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Chapter 17: Problem 101

A copper calorimeter can with mass 0.446 \(\mathrm{kg}\) contains 0.0950 \(\mathrm{kg}\) of ice. The system is initially at \(0.0^{\circ} \mathrm{C} .\) (a) If 0.0350 \(\mathrm{kg}\) of steam at \(100.0^{\circ} \mathrm{C}\) and 1.00 atm pressure is added to the can, what is the final temperature of the calorimeter can and its contents? (b) At the final temperature, how many kilograms are there of ice, how many of liquid water, and how many of steam?

Short Answer

Expert verified
The final temperature is found using energy balance, then determining the state of ice, water, and steam.

Step by step solution

01

Understand the Energy Transfer

In a calorimetry problem, we consider the energy exchanged between the substances until thermal equilibrium is reached. Assume no heat is lost to the surroundings, so the heat lost by the steam will equal the heat gained by the ice, water, and calorimeter.
02

Calculate Heat Released by Condensing Steam

The steam must first condense to water at 100掳C, releasing energy:\[Q_{ ext{condense}} = m_{ ext{steam}} \times L_v\]where \(m_{ ext{steam}} = 0.0350 \text{ kg}\) and \(L_v = 2260 \text{ kJ/kg}\). Substitute to find \(Q_{ ext{condense}}\).
03

Calculate Heat Released by Cooling Water from 100掳C to 0掳C

Next, the condensed water cools from 100掳C to 0掳C:\[Q_{ ext{cool}} = m_{ ext{steam}} \times c_w \times \Delta T\]\(c_w = 4.18 \text{ kJ/kg掳C}\) and \(\Delta T = 100掳C\). Solve for \(Q_{ ext{cool}}\).
04

Calculate Heat Required to Melt the Ice

The energy required to melt the ice is given by:\[Q_{ ext{melt}} = m_{ ext{ice}} \times L_f\]where \(m_{ ext{ice}} = 0.0950 \text{ kg}\) and \(L_f = 334 \text{ kJ/kg}\). Substitute to find \(Q_{ ext{melt}}\).
05

Calculate Heat Required to Raise Temperature of Resulting Water and Calorimeter from 0掳C to Final Temperature

Calculate for water and calorimeter:\[Q_{ ext{warm}} = (m_{ ext{water}} \times c_w + m_{ ext{calorimeter}} \times c_c) \times (T_f - 0)\]where \(m_{ ext{water}} = m_{ ext{ice}} + m_{ ext{melted ice}}\), \(m_{ ext{calorimeter}} = 0.446 \text{ kg}\), and \(c_c = 0.385 \text{ kJ/kg掳C}\).
06

Formulate the Energy Balance Equation

Using energy balance:\[Q_{ ext{condense}} + Q_{ ext{cool}} = Q_{ ext{melt}} + Q_{ ext{warm}}\]Solve for the final temperature \(T_f\).
07

Analyze Phases at Final Temperature

Assess the energy changes and compare to phase transition energies to determine if all ice has melted. If \(T_f > 0掳C\), all ice has melted and the energy distribution changes only with temperature.

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Key Concepts

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

Energy Transfer
Energy transfer plays a key role in understanding calorimetry problems. Calorimetry involves examining how energy moves between different substances. The key is to remember that energy cannot be created or destroyed, only transferred. In the problem, energy is shifting from the steam to the ice, water, and calorimeter. We assume that no energy escapes to the surroundings.

This means that all the energy lost by the steam is gained by the other elements. The steam condenses, giving off energy, which is absorbed by the ice, water, and the copper can.

By applying these principles, one can analyze how much energy is transferred through various processes such as condensation, cooling, and warming up until thermal equilibrium is reached.
Thermal Equilibrium
Thermal equilibrium is achieved when all parts of a system have the same temperature, and there is no net flow of thermal energy between them. In our calorimetry problem, this means the steam, water, ice, and calorimeter ultimately arrive at a uniform temperature.

To solve the exercise, it's important to calculate the energy exchanges that happen during the energy transfer processes mentioned. These calculations help determine the temperature at which thermal equilibrium is reached.

The process begins with steam at a high temperature losing energy, while the ice and calorimeter gain energy. The final step confirms equilibrium when the total energy lost and gained balances out, showing that the entire system has arrived at the same temperature.
Phase Transition
Phase transitions occur when substances change states, such as from solid to liquid or liquid to gas. These transitions require specific amounts of energy to occur. In this problem, the major phase transition is the melting of ice and condensation of steam.

When steam condenses into water, it releases a significant amount of latent heat, which is then used to melt the ice and warm the rest of the system. Calculating the energy involved in these phase changes is crucial for understanding how the temperature of the entire system changes.

Each phase change consumes or releases heat, affecting how energy is shared amongst the substances. By understanding the energies linked to these phase transitions, you can predict changes in states, like whether all the ice has become water by the end of the process.

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Most popular questions from this chapter

Effect of a Window in a Door. A carpenter builds a solid wood door with dimensions 2.00 \(\mathrm{m} \times 0.95 \mathrm{m} \times 5.0 \mathrm{cm} .\) Its thermal conductivity is \(k=0.120 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) . The air films on the inner and outer surfaces of the door have the same combined thermal resistance as an additional 1.8 -cm thickness of solid wood. The inside air temperature is \(20.0^{\circ} \mathrm{C},\) and the outside air temperature is \(-8.0^{\circ} \mathrm{C}\) (a) What is the rate of heat flow through the door? (b) By what factor is the heat flow increased if a window 0.500 \(\mathrm{m}\) on a side is inserted in the door? The glass is 0.450 \(\mathrm{cm}\) thick, and the glass has a thermal conductivity of 0.80 \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) . The air films on the two sides of the glass have a total thermal resistance that is the same as an additional 12.0 \(\mathrm{cm}\) of glass.

A Walk in the Sun. Consider a poor lost soul walking at 5 \(\mathrm{km} / \mathrm{h}\) on a hot day in the desert, wearing only a bathing suit. This person's skin temperature tends to rise due to four mechanisms: (i) energy is generated by metabolic reactions in the body at a rate of \(280 \mathrm{W},\) and almost all of this energy is con- verted to heat that flows to the skin; (ii) heat is delivered to the skin by convection from the outside air at a rate equal to \(k^{\prime} A_{\text { skin }}\left(T_{\text { air }}-T_{\text { skin }}\right),\) where \(k^{\prime}\) is \(54 \mathrm{J} / \mathrm{h} \cdot \mathrm{C}^{\circ} \cdot \mathrm{m}^{2},\) the exposed skin area \(A_{\text { skin }}\) is \(1.5 \mathrm{m}^{2},\) the air temperature \(T_{\mathrm{air}}\) is \(47^{\circ} \mathrm{C},\) and the skin temperature \(T_{\text { skin }}\) is \(36^{\circ} \mathrm{C} ;\) (iii) the skin absorbs radiant energy from the sun at a rate of 1400 \(\mathrm{W} / \mathrm{m}^{2}\) ; (iv) the skin absorbs radiant energy from the environment, which has temperature \(47^{\circ} \mathrm{C}\) . (a) Calculate the net rate (in watts) at which the person's skin is heated by all four of these mechanisms. Assume that the emissivity of the skin is \(e=1\) and that the skin temperature is initially \(36^{\circ} \mathrm{C}\) . Which mechanism is the most important? (b) At what rate (in \(\mathrm{L} / \mathrm{h} )\) must perspiration evaporate from this person's skin to maintain a constant skin temperature? (The heat of vaporization of water at \(36^{\circ} \mathrm{C}\) is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .\)) (c) Suppose instead the person is protected by light-colored clothing \((e \approx 0)\) so that the exposed skin area is only 0.45 \(\mathrm{m}^{2} .\) What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples.

A metal rod that is 30.0 \(\mathrm{cm}\) long expands by 0.0650 \(\mathrm{cm}\) when its temperature is raised from \(0.0^{\circ} \mathrm{C}\) to \(100.0^{\circ} \mathrm{C} .\) A rod of a different metal and of the same length expands by 0.0350 \(\mathrm{cm}\) for the same rise in temperature. A third rod, also 30.0 \(\mathrm{cm}\) long, is made up of pieces of each of the above metals placed end to end and expands 0.0580 \(\mathrm{cm}\) between \(0.0^{\circ} \mathrm{C}\) and \(100.0^{\circ} \mathrm{C} .\) Find the length of each portion of the composite rod.

Convert the following Celsius temperatures to Fahrenheit: (a) \(-62.8^{\circ} \mathrm{C},\) the lowest temperature ever recorded in North America (February \(3,1947,\) Snag, Yukon); (b) \(56.7^{\circ} \mathrm{C},\) the highest temperature ever recorded in the United States (July \(10,1913,\) Death Valley, California); \((\mathrm{c}) 31.1^{\circ} \mathrm{C},\) the world's highest average annual temperature (Lugh Ferrandi, Somalia).

Evaporation of sweat is an important mechanism for temperature control in some warm-blooded animals. (a) What mass of water must evaporate from the skin of a 70.0-kg man to cool his body 1.00 \(\mathrm{C}^{\circ} ?\) The heat of vaporization of water at body temperature \(\left(37^{\circ} \mathrm{C}\right)\) is \(2.42 \times 10^{6} \mathrm{J} / \mathrm{kg} .\) The specific heat of a typical human body is 3480 \(\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) (see Exercise 17.31\() .\) (b) What volume of water must the man drink to replenish the evaporated water? Compare to the volume of a soft-drink can \(\left(355 \mathrm{cm}^{3}\right)\).
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