/*! 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 52 Three portions of the same liqui... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 52

Three portions of the same liquid are mixed in a container that prevents the exchange of heat with the environment. Portion A has a mass \(m\) and a temperature of \(94.0^{\circ} \mathrm{C},\) portion \(\mathrm{B}\) also has a mass \(m\) but a temperature of \(78.0^{\circ} \mathrm{C},\) and portion \(\mathrm{C}\) has a mass \(m_{\mathrm{C}}\) and a temperature of \(34.0^{\circ} \mathrm{C}\) What must be the mass of portion \(C\) so that the final temperature \(T_{f}\) of the three-portion mixture is \(T_{f}=50.0^{\circ} \mathrm{C} ?\) Express your answer in terms of \(m\) for example, \(m_{\mathrm{C}}=2.20 \mathrm{m}\)

Short Answer

Expert verified
The mass of portion C should be \(4.5m\).

Step by step solution

01

Understand the Concept of Heat Transfer

The heat lost by the warmer portions of the liquid will equal the heat gained by the cooler portion due to the law of conservation of energy. No heat is exchanged with the environment as the container is insulated.
02

Determine the Heat Loss and Gain

The heat loss from portions A and B will equal the heat gain by portion C, as there's no heat exchange with the environment. Use the formula for heat transfer: For portion A: \( Q_A = m c (T_f - T_A) \), for portion B: \( Q_B = m c (T_f - T_B) \), and for portion C: \( Q_C = m_C c (T_C - T_f) \).Here, \( c \) is the specific heat capacity, which cancels out since it's the same for all portions, and \( T_f \) is the final temperature.
03

Set Up the Heat Balance Equation

According to conservation of heat, set up the equation: \( Q_A + Q_B + Q_C = 0 \). Substitute the expressions for \( Q_A, Q_B, \) and \( Q_C \): \( m (50.0 - 94.0) + m(50.0 - 78.0) + m_C (34.0 - 50.0) = 0 \).
04

Solve for the Mass \( m_C \) of Portion \( C \)

Simplify and solve for \( m_C \):\(-44m - 28m + 16m_C = 0 \)\(-72m + 16m_C = 0 \)\( 16m_C = 72m \)\( m_C = \frac{72}{16}m \)\( m_C = 4.5m \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Conservation of Energy
When discussing heat transfer, the principle of conservation of energy is essential. This principle states that energy cannot be created or destroyed but only transferred or converted from one form to another. In the context of mixing the portions of liquid in the exercise, it means that the total thermal energy in the system remains constant, even if temperatures change.

In an insulated container, there is no heat exchange with the environment. Therefore, any heat lost by the warmer parts is exactly balanced by the heat gained by the cooler parts, maintaining the overall energy balance.

This concept ensures that heat is distributed among the liquid portions until thermal equilibrium is reached, which is when the portions come to the same temperature.
Specific Heat Capacity
The specific heat capacity is a physical property of a substance that indicates how much heat energy is required to change the substance's temperature. It is usually denoted by the symbol "c".

This property is crucial in heat transfer calculations. It allows us to quantify the amount of heat energy absorbed or released by a given quantity of a substance when its temperature changes by a certain amount.

In our exercise, since all three parts of the liquid are the same, they share the same specific heat capacity. This commonality simplifies calculations by allowing us to cancel c out of the heat transfer equations, focusing only on temperature changes and masses.
  • The formula used is: \[ Q = mc(T_f - T_i) \].
  • Here, Q is the heat energy exchanged, m is the mass, c is the specific heat capacity, and \(T_f\), \(T_i\) are the final and initial temperatures, respectively.
Thermal Equilibrium
Thermal equilibrium occurs when two or more substances reach the same temperature and there is no net transfer of heat between them. In the exercise, the final goal is for all portions of the liquid to reach a final temperature of 50.0掳C.

Reaching thermal equilibrium means that the system has stabilized, with energy equally distributed according to the medium's properties and initial conditions.

In practice, thermal equilibrium helps to predict the final temperature and balance of energy within an isolated system, such as the one in the exercise.
  • The heat lost by warmer portions A and B is equal to the heat gained by the cooler portion C since there is no external heat exchange.
  • Achieving this balance ensures that the final condition is stable and measurable, following the principles of conservation of energy.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

One rod is made from lead and another from quartz. The rods are heated and experience the same change in temperature. The change in length of each rod is the same. If the initial length of the lead rod is \(0.10 \mathrm{m},\) what is the initial length of the quartz rod?

Water at \(23.0^{\circ} \mathrm{C}\) is sprayed onto \(0.180 \mathrm{kg}\) of molten gold at \(1063^{\circ} \mathrm{C}\) (its melting point). The water boils away, forming steam at \(100.0^{\circ} \mathrm{C}\) and leaving solid gold at \(1063^{\circ} \mathrm{C} .\) What is the minimum mass of water that must be used?

A thick, vertical iron pipe has an inner diameter of \(0.065 \mathrm{m}\). A thin aluminum disk, heated to a temperature of \(85^{\circ} \mathrm{C},\) has a diameter that is \(3.9 \times 10^{-5} \mathrm{m}\) greater than the pipe's inner diameter. The disk is laid on top of the open upper end of the pipe, perfectly centered on it, and allowed to cool. What is the temperature of the aluminum disk when the disk falls into the pipe? Ignore the temperature change of the pipe.

If a nonhuman civilization were to develop on Saturn's largest moon, Titan, its scientists might well devise a temperature scale based on the properties of methane, which is much more abundant on the surface than water is. Methane freezes at \(-182.6^{\circ} \mathrm{C}\) on Titan, and boils at \(-155.2^{\circ} \mathrm{C}\) Taking the boiling point of methane as \(100.0^{\circ} \mathrm{M}\) (degrees Methane) and its freezing point as \(0^{\circ} \mathrm{M},\) what temperature on the Methane scale corresponds to the absolute zero point of the Kelvin scale?

When \(4200 \mathrm{J}\) of heat are added to a \(0.15-\mathrm{m}\) -long silver bar, its length increases by \(4.3 \times 10^{-3} \mathrm{m} .\) What is the mass of the bar?
See all solutions

Recommended explanations on Physics Textbooks

Turning Points in Physics

Read Explanation

Dynamics

Read Explanation

Electrostatics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Physical Quantities And Units

Read Explanation

Engineering Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.