Question

Ethanol, \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH},\) boils at \(78.29^{\circ} \mathrm{C} .\) What quantity of heat energy, in joules, is required to raise the temperature of 1.00 kg of ethanol from \(20.0^{\circ} \mathrm{C}\) to the boiling point and then to change the liquid to vapor at that temperature? (The specific heat capacity of liquid ethanol is \(2.44 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\) and its enthalpy of vaporization is \(855 \mathrm{J} / \mathrm{g} .\) )

Short answer

997007.6 J of energy is needed.

Step-by-step solution

Calculate Mass of Ethanol

First, determine the mass of ethanol. Given that 1.00 kg is equivalent to 1000 g, we have the mass of ethanol as 1000 g.

Calculate Temperature Change

Find the change in temperature that the ethanol undergoes: it is the difference between the boiling point and the initial temperature. \[\Delta T = 78.29^{\circ}C - 20.0^{\circ}C = 58.29^{\circ}C\]

Calculate Heat for Temperature Increase

Using the formula for heat transfer: \[ q = m \times c \times \Delta T \]where - \(m\) is mass (1000 g),- \(c\) is the specific heat capacity (2.44 J/g⋅K), - \(\Delta T\) is the temperature change (58.29 K).Substitute the values: \[ q = 1000 \times 2.44 \times 58.29 \approx 142007.6 \text{ J} \]

Calculate Heat for Vaporization

Using the heat of vaporization formula: \[ q_{vap} = m \times \text{enthalpy of vaporization} \]Substitute the values:\[ q_{vap} = 1000 \times 855 = 855000 \, \text{J} \]

Calculate Total Heat Required

The total heat required is the sum of the heat for temperature increase and the heat for vaporization:\[ q_{total} = 142007.6 \, \text{J} + 855000 \, \text{J} = 997007.6 \, \text{J} \]

Key concepts

Specific Heat Capacity

In the world of thermochemistry, specific heat capacity is an essential concept to understand. It is the amount of heat required to change the temperature of one gram of a substance by one degree Celsius or Kelvin. This property varies depending on the substance, making it crucial to know the specific heat capacity when performing calculations for heat transfer.
For ethanol, the specific heat capacity is given as 2.44 J/g·K. This implies that to increase the temperature of one gram of ethanol by one degree, 2.44 joules of energy is needed. Such values allow us to calculate how much heat is required to change temperatures in everyday problems, like heating a kettle of water or in industrial settings.

• Helpful in determining the energy required for heating processes.
• Units: usually expressed in J/g·K or J/kg·K.
• Differs from substance to substance; hence, always check the specific property for calculations.

Understanding specific heat capacity helps in comprehending how different materials respond to thermal energy, allowing us to efficiently decide how much energy we need or will use.

Enthalpy of Vaporization

Enthalpy of vaporization is another significant concept in thermochemistry. It refers to the amount of energy required to convert one gram of a liquid into vapor without changing its temperature. This phase change from liquid to gas is critical in various applications, including boiling, heating, and systems operating under pressure.
For ethanol, as indicated in the problem, the enthalpy of vaporization is 855 J/g. This number is high compared to the specific heat capacity, reflecting that changing phase requires much more energy than simply heating the liquid.

• Represents the energy change during phase transition at constant temperature.
• Critical in designing processes like distillation or when managing energy consumption in cooling systems.
• Units: often expressed in J/g or kJ/mol.

This property is valuable in understanding and calculating energy requirements for processes involving phase changes, ensuring efficient energy use during chemical and physical transformations.

Heat Transfer Calculations

Heat transfer calculations are foundational in determining how much energy is required to effect temperature and phase changes in a substance. To perform these calculations accurately, integrate both the specific heat capacity and the enthalpy of vaporization.
1. **Calculating Temperature Increase**: Use the formula \[ q = m \times c \times \Delta T \] where:

• \(q\) is the heat absorbed or released,
• \(m\) is the mass of the substance,
• \(c\) is the specific heat capacity,
• \(\Delta T\) is the change in temperature.

This step calculates the energy needed to raise the temperature to the desired level without changing the physical state, ensuring that we know how much energy is consumed purely in heating.
2. **Calculating Vaporization Energy**: Use \[ q_{vap} = m \times \text{enthalpy of vaporization} \] This calculation provides the energy required to convert a liquid at its boiling point to gas, crucial for understanding processes that involve phase changes at constant temperatures.
Total energy used results from the sum of energy for heating the liquid and that for vaporizing it. Such detailed calculations ensure comprehensive knowledge of energy consumption,enabling accuracy in both scientific research and practical energy management.