrm{calorimeter}}=C_{\mathrm{calorimeter}}\times\Delta T
= 118 J/°C × 5.23°C
= 615 J
And
q_{\mathrm{fuel}}=-q_{\mathrm{calorimeter}}=-615\,\mathrm{J}
Step 3: Calculation of the energy density
Energy density =-q_{\mathrm{fuel}}/m
= –(–615 J)/0.245 g
= 2510 J/g = 2.51 kJ/g
Discussion This problem illustrates the need to be careful with signs in thermodynamic calculations. Because the burning of fuel releases heat, q for the fuel should be negative. The energy density, though, would be reported as a positive number, resulting in the additional negative sign in the final step.
Check Your Understanding The combustion of naphthalene (C_{10}H_{8}), which releases 5150.1 kJ/mol, is often used to calibrate calorimeters. A 1.05-g sample of naphthalene is burned in a calorimeter, producing a temperature rise of 3.86°C. Burning a 1.83-g sample of coal in the same calorimeter causes a temperature change of 4.90°C. What is the energy density of the coal?