Question

Under constant-volume conditions the heat of combustion of benzoic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}\right)\) is \(26.38 \mathrm{~kJ} / \mathrm{g}\). A \(1.640\) \(g\) sample of benzoic acid is burned in a bomb calorimeter. The temperature of the calorimeter increases from \(22.25^{\circ} \mathrm{C}\) to \(27.20^{\circ} \mathrm{C}\). (a) What is the total heat capacity of the calorimeter? (b) A 1.320-g sample of a new organic substance is combusted in the same calorimeter. The temperature of the calorimeter increases from \(22.14^{\circ} \mathrm{C}\) to \(26.82^{\circ} \mathrm{C}\). What is the heat of combustion per gram of the new substance? (c) Suppose that in changing samples, a portion of the water in the calorimeter were lost. In what way, if any, would this change the heat capacity of the calorimeter?

Short answer

The total heat capacity of the calorimeter is 8.738 kJ/°C. The heat of combustion per gram of the new substance is 30.98 kJ/g. Losing water in the calorimeter would decrease the overall mass of the calorimeter, and subsequently decrease its heat capacity.

Step-by-step solution

Calculate the heat released by the combustion of benzoic acid

To find the heat released by the combustion of benzoic acid, we will use the heat of combustion per gram and the mass of the benzoic acid sample. The formula for this is: Heat released = (Heat of combustion per gram) × (Mass of benzoic acid) Plugging in the given values, we get: Heat released = (26.38 kJ/g) × (1.640 g) = 43.2624 kJ

Calculate the change in temperature of the calorimeter during the combustion of benzoic acid

The change in temperature is given by the difference between the final and the initial temperature. ΔT = T(final) - T(initial) = 27.20°C - 22.25°C = 4.95°C

Determine the total heat capacity of the calorimeter

To find the total heat capacity of the calorimeter, we will use the heat released by the combustion and the change in temperature: Total heat capacity = (Heat released) / (ΔT) = 43.2624 kJ / 4.95°C = 8.738 kJ/°C The total heat capacity of the calorimeter is 8.738 kJ/°C.

Calculate the heat released by the combustion of new organic substance

First, we need to find the change in temperature during the combustion of the new substance: ΔT(new substance) = T(final) - T(initial) = 26.82°C - 22.14°C = 4.68°C Now, using the total heat capacity of the calorimeter, we can find the heat released by the combustion of the new substance: Heat released(new substance) = (Total heat capacity) × (ΔT(new substance)) = 8.738 kJ/°C × 4.68°C = 40.89744 kJ

Calculate the heat of combustion per gram of the new substance

Finally, we will find the heat of combustion per gram of the new organic substance using the heat released by the combustion and the mass of the new substance: Heat of combustion(new substance) = (Heat released(new substance)) / (Mass of new substance) = 40.89744 kJ / 1.320 g = 30.98 kJ/g The heat of combustion per gram of the new substance is 30.98 kJ/g.

Analyze the effect of losing water in the calorimeter on the heat capacity

If a portion of the water in the calorimeter is lost, the overall mass of the calorimeter will decrease. Since heat capacity is an extensive property (meaning it depends on the amount of substance or mass), the heat capacity of the calorimeter will also decrease as a result.

Key concepts

Bomb Calorimeter

A bomb calorimeter is an apparatus designed to measure the heat of combustion of a substance under constant-volume conditions. It is a sealed container, able to withstand high pressures, where a sample is combusted in an oxygen-rich environment. During the combustion process, the released heat is absorbed by the walls of the calorimeter and a surrounding water bath, causing a measurable temperature increase.

The accurate measurement of temperature changes is fundamental to determining the heat of reaction. Calorimeters are insulated to minimize heat exchange with the environment, ensuring that most of the heat from the reaction is used to raise the temperature of the system within. In the exercise, a 1.640 g sample of benzoic acid was burned, and by measuring the temperature change, the total heat capacity of the calorimeter can be determined. This total heat capacity is a crucial component for further experiments using the same calorimeter.

Heat Capacity

Heat capacity is defined as the amount of heat energy required to raise the temperature of an object or substance by one degree Celsius (or Kelvin). It is expressed in units such as joules per degree Celsius (J/°C) or kilojoules per degree Celsius (kJ/°C).

Heat capacity is an extensive property, meaning that it depends on the quantity of material: the more substance you have, the greater the heat capacity. In the context of the bomb calorimeter exercise, the heat capacity reflects the calorimeter’s ability to absorb heat without undergoing a significant change in temperature. The formula used in the exercise to calculate the total heat capacity is \( \text{Total heat capacity} = \frac{\text{Heat released}}{\Delta T} \) where \( \Delta T \) is the change in temperature of the calorimeter.

Calorimetry

Calorimetry is the science of measuring the change in heat associated with a chemical reaction. This technique is instrumental in determining thermodynamic properties such as enthalpy and entropy. In a calorimetry experiment, the heat from the chemical reaction is transferred to a known substance, and the temperature change of that substance is measured to infer the heat transfer associated with the reaction.

In the exercise provided, calorimetry is utilized to ascertain the heat of combustion of an organic substance. By observing the temperature increase in a bomb calorimeter, one can determine the amount of heat released during the combustion. This process involves accurately measuring the temperature change, ensuring constant-volume conditions, and conducting calculations that consider the heat capacity of the calorimeter. These data points culminate to provide important thermodynamic insights into the substance in question.