Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 17: Temperature and Heat

Problem 34

Chapter 17, Problem 34

You have 750 g of water at 10.0°C in a large insulated beaker. How much boiling water at 100.0°C must you add to this beaker so that the final temperature of the mixture will be 75°C?

Verified step by step guidance

First, identify the key concept involved: this is a heat transfer problem where the heat lost by the boiling water will be equal to the heat gained by the cooler water, assuming no heat is lost to the surroundings.

Use the formula for heat transfer: \( Q = mc\Delta T \), where \( Q \) is the heat energy, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature.

Set up the equation based on the principle of conservation of energy: \( m_1c\Delta T_1 = m_2c\Delta T_2 \), where \( m_1 \) and \( m_2 \) are the masses of the cooler and boiling water respectively, and \( \Delta T_1 \) and \( \Delta T_2 \) are their respective temperature changes.

Substitute the known values into the equation: \( 750 \text{ g} \times 4.18 \text{ J/g°C} \times (75°C - 10°C) = m_2 \times 4.18 \text{ J/g°C} \times (100°C - 75°C) \).

Solve for \( m_2 \), the mass of boiling water needed, by isolating \( m_2 \) on one side of the equation. This will give you the mass of boiling water required to achieve the desired final temperature.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It is crucial in calculating the heat exchange in thermal processes, as it determines how much energy is needed to raise the temperature of a given mass of a substance. For water, this value is typically 4.18 J/g°C.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the context of mixing water at different temperatures, the heat lost by the hot water must equal the heat gained by the cold water, assuming no heat is lost to the surroundings due to the insulated beaker.

Heat Transfer in Mixtures

When two substances at different temperatures are mixed, heat transfer occurs until thermal equilibrium is reached. The final temperature of the mixture depends on the masses, specific heat capacities, and initial temperatures of the substances involved. This concept is used to set up equations that allow us to solve for unknown quantities, such as the mass of boiling water needed in this problem.

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While painting the top of an antenna 225 m in height, a worker accidentally lets a 1.00-L water bottle fall from his lunchbox. The bottle lands in some bushes at ground level and does not break. If a quantity of heat equal to the magnitude of the change in mechanical energy of the water goes into the water, what is its increase in temperature?