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Chapter 5: Problem 58

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

Expert verified
By following the provided steps and calculations: 1) The heat released by the benzoic acid is calculated to be \(q = (1.640 \,g)(26.38 \, kJ/g) = 43.3028 \, kJ\). 2) The temperature change of the calorimeter is \(\Delta{T} = 27.20掳C - 22.25掳C = 4.95掳C\). 3) The heat capacity of the calorimeter is \(C = q / \Delta{T} = 43.3028 \,kJ / 4.95掳C = 8.747 \, kJ/掳C \). 4) The temperature change for the new organic substance is \(\Delta{T}_{new} = 26.82掳C - 22.14掳C = 4.68掳C\), and the heat released is \(q_{new} = C \times \Delta{T}_{new} = 8.747 \, kJ/掳C \times 4.68掳C = 40.9414 \, kJ\). 5) The heat of combustion per gram for the new organic substance is calculated to be \(q_{new}/m_{new} = 40.9414 \, kJ / 1.320 \, g =31.01 \,kJ/g\). 6) Losing water from the calorimeter would decrease its mass, which in turn would decrease its overall heat capacity.

Step by step solution

01

Calculate the heat released by the benzoic acid

To calculate the heat released by the benzoic acid, we will use the equation: Heat released (q) = Mass (m) 脳 Heat of combustion where Mass (m) = 1.640 g Heat of combustion = 26.38 kJ/g q = (1.640 g) 脳 (26.38 kJ/g)
02

Calculate the temperature change of the calorimeter

The temperature change of the calorimeter can be calculated as: 螖T = T_final - T_initial where T_final = 27.20掳C T_initial = 22.25掳C 螖T = 27.20掳C - 22.25掳C
03

Calculate the heat capacity of the calorimeter

Using the heat released by the benzoic acid and the temperature change of the calorimeter, we can find the heat capacity of the calorimeter using the formula: Heat capacity (C) = Heat released (q) / Temperature change (螖T) C = q / 螖T
04

Calculate the heat released by the new organic substance

We are given the mass of the new organic substance and the temperature change of the calorimeter when it is burned. We can use the heat capacity of the calorimeter (calculated in Step 3) and temperature change to calculate the heat released by the new organic substance: Heat released (q_new) = Heat capacity (C) 脳 Temperature change (螖T_new) where 螖T_new = (26.82掳C - 22.14掳C) q_new = C 脳 螖T_new
05

Calculate the heat of combustion per gram for the new organic substance

Finally, we can calculate the heat of combustion per gram of the new organic substance using the following formula: Heat of combustion per gram = Heat released (q_new) / Mass (m_new) where Mass (m_new) = 1.320 g Heat of combustion per gram = q_new / m_new
06

Discuss the effect of losing water on the heat capacity of the calorimeter

When a portion of the water in the calorimeter is lost, it will decrease the total mass of the system, which affects the heat capacity. Since heat capacity is proportional to mass, the heat capacity of the calorimeter will decrease as the mass (amount of water) decreases.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Heat of Combustion
The heat of combustion pertains to the energy released when a substance is burned in oxygen. It is an indication of the energy capacity of the fuel or substance. For benzoic acid, the heat of combustion is given as 26.38 kJ/g. This means when 1 gram of benzoic acid combusts, it releases 26.38 kJ of energy.
In bomb calorimetry, this concept helps determine how much energy a material can produce. When you know the heat of combustion, you can calculate how much total energy is released for a given mass of the substance. For instance, a 1.640 g sample of benzoic acid will release a certain amount of energy, which can then be used to derive other thermodynamic parameters.
Heat Capacity
Heat capacity of an object or system is the amount of heat needed to change its temperature by one degree Celsius. For a bomb calorimeter, heat capacity is crucial in understanding how much energy is needed to alter its temperature. The total heat capacity accounts for all components in the calorimeter system, including any water present.
To find the heat capacity of the calorimeter in this problem, we start with the heat released by the benzoic acid:
  • It's calculated by multiplying its mass with known heat of combustion.
  • The temperature change from burning this sample gives us another data point.
The formula used is: \[ C = \frac{q}{\Delta T} \]where \(q\) is the heat released by the substance and \(\Delta T\) is the temperature change. This gives the calorimeter's heat capacity, a constant value used in further calculations.
Temperature Change
Temperature change is a key measure in calorimetry, as it reflects how much the system's temperature fluctuates due to combustion. It is computed by subtracting the initial temperature from the final temperature of the calorimeter.
The formula is:
  • \( \Delta T = T_{\text{final}} - T_{\text{initial}} \)
For benzoic acid burning, this temperature change reflects the energy transferred to the calorimeter. The larger the temperature change, the more energy has been absorbed by the calorimeter.
By understanding this change, one can further compute the calorimetric analysis of another substance using the same calorimeter, provided its initial and final temperatures are known.
Calorimetric Analysis
Calorimetric analysis involves calculating various thermal properties of substances using a calorimeter. This analysis gives insights into energy transformations during combustion. In this exercise, with calorimetric analysis, you're finding values like heat of combustion for given samples.
Steps in this process include:
  • Burning the sample and recording temperature changes.
  • Using known properties like heat of combustion or heat capacity.
  • Calculating unknowns for new substances using the same calorimeter setup.
The analysis becomes more complex when alterations in the calorimeter environment, such as water loss, occur. Loss of water alters the system's mass, decreasing its heat capacity, thereby affecting subsequent analyses. Understanding these factors ensures accurate measurements and interpretations in thermal studies.

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Most popular questions from this chapter

Meals-ready-to-eat (MREs) are military meals that can be heated on a flameless heater. The heat is produced by the following reaction: \(\mathrm{Mg}(s)+2 \mathrm{H}_{2} \mathrm{O}(l) \longrightarrow\) \(\mathrm{Mg}(\mathrm{OH})_{2}(s)+\mathrm{H}_{2}(g)\) (a) Calculate the standard enthalpy change for this reaction. (b) Calculate the number of grams of \(\mathrm{Mg}\) needed for this reaction to release enough energy to increase the temperature of \(25 \mathrm{~mL}\) of water from \(15^{\circ} \mathrm{C}\) to \(85^{\circ} \mathrm{C}\).

It is estimated that the net amount of carbon dioxide fixed by photosynthesis on the landmass of Earth is \(5.5 \times\) \(10^{16} \mathrm{~g} / \mathrm{yr}\) of \(\mathrm{CO}_{2}\). Assume that all this carbon is converted into glucose. (a) Calculate the energy stored by photosynthesis on land per year in \(\mathrm{kJ}\). (b) Calculate the average rate of conversion of solar energy into plant energy in MW \((1 \mathrm{~W}=1 \mathrm{~J} / \mathrm{s})\). A large nuclear power plant produces about \(10^{3} \mathrm{MW}\). The energy of how many such nuclear power plants is equivalent to the solar energy conversion?

What is the connection between Hess's law and the fact that \(H\) is a state function?

A \(2.200-g\) sample of quinone \(\left(\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{O}_{2}\right)\) is burned in a bomb calorimeter whose total heat capacity is \(7.854 \mathrm{~kJ} /{ }^{\circ} \mathrm{C}\). The temperature of the calorimeter increases from \(23.44^{\circ} \mathrm{C}\) to \(30.57^{\circ} \mathrm{C}\). What is the heat of combustion per gram of quinone? Per mole of quinone?

In a thermodynamic study a scientist focuses on the properties of a solution in an apparatus as illustrated. A solu- tion is continuously flowing into the apparatus at the top and out at the bottom, such that the amount of solution in the apparatus is constant with time. (a) Is the solution in the apparatus a closed system, open system, or isolated system? Explain your choice. (b) If it is not a closed system, what could be done to make it a closed system?
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