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Steam at \(100^{\circ} \mathrm{C}\) was passed into a flask containing \(275 \mathrm{~g}\) of water at \(21^{\circ} \mathrm{C}\), where the steam condensed. How many grams of steam must have condensed if the temperature of the water in the flask was raised to \(83{ }^{\circ} \mathrm{C} ?\) The heat of vaporization of water at \(100^{\circ} \mathrm{C}\) is \(40.7 \mathrm{~kJ} / \mathrm{mol}\) and the specific heat is \(4.18 \mathrm{~J} /\left(\mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\)

Short Answer

Expert verified
4.12 g of steam condensed.

Step by step solution

01

Identify the Energy Change Required

First, calculate the energy required to raise the temperature of the water from its initial temperature of 21°C to 83°C. Use the formula for heat transfer: \( q = m \cdot c \cdot \Delta T \), where \( m \) is the mass of water, \( c \) is the specific heat of water, and \( \Delta T \) is the change in temperature.
02

Use the Heat Transfer Formula

Substitute the given values into the heat transfer formula: \( q = 275 \text{ g} \times 4.18 \text{ J/g°C} \times (83 - 21)°C \). Calculate the energy \( q \) in Joules.
03

Convert Joules to Kilojoules

Convert the energy calculated in Joules to Kilojoules to match the units of the heat of vaporization: \( 1 \text{ kJ} = 1000 \text{ J} \).
04

Relate Condensed Steam to Energy Change

Determine the number of moles of steam that condensed by relating the energy change to the heat of vaporization. Use the equation: \( q = n \cdot L_v \), where \( n \) is the number of moles and \( L_v \) is the heat of vaporization.
05

Determine Mass of Condensed Steam

Calculate the mass of the condensed steam using the number of moles calculated and the molar mass of water (18 g/mol): \( m = n \cdot 18 \text{ g/mol} \).

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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 a fascinating concept in calorimetry that involves the energy required to turn a liquid into a gas at its boiling point. For water, this is a high value, demonstrating that it takes significant energy to change water from a liquid to a gas. At 100°C, the heat of vaporization for water is 40.7 kJ/mol. This particular exercise demonstrates how energy must be supplied to change the state of water functioning purely with latent heat.

When steam condenses back into water, this energy, which was previously invested to convert water into steam, is released. This release of energy is why steam can be so effective at transferring heat. It can release a lot of energy into the surrounding environment, affecting temperature changes, as seen in this problem where steam heats water dramatically.
  • Heat of vaporization is involved in the state change between liquid and gas.
  • Release of latent heat during condensation can warm substances efficiently.
Specific Heat Capacity
Specific heat capacity is a crucial concept in understanding how substances absorb and release heat. Simply put, it refers to the amount of heat a substance needs to raise its temperature by 1°C per unit mass. The specific heat capacity helps explain why different substances heat up or cool down at different rates when equal amounts of energy are supplied.

In the given exercise, water has a specific heat capacity of 4.18 J/(g°C). This means it takes 4.18 Joules of energy to increase the temperature of 1 gram of water by 1°C. So for the entire 275 g of water in the flask, this specific heat capacity determines the total energy required to raise the temperature from 21°C to 83°C. This tells us about both the energy interaction with its environment and how water distributes heat internally.
  • Allows calculation of heat required for a temperature change.
  • Plays a key role in determining energy exchange.
Condensation
Condensation is the process where vapor changes back to liquid. It's the reverse of vaporization and is accompanied by the release of latent heat. When steam condenses, it releases the energy absorbed during vaporization, allowing the surrounding environment to absorb this newly liberated heat.

In this particular problem, steam at 100°C was introduced to water in a flask, and through condensation, it released energy that heated the water. The condensed steam implicates how energy transfers occur due to changes in state, as the latent heat released during condensation is used to raise the temperature of the surrounding water.
  • Occurs when a vapor turns into a liquid.
  • Heat is released during condensation, impacting temperature.

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Most popular questions from this chapter

Greater variation exists between the properties of the first and second members of a family in the periodic table than between other members. Discuss this observation for the oxygen family using the following data. \(\begin{array}{llll} & \text { Boiling } & & \text { Boiling } \\ \text { Element } & \text { Point, }^{\circ} \mathbf{C} & \text { Compound } & \text { Point, }^{\circ} \mathbf{C} \\ \mathrm{O}_{2} & -183 & \mathrm{H}_{2} \mathrm{O} & 100 \\ \mathrm{~S}_{8} & 445 & \mathrm{H}_{2} \mathrm{~S} & -61 \\\ \mathrm{Se}_{8} & 685 & \mathrm{H}_{2} \mathrm{Se} & -42\end{array}\)

The vapor pressure of a volatile liquid can be determined by slowly bubbling a known volume of gas through the liquid at a given temperature and pressure. In an experiment, a 5.40-L sample of nitrogen gas, \(\mathrm{N}_{2}\), at \(20.0^{\circ} \mathrm{C}\) and \(745 \mathrm{mmHg}\) is bubbled through liquid isopropyl alcohol, \(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}\), at \(20.0^{\circ} \mathrm{C}\). Nitrogen containing the vapor of \(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}\) at its vapor pressure leaves the vessel at \(20.0^{\circ} \mathrm{C}\) and \(745 \mathrm{mmHg} .\) It is found that \(0.6149 \mathrm{~g} \mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}\) has evaporated. How many moles of \(\mathrm{N}_{2}\) are in the gas leaving the liquid? How many moles of alcohol are in this gaseous mixture? What is the mole fraction of alcohol vapor in the gaseous mixture? What is the partial pressure of the alcohol in the gaseous mixture? What is the vapor pressure of \(\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}\) at \(20.0^{\circ} \mathrm{C} ?\)

Explain the surface tension of a liquid in molecular terms. How does the surface tension make a liquid act as though it had a "skin"?

Metallic barium has a body-centered cubic structure (all atoms at the lattice points) and a density of \(3.51 \mathrm{~g} / \mathrm{cm}^{3}\). Assume barium atoms to be spheres. The spheres in a bodycentered array occupy \(68.0 \%\) of the total space. Find the atomic radius of barium. (See Problem 11.87.)

If you place room temperature water in a wellinsulated cup and allow some of the water to evaporate, the temperature of the water in the cup will drop lower than room temperature. Come up with an explanation for this observation.

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