/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Ethyl alcohol has a boiling poin... [FREE SOLUTION] | 91影视

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Chapter 18: Problem 15

Ethyl alcohol has a boiling point of \(78.0^{\circ} \mathrm{C}\), a freezing point of \(-114^{\circ} \mathrm{C}\), a heat of vaporization of \(879 \mathrm{~kJ} / \mathrm{kg}\), a heat of fusion of \(109 \mathrm{~kJ} / \mathrm{kg}_{n}\) and a specific heat of \(2.43 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\). How much energy must be removed from \(0.720 \mathrm{~kg}\) of ethyl alcohol that is initially a gas at \(78.0^{\circ} \mathrm{C}\) so that it becomes a solid at \(-114^{\circ} \mathrm{C}\) ?

Short Answer

Expert verified
1046.76 kJ of energy must be removed.

Step by step solution

01

Cool the gas to boiling point

First, we need to cool the ethyl alcohol gas from its initial temperature of \(78.0^{\circ} \mathrm{C}\) to its boiling point of \(78.0^{\circ} \mathrm{C}\). Since this is already at the boiling point, no cooling is actually needed in this step for the temperature to change, but this helps us organize the process.
02

Condense the gas at boiling point

Next, we convert the gaseous alcohol to liquid by removing the latent heat of vaporization. The heat removed (\(Q_1\)) is calculated using:\[ Q_1 = m \times L_v \]where \(m = 0.720\, \text{kg}\) and \(L_v = 879\, \text{kJ/kg}\).\[ Q_1 = 0.720 \times 879 = 632.88\, \text{kJ} \]
03

Cool liquid alcohol to freezing point

Next, the liquid needs to cool from its boiling point \(78.0^{\circ} \mathrm{C}\) to its freezing point \(-114^{\circ} \mathrm{C}\). The heat removed (\(Q_2\)) is calculated using:\[ Q_2 = m \times c \times \Delta T \]where \(c = 2.43\, \text{kJ/kg} \cdot \text{K}\) and \(\Delta T = 78.0 + 114 = 192\, \text{K}\).\[ Q_2 = 0.720 \times 2.43 \times 192 = 335.4048\, \text{kJ} \]
04

Freeze the liquid at freezing point

Now we need to freeze the liquid into a solid by removing the latent heat of fusion. The heat removed (\(Q_3\)) is calculated using:\[ Q_3 = m \times L_f \]where \(L_f = 109\, \text{kJ/kg}\).\[ Q_3 = 0.720 \times 109 = 78.48\, \text{kJ} \]
05

Total energy removed

Finally, sum up all the energy removed during each phase:\[ Q_\text{total} = Q_1 + Q_2 + Q_3 \]\[ Q_\text{total} = 632.88 + 335.4048 + 78.48 = 1046.7648\, \text{kJ} \]

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Key Concepts

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

Heat of Vaporization
The heat of vaporization is the amount of energy required to convert a unit mass of a liquid into a gas at its boiling point, without changing its temperature. In our exercise, ethyl alcohol requires 879 kJ/kg for this phase transition. This energy's primary purpose is to overcome the molecular attractions within the liquid.
  • When ethyl alcohol is at its boiling point of 78.0掳C, it absorbs heat to become a gas.
  • This process is endothermic, meaning it requires heat input to occur.
  • The energy is used to provide the molecules enough energy to escape into the vapor phase.
Understanding this concept is vital because the same amount of energy is released when the gas condenses back into a liquid. This "latent heat" does not change the temperature during the phase change, but is crucial for understanding energy balance in phase transitions.
Heat of Fusion
The heat of fusion is the energy needed to change a unit mass of a solid to a liquid at its melting point, without changing the actual temperature. Ethyl alcohol has a heat of fusion of 109 kJ/kg, which tells us how much energy is either absorbed during melting or released during freezing.
  • For ethyl alcohol, as it freezes from a liquid to a solid, it releases heat equivalent to its heat of fusion.
  • This occurs at a constant temperature, specifically its freezing point of -114掳C.
  • The concept of latent heat of fusion is important in understanding how substances transition between solid and liquid states while maintaining a temperature equilibrium.
Recognizing this energy flow is essential for calculating the total energy changes during processes involving freezing or melting, like those tackled in the present exercise.
Specific Heat Capacity
Specific heat capacity is the amount of heat required to raise the temperature of a unit mass of a substance by one degree Kelvin (or Celsius). The specific heat capacity of ethyl alcohol is 2.43 kJ/kg路K, which impacts how much heat is needed or released to change its temperature.
  • This property is crucial when cooling the ethyl alcohol liquid from 78掳C to -114掳C.
  • The formula used is: \[ Q = m \times c \times \Delta T \]where \( m \) is mass, \( c \) is specific heat capacity, and \( \Delta T \) is change in temperature.
  • Understanding specific heat capacity helps in calculating how much energy (heat) is necessary to change the temperature of a material in non-phase change context.
This concept is foundational in thermodynamics, as it helps us quantify thermal energy transitions required to achieve the desired temperature changes in materials.

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Most popular questions from this chapter

In a solar water heater, energy from the Sun is gathered by water that circulates through tubes in a rooftop collector. The solar radiation enters the collector through a transparent cover and warms the water in the tubes; this water is pumped into a holding tank. Assume that the efficiency of the overall system is \(25 \%\) (that is, \(80 \%\) of the incident solar energy is lost from the system). What collector area is necessary to raise the temperature of 200 L of water in the tank from \(20^{\circ} \mathrm{C}\) to \(40^{\circ} \mathrm{C}\) in \(1.0 \mathrm{~h}\) when the intensity of incident sunlight is \(750 \mathrm{~W} / \mathrm{m}^{2} ?\)

Suppose the temperature of a gas is \(372.90 \mathrm{~K}\) when it is at the boiling point of water. What then is the limiting value of the ratio of the pressure of the gas at that boiling point to its pressure at the triple point of water? (Assume the volume of the gas is the same at both temperatures.)

A sphere of radius \(0.350 \mathrm{~m}\), temperature \(27.0^{\circ} \mathrm{C}\), and emissivity \(0.850\) is located in an environment of temperature \(77.0^{\circ} \mathrm{C}\). At what rate does the sphere (a) emit and (b) absorb thermal radiation? (c) What is the sphere's net change in energy in \(3.50 \mathrm{~min}\) ?

At \(20^{\circ} \mathrm{C}\), a rod is exactly \(20.05 \mathrm{~cm}\) long on a steel ruler. Both are placed in an oven at \(250^{\circ} \mathrm{C}\), where the rod now measures \(20.11\) \(\mathrm{cm}\) on the same ruler. What is the coefficient of linear expansion for the material of which the rod is made?

A \(150 \mathrm{~g}\) copper bowl contains \(260 \mathrm{~g}\) of water, both at \(20.0^{\circ} \mathrm{C}\) A very hot \(300 \mathrm{~g}\) copper cylinder is dropped into the water, causing the water to boil, with \(5.00 \mathrm{~g}\) being converted to steam. The final temperature of the system is \(100^{\circ} \mathrm{C}\). Neglect energy transfers with the environment. (a) How much energy (in calories) is transferred to the water as heat? (b) How much to the bowl? (c) What is the original temperature of the cylinder?
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